{"id":347,"date":"2026-07-25T23:30:26","date_gmt":"2026-07-25T23:30:26","guid":{"rendered":"https:\/\/summergeometry.org\/sgi2026\/?p=347"},"modified":"2026-07-25T23:30:28","modified_gmt":"2026-07-25T23:30:28","slug":"geometric-deep-learning-for-fluids","status":"publish","type":"post","link":"https:\/\/summergeometry.org\/sgi2026\/geometric-deep-learning-for-fluids\/","title":{"rendered":"Geometric Deep Learning for Fluids"},"content":{"rendered":"\n

SGI Mentor: Akhil Sadam<\/p>\n\n\n\n

SGI Fellows: Santoshi Yadagiri, Pietro Palombini<\/p>\n\n\n\n

1. Introduction<\/h2>\n\n\n\n

Many geophysical inverse problems require reconstruction of a high-dimensional physical state from observations that are incomplete, noisy, or available only over part of the spatial domain. In oceanic and atmospheric applications, observations may be coarse, sparse, or separated by large unmeasured regions. The objective is not only to produce a plausible reconstruction, but also to characterize how much information the measurements provide about the unobserved portion of the state.<\/p>\n\n\n\n

Let the clean physical state be partitioned as<\/p>\n\n\n\n

U<\/mi>0<\/mn><\/msub>=<\/mo>(<\/mo>C<\/mi><\/mtd><\/mtr>D<\/mi><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>,<\/mo><\/mrow>U_0\n=\n\\begin{pmatrix}\nC\\\\\nD\n\\end{pmatrix},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where C<\/mi>C<\/annotation><\/semantics><\/math> denotes the observed or near component and D<\/mi>D<\/annotation><\/semantics><\/math> denotes the unobserved or far component. Measurements are assumed to depend directly only on C<\/mi>C<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
Y<\/mi>=<\/mo>A<\/mi>C<\/mi>+<\/mo>N<\/mi>,<\/mo><\/mspace>N<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>0<\/mn>,<\/mo>R<\/mi>)<\/mo>.<\/mi><\/mrow>Y\n=\nAC+N,\n\\qquad\nN\\sim\\mathcal{N}(0,R).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Equivalently, defining<\/p>\n\n\n\n

H<\/mi>=<\/mo>(<\/mo>A<\/mi><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>,<\/mo><\/mrow>H\n=\n\\begin{pmatrix}\nA&amp;0\n\\end{pmatrix},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

the observation model becomes<\/p>\n\n\n\n

Y<\/mi>=<\/mo>H<\/mi>U<\/mi>0<\/mn><\/msub>+<\/mo>N<\/mi>.<\/mi><\/mrow>Y\n=\nHU_0+N.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The operator A<\/mi>A<\/annotation><\/semantics><\/math> determines which spatial directions or scales of the near state are measured, while R<\/mi>R<\/annotation><\/semantics><\/math> describes the measurement-noise covariance. No component of D<\/mi>D<\/annotation><\/semantics><\/math> is observed directly. Information about the far region can therefore be recovered only through statistical or dynamical coupling between C<\/mi>C<\/annotation><\/semantics><\/math> and D<\/mi>D<\/annotation><\/semantics><\/math>. <\/p>\n\n\n\n

This problem is motivated by the reconstruction of quasi-geostrophic flow fields from coarse, sparse, and gappy observations. Diffusion-based generative models can provide probabilistic reconstructions without explicitly solving for a single deterministic inverse. However, recent numerical results show that guided unconditional methods such as diffusion posterior sampling may have difficulty propagating observational information into unobserved regions [1<\/mn>1<\/annotation><\/semantics><\/math>]. This raises the question of how information from a partial observation influences uncertainty and reconstruction quality outside the observed region.<\/p>\n\n\n\n

First, a linear Gaussian model is used to derive the conditional distribution, uncertainty reduction, and information transfer exactly. Second, the same structure is extended to a nonlinear flow-matching model through local linearization of the denoising map and the guided velocity field.<\/p>\n\n\n\n

2. Background<\/h2>\n\n\n\n

2.1 Geophysical Inverse Problems<\/h4>\n\n\n\n

An inverse problem seeks to infer an unknown physical state from indirect measurements. In the present setting, the forward observation process maps the clean state U<\/mi>0<\/mn><\/msub>U_0<\/annotation><\/semantics><\/math> to data Y<\/mi>Y<\/annotation><\/semantics><\/math>:<\/p>\n\n\n\n
Y<\/mi>=<\/mo>\ud835\udc9c<\/mi>(<\/mo>U<\/mi>0<\/mn><\/msub>)<\/mo>+<\/mo>N<\/mi>.<\/mi><\/mrow>Y\n=\n\\mathcal{A}(U_0)+N.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

For a linear observation operator, this reduces to<\/p>\n\n\n\n

Y<\/mi>=<\/mo>H<\/mi>U<\/mi>0<\/mn><\/msub>+<\/mo>N<\/mi>.<\/mi><\/mrow>Y\n=\nHU_0+N.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The inverse problem is generally ill posed because multiple clean states may produce similar observations, particularly when the data are low resolution, spatially incomplete, or noisy. Rather than selecting a single reconstruction, a probabilistic method seeks the posterior distribution<\/p>\n\n\n\n

p<\/mi>(<\/mo>U<\/mi>0<\/mn><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>p(U_0\\mid Y=y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

This posterior represents both the states that are compatible with the measurement and the remaining uncertainty after conditioning. <\/p>\n\n\n\n

In the near-far decomposition, the corresponding far-state posterior is <\/p>\n\n\n\n

p<\/mi>(<\/mo>D<\/mi>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>p(D\\mid Y=y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

A principal objective is to determine when this distribution differs meaningfully from the prior distribution of D<\/mi>D<\/annotation><\/semantics><\/math>. If it does not, then the observation provides no usable information about the unobserved region.<\/p>\n\n\n\n

2.2 Diffusion-Based Posterior Sampling<\/h4>\n\n\n\n

Diffusion models introduce a family of noisy states connecting clean data to an approximately Gaussian terminal distribution. A standard linear forward-noising model is<\/p>\n\n\n\n

U<\/mi>t<\/mi><\/msub>=<\/mo>\u03b1<\/mi>t<\/mi><\/msub>U<\/mi>0<\/mn><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi><\/msub>\u03b5<\/mi>,<\/mo><\/mspace>\u03b5<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>0<\/mn>,<\/mo>I<\/mi>n<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>U_t\n=\n\\alpha_tU_0+\\sigma_t\\varepsilon,\n\\qquad\n\\varepsilon\\sim\\mathcal{N}(0,I_n).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The reverse process is governed by a score function. The unconditional score is<\/p>\n\n\n\n

s<\/mi>t<\/mi><\/msub>(<\/mo>u<\/mi>)<\/mo>=<\/mo>\u2207<\/mo>u<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>t<\/mi><\/msub>(<\/mo>u<\/mi>)<\/mo>,<\/mo><\/mrow>s_t(u)\n=\n\\nabla_u\\log p_t(u),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

while the conditional score is<\/p>\n\n\n\n

s<\/mi>t<\/mi>\u2217<\/mo><\/msubsup>(<\/mo>u<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2207<\/mo>u<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>t<\/mi><\/msub>(<\/mo>u<\/mi>|<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>s_t^\\ast(u,y)\n=\n\\nabla_u\\log p_t(u\\mid y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Bayes’ rule gives the score decomposition<\/p>\n\n\n\n

s<\/mi>t<\/mi>\u2217<\/mo><\/msubsup>(<\/mo>u<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>s<\/mi>t<\/mi><\/msub>(<\/mo>u<\/mi>)<\/mo>+<\/mo>\u2207<\/mo>u<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>t<\/mi><\/msub>(<\/mo>y<\/mi>|<\/mo>u<\/mi>)<\/mo>.<\/mi><\/mrow>s_t^\\ast(u,y)\n=\ns_t(u)\n+\n\\nabla_u\\log p_t(y\\mid u).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The first term is given by an unconditional generative model. The second term incorporates the measurement and directs sampling toward states that are compatible with the observation.<\/p>\n\n\n\n

The likelihood term is usually intractable because the observation is defined on the clean state U<\/mi>0<\/mn><\/msub>U_0<\/annotation><\/semantics><\/math>, while the reverse process evolves through the noisy state U<\/mi>t<\/mi><\/msub>U_t<\/annotation><\/semantics><\/math>. Diffusion posterior sampling approximates this term by applying the observation operator to a denoised estimate of the clean state and differentiating the resulting data-fidelity loss [2<\/mn>2<\/annotation><\/semantics><\/math>].<\/p>\n\n\n\n

2.3 QG Sampler and Observation Operator<\/h4>\n\n\n\n

The quasi-geostrophic (QG) implementation provides a concrete realization of the likelihood-guided reverse process. The sampler assumes a variance-preserving stochastic differential equation and supports unconditional sampling, conditional sampling, classifier-free guidance, SDEdit, and diffusion posterior sampling [3<\/mn>3<\/annotation><\/semantics><\/math>].<\/p>\n\n\n\n

At diffusion time t<\/mi>t<\/annotation><\/semantics><\/math>, the unconditional network predicts a noise field<\/p>\n\n\n\n
\u03b5<\/mi>\u03b8<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>t<\/mi>)<\/mo>.<\/mi><\/mrow>\\varepsilon_\\theta(U_t,t).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Using the variance-preserving parameterization, the corresponding estimate of the clean field is<\/p>\n\n\n\n

U<\/mi>^<\/mo><\/mover>0<\/mn><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>t<\/mi>)<\/mo>=<\/mo>1<\/mn>\u03bc<\/mi>t<\/mi><\/msub><\/mfrac>(<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>\u03c3<\/mi>t<\/mi><\/msub>\u03b5<\/mi>\u03b8<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>t<\/mi>)<\/mo>)<\/mo><\/mrow>.<\/mi><\/mrow>\\widehat{U}_0(U_t,t)\n=\n\\frac{1}{\\mu_t}\n\\left(\nU_t-\\sigma_t\\varepsilon_\\theta(U_t,t)\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The implementation then applies a low-resolution observation operator to this clean-state estimate. In Fourier space, the field I first multiplied by a Gaussian filter of the form<\/p>\n\n\n\n

G<\/mi>(<\/mo>k<\/mi>)<\/mo>=<\/mo>exp<\/mi>\u2061<\/mo><\/mrow>[<\/mo>\u2212<\/mo>4<\/mn>k<\/mi>r<\/mi>2<\/mn><\/msubsup>(<\/mo>s<\/mi>\u0394<\/mi><\/mrow>x<\/mi>)<\/mo>2<\/mn><\/msup><\/mrow>24<\/mn><\/mfrac>]<\/mo><\/mrow>,<\/mo><\/mrow>G(k)\n=\n\\exp\n\\left[\n-\\frac{4k_r^2(s\\Delta x)^2}{24}\n\\right],<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where s<\/mi>s<\/annotation><\/semantics><\/math> is the coarsening scale, \u0394<\/mi><\/mrow>x<\/mi><\/mrow>\\Delta x<\/annotation><\/semantics><\/math> is the grid spacing, and k<\/mi>r<\/mi><\/msub>k_r<\/annotation><\/semantics><\/math> is the radial wavenumber. Additional spectral cutoffs remove modes above the coarse-grid Nyquist limit. The filtered field is transformed back to physical space, average pooled on an s<\/mi>\u00d7<\/mo>s<\/mi><\/mrow>s\\times s<\/annotation><\/semantics><\/math> grid, and repeated to the original resolution.<\/p>\n\n\n\n

The resulting operator may be represented abstractly as<\/p>\n\n\n\n

\ud835\udc9c<\/mi><\/mtext>LES<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>^<\/mo><\/mover>0<\/mn><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>\\mathcal{A}_{\\mathrm{LES}}\n\\left(\n\\widehat{U}_0\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The implementation also includes an optional gappy-observation mask that sets selected spatial swaths of the coarsened field to zero [3<\/mn>3<\/annotation><\/semantics><\/math>]. <\/p>\n\n\n\n

This construction makes the spatial bandwidth of the observation operator explicit. The Gaussian spectral filer, hard spectral cutoff, pooling scale, and spatial mask collectively determine which directions of the high-resolution state are visible to the likelihood.<\/p>\n\n\n\n

2.4 Observation Support and Information Propagation<\/h4>\n\n\n\n

For a partial observation,<\/p>\n\n\n\n

H<\/mi>=<\/mo>(<\/mo>A<\/mi><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>,<\/mo><\/mrow>H\n=\n\\begin{pmatrix}\nA&amp;0\n\\end{pmatrix},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

so the likelihood depends directly only on the near component. The spatial support of bandwidth of the observation kernel is encoded by A<\/mi>A<\/annotation><\/semantics><\/math>. A wider kernel may observe more spatial directions, while a restricted kernel may leave large subspaces unmeasured. <\/p>\n\n\n\n

Direct observation is not the only mechanism through which information can propagate. If the near and far components are statistically correlated, observing C<\/mi>C<\/annotation><\/semantics><\/math> can reduce uncertainty in D<\/mi>D<\/annotation><\/semantics><\/math>. In a nonlinear generative model, an analogous effect occurs when the predicted observed region depends on the hidden coordinates of the current state. The linear Gaussian model isolates this mechanism in a form that can be derived exactly.<\/p>\n\n\n\n

3. Linear Gaussian Model<\/h2>\n\n\n\n

3.1 State and Observation Model<\/h4>\n\n\n\n

Let<\/p>\n\n\n\n

U<\/mi>0<\/mn><\/msub>=<\/mo>(<\/mo>C<\/mi><\/mtd><\/mtr>D<\/mi><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>\u2208<\/mo>\u211d<\/mi>n<\/mi><\/msup>,<\/mo><\/mspace>n<\/mi>=<\/mo>n<\/mi>C<\/mi><\/msub>+<\/mo>n<\/mi>D<\/mi><\/msub>,<\/mo><\/mrow>U_0\n=\n\\begin{pmatrix}\nC\\\\\nD\n\\end{pmatrix}\n\\in\\mathbb{R}^{n},\n\\qquad\nn=n_C+n_D,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

with<\/p>\n\n\n\n

C<\/mi>\u2208<\/mo>\u211d<\/mi>n<\/mi>C<\/mi><\/msub><\/msup>,<\/mo><\/mspace>D<\/mi>\u2208<\/mo>\u211d<\/mi>n<\/mi>D<\/mi><\/msub><\/msup>.<\/mi><\/mrow>C\\in\\mathbb{R}^{n_C},\n\\qquad\nD\\in\\mathbb{R}^{n_D}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Assume a centered Gaussian prior<\/p>\n\n\n\n

U<\/mi>0<\/mn><\/msub>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>0<\/mn>,<\/mo>\u03a3<\/mi><\/mrow>)<\/mo>,<\/mo><\/mrow>U_0\n\\sim\n\\mathcal{N}(0,\\Sigma),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

with block covariance<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>=<\/mo>(<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>C<\/mi><\/mrow><\/msub><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>\u227b<\/mo>0.<\/mn><\/mrow>\\Sigma\n=\n\\begin{pmatrix}\n\\Sigma_{CC}&amp;\\Sigma_{CD}\\\\\n\\Sigma_{DC}&amp;\\Sigma_{DD}\n\\end{pmatrix}\n\\succ0.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The diagonal blocks are the marginal covariances<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>C<\/mi>C<\/mi><\/mrow><\/msub>=<\/mo>Cov<\/mi>\u2061<\/mo>(<\/mo>C<\/mi>)<\/mo>,<\/mo><\/mspace>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>=<\/mo>Cov<\/mi>\u2061<\/mo>(<\/mo>D<\/mi>)<\/mo>,<\/mo><\/mrow>\\Sigma_{CC}\n=\n\\operatorname{Cov}(C),\n\\qquad\n\\Sigma_{DD}\n=\n\\operatorname{Cov}(D),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and the off-diagonal blocks are the cross-covariances<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow><\/msub>=<\/mo>Cov<\/mi>\u2061<\/mo>(<\/mo>C<\/mi>,<\/mo>D<\/mi>)<\/mo>,<\/mo><\/mspace>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>.<\/mi><\/mrow>\\Sigma_{CD}\n=\n\\operatorname{Cov}(C,D),\n\\qquad\n\\Sigma_{DC}\n=\n\\Sigma_{CD}^{\\top}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The observation is<\/p>\n\n\n\n

Y<\/mi>=<\/mo>A<\/mi>C<\/mi>+<\/mo>N<\/mi>,<\/mo><\/mspace>A<\/mi>\u2208<\/mo>\u211d<\/mi>m<\/mi>\u00d7<\/mo>n<\/mi>C<\/mi><\/msub><\/mrow><\/msup>,<\/mo><\/mrow>Y\n=\nAC+N,\n\\qquad\nA\\in\\mathbb{R}^{m\\times n_C},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where<\/p>\n\n\n\n

N<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>0<\/mn>,<\/mo>R<\/mi>)<\/mo>,<\/mo><\/mspace>R<\/mi>\u227b<\/mo>0<\/mn>,<\/mo><\/mrow>N\n\\sim\n\\mathcal{N}(0,R),\n\\qquad\nR\\succ0,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and N<\/mi>N<\/annotation><\/semantics><\/math> is independent of U<\/mi>0<\/mn><\/msub>U_0<\/annotation><\/semantics><\/math>. With<\/p>\n\n\n\n
H<\/mi>=<\/mo>(<\/mo>A<\/mi><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>\u2208<\/mo>\u211d<\/mi>m<\/mi>\u00d7<\/mo>n<\/mi><\/mrow><\/msup>,<\/mo><\/mrow>H\n=\n\\begin{pmatrix}\nA&amp;0\n\\end{pmatrix}\n\\in\\mathbb{R}^{m\\times n},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

the observation equation is<\/p>\n\n\n\n

Y<\/mi>=<\/mo>H<\/mi>U<\/mi>0<\/mn><\/msub>+<\/mo>N<\/mi>.<\/mi><\/mrow>Y\n=\nHU_0+N.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

3.2 Gaussian Conditioning<\/h4>\n\n\n\n

The conditioning calculation uses the following standard result. Let X<\/mi>X<\/annotation><\/semantics><\/math> and Z<\/mi>Z<\/annotation><\/semantics><\/math> be jointly Gaussian with zero means,<\/p>\n\n\n\n
Cov<\/mi>\u2061<\/mo>(<\/mo>X<\/mi>)<\/mo>=<\/mo>S<\/mi>X<\/mi><\/msub>,<\/mo><\/mspace>Cov<\/mi>\u2061<\/mo>(<\/mo>Z<\/mi>)<\/mo>=<\/mo>S<\/mi>Z<\/mi><\/msub>\u227b<\/mo>0<\/mn>,<\/mo><\/mspace>Cov<\/mi>\u2061<\/mo>(<\/mo>X<\/mi>,<\/mo>Z<\/mi>)<\/mo>=<\/mo>S<\/mi>X<\/mi>Z<\/mi><\/mrow><\/msub>.<\/mi><\/mrow>\\operatorname{Cov}(X)=S_X,\n\\qquad\n\\operatorname{Cov}(Z)=S_Z\\succ0,\n\\qquad\n\\operatorname{Cov}(X,Z)=S_{XZ}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Then [4<\/mn>4<\/annotation><\/semantics><\/math>]<\/p>\n\n\n\n
X<\/mi>|<\/mo>Z<\/mi>=<\/mo>z<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>S<\/mi>X<\/mi>Z<\/mi><\/mrow><\/msub>S<\/mi>Z<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>z<\/mi>,<\/mo>S<\/mi>X<\/mi><\/msub>\u2212<\/mo>S<\/mi>X<\/mi>Z<\/mi><\/mrow><\/msub>S<\/mi>Z<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>S<\/mi>X<\/mi>Z<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>)<\/mo><\/mrow>.<\/mi><\/mrow>X\\mid Z=z\n\\sim\n\\mathcal{N}\n\\left(\nS_{XZ}S_Z^{-1}z,\nS_X-S_{XZ}S_Z^{-1}S_{XZ}^{\\top}\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Because Y<\/mi>Y<\/annotation><\/semantics><\/math> is a linear function of independent Gaussian variables, the pair (<\/mo>U<\/mi>0<\/mn><\/msub>,<\/mo>Y<\/mi>)<\/mo><\/mrow>(U_0,Y)<\/annotation><\/semantics><\/math> is jointly Gaussian. Its observation covariance is<\/p>\n\n\n\n
S<\/mi>Y<\/mi><\/msub>:<\/mo>=<\/mo>Cov<\/mi>\u2061<\/mo>(<\/mo>Y<\/mi>)<\/mo>=<\/mo>H<\/mi>\u03a3<\/mi><\/mrow>H<\/mi>\u22a4<\/mi><\/msup>+<\/mo>R<\/mi>=<\/mo>A<\/mi>\u03a3<\/mi><\/mrow>C<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>+<\/mo>R<\/mi>,<\/mo><\/mrow>S_Y\n:=\n\\operatorname{Cov}(Y)\n=\nH\\Sigma H^\\top+R\n=\nA\\Sigma_{CC}A^\\top+R,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and the state-observation cross-covariance is<\/p>\n\n\n\n

Cov<\/mi>\u2061<\/mo>(<\/mo>U<\/mi>0<\/mn><\/msub>,<\/mo>Y<\/mi>)<\/mo>=<\/mo>\u03a3<\/mi><\/mrow>H<\/mi>\u22a4<\/mi><\/msup>.<\/mi><\/mrow>\\operatorname{Cov}(U_0,Y)\n=\n\\Sigma H^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Applying the Gaussian conditioning formula [4<\/mn>4<\/annotation><\/semantics><\/math>] gives<\/p>\n\n\n\n
U<\/mi>0<\/mn><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>m<\/mi>0<\/mn>|<\/mo>y<\/mi><\/mrow><\/msub>,<\/mo>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>Y<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>,<\/mo><\/mrow>U_0\\mid Y=y\n\\sim\n\\mathcal{N}\n\\left(\nm_{0\\mid y},\n\\Sigma_{0\\mid Y}\n\\right),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where<\/p>\n\n\n\n

m<\/mi>0<\/mn>|<\/mo>y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>H<\/mi>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>y<\/mi><\/mrow>m_{0\\mid y}\n=\n\\Sigma H^\\top S_Y^{-1}y<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>Y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>\u2212<\/mo>\u03a3<\/mi><\/mrow>H<\/mi>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>H<\/mi>\u03a3<\/mi><\/mrow>.<\/mi><\/mrow>\\Sigma_{0\\mid Y}\n=\n\\Sigma\n–\n\\Sigma H^\\top S_Y^{-1}H\\Sigma.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The posterior mean depends linearly on the realized observation y<\/mi>y<\/annotation><\/semantics><\/math>. The posterior covariance does not depend on the particular observed value because the model is linear Gaussian.<\/p>\n\n\n\n

4. Information Transfer from C<\/em> to D<\/em><\/h2>\n\n\n\n

4.1 Far-State Posterior<\/h4>\n\n\n\n

The cross-covariance between the far state and the observation is<\/p>\n\n\n\n

Cov<\/mi>\u2061<\/mo>(<\/mo>D<\/mi>,<\/mo>Y<\/mi>)<\/mo>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>.<\/mi><\/mrow>\\operatorname{Cov}(D,Y)\n=\n\\Sigma_{DC}A^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Applying Gaussian conditioning directly to (<\/mo>D<\/mi>,<\/mo>Y<\/mi>)<\/mo><\/mrow>(D,Y)<\/annotation><\/semantics><\/math> gives<\/p>\n\n\n\n
D<\/mi>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>,<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>,<\/mo><\/mrow>D\\mid Y=y\n\\sim\n\\mathcal{N}\n\\left(\nm_{D\\mid y},\n\\Sigma_{D\\mid Y}\n\\right),\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

with posterior mean<\/p>\n\n\n\n

m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>y<\/mi><\/mrow>m_{D\\mid y}\n=\n\\Sigma_{DC}A^\\top S_Y^{-1}y<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and posterior covariance<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>\u2212<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>A<\/mi>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow><\/msub>.<\/mi><\/mrow>\\Sigma_{D\\mid Y}\n=\n\\Sigma_{DD}\n–\n\\Sigma_{DC}A^\\top S_Y^{-1}A\\Sigma_{CD}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

These expressions separate the two mechanism that determine information transfer. The observation operator A<\/mi>A<\/annotation><\/semantics><\/math> selects directions of the near state, while the cross-covariance \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>\\Sigma_{DC}<\/annotation><\/semantics><\/math> determines which of those observed directions are correlated with the far state.<\/p>\n\n\n\n

If<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>=<\/mo>0<\/mn>,<\/mo><\/mrow>\\Sigma_{DC}A^\\top\n=\n0,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

then<\/p>\n\n\n\n

m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>=<\/mo>0<\/mn>,<\/mo><\/mspace>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>.<\/mi><\/mrow>m_{D\\mid y}\n=\n0,\n\\qquad\n\\Sigma_{D\\mid Y}\n=\n\\Sigma_{DD}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

In this case, conditioning on the observation does not change the distribution of D<\/mi>D<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

4.2 Reduction in Far-State Uncertainty<\/h4>\n\n\n\n

Define<\/p>\n\n\n\n

Q<\/mi>D<\/mi><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>A<\/mi>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow><\/msub>.<\/mi><\/mrow>Q_D\n=\n\\Sigma_{DC}A^\\top S_Y^{-1}A\\Sigma_{CD}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Then<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>\u2212<\/mo>Q<\/mi>D<\/mi><\/msub>.<\/mi><\/mrow>\\Sigma_{D\\mid Y}\n=\n\\Sigma_{DD}-Q_D.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

For any v<\/mi>\u2208<\/mo>\u211d<\/mi>n<\/mi>D<\/mi><\/msub><\/msup><\/mrow>v\\in\\mathbb{R}^{n_D}<\/annotation><\/semantics><\/math>, <\/p>\n\n\n\n
v<\/mi>\u22a4<\/mi><\/msup>Q<\/mi>D<\/mi><\/msub>v<\/mi>=<\/mo>(<\/mo>A<\/mi>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow><\/msub>v<\/mi>)<\/mo><\/mrow>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>(<\/mo>A<\/mi>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow><\/msub>v<\/mi>)<\/mo><\/mrow>\u2265<\/mo>0.<\/mn><\/mrow>v^\\top Q_Dv\n=\n\\left(A\\Sigma_{CD}v\\right)^\\top\nS_Y^{-1}\n\\left(A\\Sigma_{CD}v\\right)\n\\geq0.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Therefore,<\/p>\n\n\n\n

Q<\/mi>D<\/mi><\/msub>\u2ab0<\/mo>0<\/mn><\/mrow>Q_D\\succeq0<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>\u2aaf<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>.<\/mi><\/mrow>\\Sigma_{D\\mid Y}\n\\preceq\n\\Sigma_{DD}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Thus, conditioning cannot increase posterior uncertainty in any linear direction of the far state.<\/p>\n\n\n\n

Let<\/p>\n\n\n\n

L<\/mi>D<\/mi><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>.<\/mi><\/mrow>L_D\n=\n\\Sigma_{DC}A^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Since<\/p>\n\n\n\n

Q<\/mi>D<\/mi><\/msub>=<\/mo>L<\/mi>D<\/mi><\/msub>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>L<\/mi>D<\/mi>\u22a4<\/mi><\/msubsup>,<\/mo><\/mrow>Q_D\n=\nL_DS_Y^{-1}L_D^\\top,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

the rank of the covariance reduction satisfies<\/p>\n\n\n\n

rank<\/mi>\u2061<\/mo>(<\/mo>Q<\/mi>D<\/mi><\/msub>)<\/mo>=<\/mo>rank<\/mi>\u2061<\/mo>(<\/mo>L<\/mi>D<\/mi><\/msub>)<\/mo>\u2264<\/mo>rank<\/mi>\u2061<\/mo>(<\/mo>A<\/mi>)<\/mo>\u2264<\/mo>m<\/mi>.<\/mi><\/mrow>\\operatorname{rank}(Q_D)\n=\n\\operatorname{rank}(L_D)\n\\leq\n\\operatorname{rank}(A)\n\\leq\nm.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Consequently, m<\/mi>m<\/annotation><\/semantics><\/math> measurements can reduce far-state uncertainty in at most m<\/mi>m<\/annotation><\/semantics><\/math> independent directions. In addition,<\/p>\n\n\n\n
m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>\u2208<\/mo>Range<\/mi>\u2061<\/mo>(<\/mo>L<\/mi>D<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>m_{D\\mid y}\n\\in\n\\operatorname{Range}(L_D).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Both the posterior-mean update and covariance reduction are therefore restricted to directions selected by \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup><\/mrow>\\Sigma_{DC}A^\\top<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

4.3 Mutual Information<\/h4>\n\n\n\n

For a Gaussian random variable X<\/mi>\u2208<\/mo>\u211d<\/mi>k<\/mi><\/msup><\/mrow>X\\in\\mathbb{R}^k<\/annotation><\/semantics><\/math> with covariance S<\/mi>\u227b<\/mo>0<\/mn><\/mrow>S\\succ0<\/annotation><\/semantics><\/math>, the differential entropy is<\/p>\n\n\n\n
h<\/mi>(<\/mo>X<\/mi>)<\/mo>=<\/mo>1<\/mn>2<\/mn><\/mfrac>log<\/mi>\u2061<\/mo><\/mrow>(<\/mo>(<\/mo>2<\/mn>\u03c0<\/mi>e<\/mi>)<\/mo>k<\/mi><\/msup>det<\/mi>\u2061<\/mo><\/mspace><\/mrow>S<\/mi>)<\/mo><\/mrow>.<\/mi><\/mrow>h(X)\n=\n\\frac{1}{2}\n\\log\n\\left(\n(2\\pi e)^k\\det S\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Applying<\/p>\n\n\n\n

I<\/mi>(<\/mo>D<\/mi>;<\/mo>Y<\/mi>)<\/mo>=<\/mo>h<\/mi>(<\/mo>D<\/mi>)<\/mo>\u2212<\/mo>h<\/mi>(<\/mo>D<\/mi>|<\/mo>Y<\/mi>)<\/mo><\/mrow>I(D;Y)\n=\nh(D)-h(D\\mid Y)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

causes the constant terms to cancel and gives the determinant ratio below.<\/p>\n\n\n\n

The reduction in uncertainty can also be expressed through mutual information. For Gaussian variables.<\/p>\n\n\n\n

I<\/mi>(<\/mo>D<\/mi>;<\/mo>Y<\/mi>)<\/mo>=<\/mo>1<\/mn>2<\/mn><\/mfrac>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>det<\/mi>\u2061<\/mo><\/mspace><\/mrow>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub><\/mrow>det<\/mi>\u2061<\/mo><\/mspace><\/mrow>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub><\/mrow><\/mfrac>.<\/mi><\/mrow>I(D;Y)\n=\n\\frac{1}{2}\n\\log\n\\frac{\\det\\Sigma_{DD}}\n{\\det\\Sigma_{D\\mid Y}}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

This quantity measures the information about the far state contained in the observation. It vanishes exactly when<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>=<\/mo>0.<\/mn><\/mrow>\\Sigma_{DC}A^\\top\n=\n0.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Therefore, the observation carries information about D<\/mi>D<\/annotation><\/semantics><\/math> only through the near-far covariance directions that are also visible to A<\/mi>A<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

Define the normalized covariance reduction<\/p>\n\n\n\n

K<\/mi>D<\/mi><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow>\u2212<\/mo>1<\/mn>\/<\/mi>2<\/mn><\/mrow><\/msubsup>Q<\/mi>D<\/mi><\/msub>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow>\u2212<\/mo>1<\/mn>\/<\/mi>2<\/mn><\/mrow><\/msubsup>.<\/mi><\/mrow>K_D\n=\n\\Sigma_{DD}^{-1\/2}\nQ_D\n\\Sigma_{DD}^{-1\/2}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Then<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow>1<\/mn>\/<\/mi>2<\/mn><\/mrow><\/msubsup>(<\/mo>I<\/mi>\u2212<\/mo>K<\/mi>D<\/mi><\/msub>)<\/mo><\/mrow>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow>1<\/mn>\/<\/mi>2<\/mn><\/mrow><\/msubsup>,<\/mo><\/mrow>\\Sigma_{D\\mid Y}\n=\n\\Sigma_{DD}^{1\/2}\n\\left(I-K_D\\right)\n\\Sigma_{DD}^{1\/2},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and<\/p>\n\n\n\n

I<\/mi>(<\/mo>D<\/mi>;<\/mo>Y<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>1<\/mn>2<\/mn><\/mfrac>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>det<\/mi>\u2061<\/mo><\/mrow>(<\/mo>I<\/mi>\u2212<\/mo>K<\/mi>D<\/mi><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>I(D;Y)\n=\n-\\frac{1}{2}\n\\log\\det\\left(I-K_D\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The eigenvalues of K<\/mi>D<\/mi><\/msub>K_D<\/annotation><\/semantics><\/math> quantify the fractional uncertainty reduction along informative far-state directions.<\/p>\n\n\n\n

4.4 Information Under Forward Noising<\/h4>\n\n\n\n

For the forward-noised far state<\/p>\n\n\n\n

D<\/mi>t<\/mi><\/msub>=<\/mo>\u03b1<\/mi>t<\/mi><\/msub>D<\/mi>+<\/mo>\u03c3<\/mi>t<\/mi><\/msub>\u03b5<\/mi>D<\/mi><\/msub>,<\/mo><\/mrow>D_t\n=\n\\alpha_tD+\\sigma_t\\varepsilon_D,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

the unconditional and conditional covariances are<\/p>\n\n\n\n

Cov<\/mi>\u2061<\/mo>(<\/mo>D<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi>D<\/mi><\/msub><\/msub><\/mrow>\\operatorname{Cov}(D_t)\n=\n\\alpha_t^2\\Sigma_{DD}\n+\n\\sigma_t^2I_{n_D}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and<\/p>\n\n\n\n

Cov<\/mi>\u2061<\/mo>(<\/mo>D<\/mi>t<\/mi><\/msub>|<\/mo>Y<\/mi>)<\/mo>=<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi>D<\/mi><\/msub><\/msub>.<\/mi><\/mrow>\\operatorname{Cov}(D_t\\mid Y)\n=\n\\alpha_t^2\\Sigma_{D\\mid Y}\n+\n\\sigma_t^2I_{n_D}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Hence,<\/p>\n\n\n\n

I<\/mi>(<\/mo>D<\/mi>t<\/mi><\/msub>;<\/mo>Y<\/mi>)<\/mo>=<\/mo>1<\/mn>2<\/mn><\/mfrac>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>det<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi>D<\/mi><\/msub><\/msub>)<\/mo><\/mrow><\/mrow>det<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi>D<\/mi><\/msub><\/msub>)<\/mo><\/mrow><\/mrow><\/mfrac>.<\/mi><\/mrow>I(D_t;Y)\n=\n\\frac{1}{2}\n\\log\n\\frac{\n\\det\\left(\n\\alpha_t^2\\Sigma_{DD}\n+\n\\sigma_t^2I_{n_D}\n\\right)\n}{\n\\det\\left(\n\\alpha_t^2\\Sigma_{D\\mid Y}\n+\n\\sigma_t^2I_{n_D}\n\\right)\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

For \u03c3<\/mi>t<\/mi><\/msub>><\/mo>0<\/mn><\/mrow>\\sigma_t>0<\/annotation><\/semantics><\/math>, define the signal-to-noise ratio<\/p>\n\n\n\n
\u03bb<\/mi>t<\/mi><\/msub>=<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup><\/mfrac>.<\/mi><\/mrow>\\lambda_t\n=\n\\frac{\\alpha_t^2}{\\sigma_t^2}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Scaling by a nonzero constant preserves mutual information, so<\/p>\n\n\n\n

I<\/mi>(<\/mo>D<\/mi>t<\/mi><\/msub>;<\/mo>Y<\/mi>)<\/mo>=<\/mo>I<\/mi>(<\/mo>\u03bb<\/mi>t<\/mi><\/msub><\/msqrt>D<\/mi>+<\/mo>\u03b5<\/mi>D<\/mi><\/msub>;<\/mo>Y<\/mi>)<\/mo><\/mrow>.<\/mi><\/mrow>I(D_t;Y)\n=\nI\n\\left(\n\\sqrt{\\lambda_t}D+\\varepsilon_D;\nY\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

To compare two signal-to-noise ratios, suppose<\/p>\n\n\n\n

\u03bb<\/mi>1<\/mn><\/msub>\u2265<\/mo>\u03bb<\/mi>2<\/mn><\/msub>\u2265<\/mo>0<\/mn><\/mrow>\\lambda_1\\geq\\lambda_2\\geq0<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and define<\/p>\n\n\n\n

a<\/mi>=<\/mo>\u03bb<\/mi>2<\/mn><\/msub>\u03bb<\/mi>1<\/mn><\/msub><\/mfrac><\/msqrt>,<\/mo><\/mrow>a\n=\n\\sqrt{\n\\frac{\\lambda_2}{\\lambda_1}\n},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
Z<\/mi>1<\/mn><\/msub>=<\/mo>\u03bb<\/mi>1<\/mn><\/msub><\/msqrt>D<\/mi>+<\/mo>\u03b5<\/mi>1<\/mn><\/msub>.<\/mi><\/mrow>Z_1\n=\n\\sqrt{\\lambda_1}D+\\varepsilon_1.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Let<\/p>\n\n\n\n

Z<\/mi>2<\/mn><\/msub>=<\/mo>a<\/mi>Z<\/mi>1<\/mn><\/msub>+<\/mo>1<\/mn>\u2212<\/mo>a<\/mi>2<\/mn><\/msup><\/mrow><\/msqrt><\/mspace>\u03b5<\/mi>\u2032<\/mo><\/msup>,<\/mo><\/mrow>Z_2\n=\naZ_1\n+\n\\sqrt{1-a^2}\\,\n\\varepsilon’,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where \u03b5<\/mi>1<\/mn><\/msub>\\varepsilon_1<\/annotation><\/semantics><\/math> and \u03b5<\/mi>\u2032<\/mo><\/msup>\\varepsilon’<\/annotation><\/semantics><\/math> are independent standard Gaussian variables independent of (<\/mo>D<\/mi>,<\/mo>Y<\/mi>)<\/mo><\/mrow>(D,Y)<\/annotation><\/semantics><\/math>. Expanding Z<\/mi>2<\/mn><\/msub>Z_2<\/annotation><\/semantics><\/math> gives<\/p>\n\n\n\n
Z<\/mi>2<\/mn><\/msub>=<\/mo>\u03bb<\/mi>2<\/mn><\/msub><\/msqrt>D<\/mi>+<\/mo>(<\/mo>a<\/mi>\u03b5<\/mi>1<\/mn><\/msub>+<\/mo>1<\/mn>\u2212<\/mo>a<\/mi>2<\/mn><\/msup><\/mrow><\/msqrt><\/mspace>\u03b5<\/mi>\u2032<\/mo><\/msup>)<\/mo><\/mrow>.<\/mi><\/mrow>Z_2\n=\n\\sqrt{\\lambda_2}D\n+\n\\left(\na\\varepsilon_1\n+\n\\sqrt{1-a^2}\\,\n\\varepsilon’\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The Gaussian noise in parentheses is standard, so Z<\/mi>2<\/mn><\/msub>Z_2<\/annotation><\/semantics><\/math> has the lower-signal-to-noise law and is obtained from Z<\/mi>1<\/mn><\/msub>Z_1<\/annotation><\/semantics><\/math> by adding noise. Therefore,<\/p>\n\n\n\n
Y<\/mi>\u27f6<\/mo>Z<\/mi>1<\/mn><\/msub>\u27f6<\/mo>Z<\/mi>2<\/mn><\/msub><\/mrow>Y\n\\longrightarrow\nZ_1\n\\longrightarrow\nZ_2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

is a Markov chain. By the data-processing inequality [5],<\/p>\n\n\n\n

I<\/mi>(<\/mo>Z<\/mi>2<\/mn><\/msub>;<\/mo>Y<\/mi>)<\/mo>\u2264<\/mo>I<\/mi>(<\/mo>Z<\/mi>1<\/mn><\/msub>;<\/mo>Y<\/mi>)<\/mo>.<\/mi><\/mrow>I(Z_2;Y)\n\\leq\nI(Z_1;Y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Hence I<\/mi>(<\/mo>D<\/mi>t<\/mi><\/msub>;<\/mo>Y<\/mi>)<\/mo><\/mrow>I(D_t;Y)<\/annotation><\/semantics><\/math> is nondecreasing in \u03bb<\/mi>t<\/mi><\/msub>=<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\/<\/mi>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup><\/mrow>\\lambda_t=\\alpha_t^2\/\\sigma_t^2<\/annotation><\/semantics><\/math>. Equivalently, forward noising cannot increase the information about Y<\/mi>Y<\/annotation><\/semantics><\/math> available in the far state. This gives an information-theoretic interpretation of the reverse process. As the signal-t0-noise ratio increases, the process can progressively recover the information present in the conditional clean-state distribution.<\/p>\n\n\n\n

5. Gaussian Score and DPS<\/h2>\n\n\n\n

5.1 Forward-Noised Posterior<\/h4>\n\n\n\n

The forward-noised state is<\/p>\n\n\n\n

U<\/mi>t<\/mi><\/msub>=<\/mo>\u03b1<\/mi>t<\/mi><\/msub>U<\/mi>0<\/mn><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi><\/msub>\u03b5<\/mi>,<\/mo><\/mspace>\u03b5<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>0<\/mn>,<\/mo>I<\/mi>n<\/mi><\/msub>)<\/mo>,<\/mo><\/mrow>U_t\n=\n\\alpha_tU_0+\\sigma_t\\varepsilon,\n\\qquad\n\\varepsilon\\sim\\mathcal{N}(0,I_n),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

with independent of (<\/mo>U<\/mi>0<\/mn><\/msub>,<\/mo>Y<\/mi>)<\/mo><\/mrow>(U_0,Y)<\/annotation><\/semantics><\/math>. Since U<\/mi>0<\/mn><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi><\/mrow>U_0\\mid Y=y<\/annotation><\/semantics><\/math> is Gaussian,<\/p>\n\n\n\n
U<\/mi>t<\/mi><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>\u03b1<\/mi>t<\/mi><\/msub>m<\/mi>0<\/mn>|<\/mo>y<\/mi><\/mrow><\/msub>,<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>Y<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>U_t\\mid Y=y\n\\sim\n\\mathcal{N}\n\\left(\n\\alpha_tm_{0\\mid y},\n\\alpha_t^2\\Sigma_{0\\mid Y}\n+\n\\sigma_t^2I_n\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The far marginal satisfies<\/p>\n\n\n\n

D<\/mi>t<\/mi><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>\u03b1<\/mi>t<\/mi><\/msub>m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>,<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi>D<\/mi><\/msub><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>D_t\\mid Y=y\n\\sim\n\\mathcal{N}\n\\left(\n\\alpha_tm_{D\\mid y},\n\\alpha_t^2\\Sigma_{D\\mid Y}\n+\n\\sigma_t^2I_{n_D}\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

5.2 Exact Conditional Score<\/h4>\n\n\n\n

For a Gaussian random variable<\/p>\n\n\n\n

X<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>m<\/mi>,<\/mo>S<\/mi>)<\/mo>,<\/mo><\/mrow>X\\sim\\mathcal{N}(m,S),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

the score is<\/p>\n\n\n\n

\u2207<\/mo>x<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>X<\/mi><\/msub>(<\/mo>x<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>S<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>x<\/mi>\u2212<\/mo>m<\/mi>)<\/mo>.<\/mi><\/mrow>\\nabla_x\\log p_X(x)\n=\n-S^{-1}(x-m).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Applying this identity to the conditional distribution of U<\/mi>t<\/mi><\/msub>U_t<\/annotation><\/semantics><\/math> gives<\/p>\n\n\n\n
s<\/mi>t<\/mi>\u2217<\/mo><\/msubsup>(<\/mo>u<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>(<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>Y<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi><\/msub>)<\/mo><\/mrow>\u2212<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>u<\/mi>\u2212<\/mo>\u03b1<\/mi>t<\/mi><\/msub>m<\/mi>0<\/mn>|<\/mo>y<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>s_t^\\ast(u,y)\n=\n–\n\\left(\n\\alpha_t^2\\Sigma_{0\\mid Y}\n+\n\\sigma_t^2I_n\n\\right)^{-1}\n\\left(\nu-\\alpha_tm_{0\\mid y}\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Unconditionally,<\/p>\n\n\n\n

U<\/mi>t<\/mi><\/msub>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>0<\/mn>,<\/mo>S<\/mi>t<\/mi><\/msub>)<\/mo>,<\/mo><\/mrow>U_t\n\\sim\n\\mathcal{N}(0,S_t),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where<\/p>\n\n\n\n

S<\/mi>t<\/mi><\/msub>=<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi><\/msub>.<\/mi><\/mrow>S_t\n=\n\\alpha_t^2\\Sigma+\\sigma_t^2I_n.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The unconditional score is therefore<\/p>\n\n\n\n

s<\/mi>t<\/mi><\/msub>(<\/mo>u<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>S<\/mi>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>u<\/mi>.<\/mi><\/mrow>s_t(u)\n=\n-S_t^{-1}u.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The score of the far marginal is<\/p>\n\n\n\n

\u2207<\/mo>d<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>t<\/mi>D<\/mi><\/msubsup>(<\/mo>d<\/mi>|<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>(<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>+<\/mo>\u03c3<\/mi>t<\/mi>2<\/mn><\/msubsup>I<\/mi>n<\/mi>D<\/mi><\/msub><\/msub>)<\/mo><\/mrow>\u2212<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>d<\/mi>\u2212<\/mo>\u03b1<\/mi>t<\/mi><\/msub>m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>\\nabla_d\\log p_t^D(d\\mid y)\n=\n–\n\\left(\n\\alpha_t^2\\Sigma_{D\\mid Y}\n+\n\\sigma_t^2I_{n_D}\n\\right)^{-1}\n\\left(\nd-\\alpha_tm_{D\\mid y}\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

This far-marginal score is not generally equal to the far coordinates of the full-state score. The full-state score may depend jointly on the current near and far coordinates, while the marginal score depends only on d<\/mi>d<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

5.3 Exact Denoiser and Noisy Likelihood<\/h4>\n\n\n\n

The pair (<\/mo>U<\/mi>0<\/mn><\/msub>,<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow>(U_0,U_t)<\/annotation><\/semantics><\/math> is jointly Gaussian, with<\/p>\n\n\n\n
Cov<\/mi>\u2061<\/mo>(<\/mo>U<\/mi>0<\/mn><\/msub>,<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>\u03b1<\/mi>t<\/mi><\/msub>\u03a3<\/mi><\/mrow><\/mrow>\\operatorname{Cov}(U_0,U_t)\n=\n\\alpha_t\\Sigma<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and<\/p>\n\n\n\n

Cov<\/mi>\u2061<\/mo>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>S<\/mi>t<\/mi><\/msub>.<\/mi><\/mrow>\\operatorname{Cov}(U_t)\n=\nS_t.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Conditioning gives<\/p>\n\n\n\n

U<\/mi>0<\/mn><\/msub>|<\/mo>U<\/mi>t<\/mi><\/msub>=<\/mo>u<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>B<\/mi>t<\/mi><\/msub>u<\/mi>,<\/mo>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>t<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>,<\/mo><\/mrow>U_0\\mid U_t=u\n\\sim\n\\mathcal{N}\n\\left(\nB_tu,\n\\Sigma_{0\\mid t}\n\\right),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where<\/p>\n\n\n\n

B<\/mi>t<\/mi><\/msub>=<\/mo>\u03b1<\/mi>t<\/mi><\/msub>\u03a3<\/mi><\/mrow>S<\/mi>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup><\/mrow>B_t\n=\n\\alpha_t\\Sigma S_t^{-1}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and<\/p>\n\n\n\n

\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>t<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>\u2212<\/mo>\u03b1<\/mi>t<\/mi>2<\/mn><\/msubsup>\u03a3<\/mi><\/mrow>S<\/mi>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>\u03a3<\/mi><\/mrow>.<\/mi><\/mrow>\\Sigma_{0\\mid t}\n=\n\\Sigma\n–\n\\alpha_t^2\n\\Sigma S_t^{-1}\\Sigma.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The exact unconditional denoiser is therefore<\/p>\n\n\n\n

\ud835\udd3c<\/mi>[<\/mo>U<\/mi>0<\/mn><\/msub>|<\/mo>U<\/mi>t<\/mi><\/msub>=<\/mo>u<\/mi>]<\/mo>=<\/mo>B<\/mi>t<\/mi><\/msub>u<\/mi>.<\/mi><\/mrow>\\mathbb{E}[U_0\\mid U_t=u]\n=\nB_tu.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Given U<\/mi>t<\/mi><\/msub>=<\/mo>u<\/mi><\/mrow>U_t=u<\/annotation><\/semantics><\/math>, the observation distribution is<\/p>\n\n\n\n
Y<\/mi>|<\/mo>U<\/mi>t<\/mi><\/msub>=<\/mo>u<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>H<\/mi>B<\/mi>t<\/mi><\/msub>u<\/mi>,<\/mo>\u0393<\/mi><\/mrow>t<\/mi><\/msub>)<\/mo><\/mrow>,<\/mo><\/mrow>Y\\mid U_t=u\n\\sim\n\\mathcal{N}\n\\left(\nHB_tu,\n\\Gamma_t\n\\right),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where<\/p>\n\n\n\n

\u0393<\/mi><\/mrow>t<\/mi><\/msub>=<\/mo>R<\/mi>+<\/mo>H<\/mi>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>t<\/mi><\/mrow><\/msub>H<\/mi>\u22a4<\/mi><\/msup>.<\/mi><\/mrow>\\Gamma_t\n=\nR\n+\nH\\Sigma_{0\\mid t}H^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The matrix \u0393<\/mi><\/mrow>t<\/mi><\/msub>\\Gamma_t<\/annotation><\/semantics><\/math> is the effective observation covariance at time t<\/mi>t<\/annotation><\/semantics><\/math>. It combines the original measurement noise with the remaining uncertainty about the clean state after conditioning on U<\/mi>t<\/mi><\/msub>=<\/mo>u<\/mi><\/mrow>U_t=u<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

Differentiating the exact noisy likelihood gives<\/p>\n\n\n\n

\u2207<\/mo>u<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>t<\/mi><\/msub>(<\/mo>y<\/mi>|<\/mo>u<\/mi>)<\/mo>=<\/mo>B<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>\u0393<\/mi><\/mrow>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>(<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>B<\/mi>t<\/mi><\/msub>u<\/mi>)<\/mo><\/mrow>.<\/mi><\/mrow>\\nabla_u\\log p_t(y\\mid u)\n=\nB_t^\\top H^\\top\\Gamma_t^{-1}\n\\left(\ny-HB_tu\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

5.4 Exact Gaussian DPS Identity<\/h4>\n\n\n\n

Combining the unconditional score with the noisy likelihood gradient yields<\/p>\n\n\n\n

s<\/mi>t<\/mi>\u2217<\/mo><\/msubsup>(<\/mo>u<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>s<\/mi>t<\/mi><\/msub>(<\/mo>u<\/mi>)<\/mo>+<\/mo>B<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>\u0393<\/mi><\/mrow>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>(<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>B<\/mi>t<\/mi><\/msub>u<\/mi>)<\/mo><\/mrow>.<\/mi><\/mrow>s_t^\\ast(u,y)\n=\ns_t(u)\n+\nB_t^\\top H^\\top\\Gamma_t^{-1}\n\\left(\ny-HB_tu\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

In the linear Gaussian model, this identity is exact. It has the same structure as diffusion posterior sampling:<\/p>\n\n\n\n

    \n
  1. estimate the clean state from the current noisy state<\/li>\n\n\n\n
  2. apply the observation operator to the estimate<\/li>\n\n\n\n
  3. compute the measurement residual<\/li>\n\n\n\n
  4. propagate the residual back to the current state.<\/li>\n<\/ol>\n\n\n\n

    A standard point-estimate DPS approximation replaces the effective covariance \u0393<\/mi><\/mrow>t<\/mi><\/msub>\\Gamma_t<\/annotation><\/semantics><\/math> with a simpler measurement weighting and replaces the exact linear denoiser B<\/mi>t<\/mi><\/msub>u<\/mi><\/mrow>B_tu<\/annotation><\/semantics><\/math> with a learned denoising estimate. The Gaussian analysis therefore provides both an exact benchmark and a direct motivation for the nonlinear flow-matching extension.<\/p>\n\n\n\n

    5.5 Exact Reverse-Process Distribution<\/h4>\n\n\n\n

    Let U<\/mi>^<\/mo><\/mover>\\widehat{U}<\/annotation><\/semantics><\/math> denote the output of the exact continous reverse process initialized from the exact terminal distribution, driven by the exact conditional score, and simulated without numerical error. Because this is the exact reverse process,<\/p>\n\n\n\n
    P<\/mi>(<\/mo>U<\/mi>^<\/mo><\/mover>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>)<\/mo><\/mrow>=<\/mo>P<\/mi>(<\/mo>U<\/mi>0<\/mn><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>)<\/mo><\/mrow>.<\/mi><\/mrow>P\n\\left(\n\\widehat{U}\\mid Y=y\n\\right)\n=\nP\n\\left(\nU_0\\mid Y=y\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    In the linear Gaussian setting, this gives<\/p>\n\n\n\n

    U<\/mi>^<\/mo><\/mover>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>m<\/mi>0<\/mn>|<\/mo>y<\/mi><\/mrow><\/msub>,<\/mo>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>Y<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>\\widehat{U}\\mid Y=y\n\\sim\n\\mathcal{N}\n\\left(\nm_{0\\mid y},\n\\Sigma_{0\\mid Y}\n\\right).\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    Let P<\/mi>D<\/mi><\/msub>P_D<\/annotation><\/semantics><\/math> denote projection onto the far-state coordinates and define<\/p>\n\n\n\n
    D<\/mi>^<\/mo><\/mover>=<\/mo>P<\/mi>D<\/mi><\/msub>U<\/mi>^<\/mo><\/mover>.<\/mi><\/mrow>\\widehat{D}\n=\nP_D\\widehat{U}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    Then<\/p>\n\n\n\n

    D<\/mi>^<\/mo><\/mover>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>,<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>\\widehat{D}\\mid Y=y\n\\sim\n\\mathcal{N}\n\\left(\nm_{D\\mid y},\n\\Sigma_{D\\mid Y}\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    Thus, under exact sampling assumptions, the reverse process reproduces both the complete conditional distribution and its far-state marginal.<\/p>\n\n\n\n

    5.6 Relation to the QG DPS Implementation<\/h4>\n\n\n\n

    The exact Gaussian identity can be compared directly with the quasi-geostrophic sampler. The exact linear denoiser<\/p>\n\n\n\n

    B<\/mi>t<\/mi><\/msub>u<\/mi><\/mrow>B_tu<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    is replaced by the learned clean-state estimate<\/p>\n\n\n\n

    U<\/mi>^<\/mo><\/mover>0<\/mn><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>t<\/mi>)<\/mo>=<\/mo>1<\/mn>\u03bc<\/mi>t<\/mi><\/msub><\/mfrac>(<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>\u03c3<\/mi>t<\/mi><\/msub>\u03b5<\/mi>\u03b8<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>t<\/mi>)<\/mo>)<\/mo><\/mrow>.<\/mi><\/mrow>\\widehat{U}_0(U_t,t)\n=\n\\frac{1}{\\mu_t}\n\\left(\nU_t-\\sigma_t\\varepsilon_\\theta(U_t,t)\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    The theoretical observation operator H<\/mi>H<\/annotation><\/semantics><\/math> is replaced by the filtered and coarsened operator<\/p>\n\n\n\n
    \ud835\udc9c<\/mi><\/mtext>LES<\/mi><\/mrow><\/msub>.<\/mi><\/mrow>\\mathcal{A}_{\\mathrm{LES}}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    The implementation defines a mean-squared measurement loss<\/p>\n\n\n\n

    \u2112<\/mi><\/mtext>meas<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>MSE<\/mi>\u2061<\/mo><\/mspace>(<\/mo>\ud835\udc9c<\/mi><\/mtext>LES<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>^<\/mo><\/mover>0<\/mn><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>t<\/mi>)<\/mo>)<\/mo><\/mrow>,<\/mo>y<\/mi>)<\/mo><\/mrow>.<\/mi><\/mrow>\\mathcal{L}_{\\mathrm{meas}}(U_t)\n=\n\\operatorname{MSE}\n\\left(\n\\mathcal{A}_{\\mathrm{LES}}\n\\left(\n\\widehat{U}_0(U_t,t)\n\\right),\ny\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    Automatic differentiation is used to compute<\/p>\n\n\n\n

    \u2207<\/mo>U<\/mi>t<\/mi><\/msub><\/msub>\u2112<\/mi><\/mtext>meas<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>\\nabla_{U_t}\n\\mathcal{L}_{\\mathrm{meas}}(U_t).\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    The implemented DPS correction is<\/p>\n\n\n\n

    g<\/mi>t<\/mi><\/mtext>DPS<\/mi><\/mrow><\/msubsup>=<\/mo>\u2212<\/mo>1<\/mn>2<\/mn>\u03c3<\/mi><\/mtext>scaled<\/mi><\/mrow><\/msub><\/mrow><\/mfrac>\u2207<\/mo>U<\/mi>t<\/mi><\/msub><\/msub>\u2112<\/mi><\/mtext>meas<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>,<\/mo><\/mrow>g_t^{\\mathrm{DPS}}\n=\n-\\frac{1}{2\\sigma_{\\mathrm{scaled}}}\n\\nabla_{U_t}\n\\mathcal{L}_{\\mathrm{meas}}(U_t),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    with time-dependent scale<\/p>\n\n\n\n

    \u03c3<\/mi><\/mtext>scaled<\/mi><\/mrow><\/msub>=<\/mo>\u03c3<\/mi><\/mtext>measure<\/mi><\/mrow>2<\/mn><\/msubsup>+<\/mo>C<\/mi><\/mtext>DPS<\/mi><\/mrow><\/msub>(<\/mo>\u03c3<\/mi>t<\/mi><\/msub>\u03bc<\/mi>t<\/mi><\/msub><\/mfrac>)<\/mo><\/mrow>2<\/mn><\/msup>.<\/mi><\/mrow>\\sigma_{\\mathrm{scaled}}\n=\n\\sigma_{\\mathrm{measure}}^2\n+\nC_{\\mathrm{DPS}}\n\\left(\n\\frac{\\sigma_t}{\\mu_t}\n\\right)^2.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    The correction is added to the model score during both the reverse updated and the Langevin correction steps [3].<\/p>\n\n\n\n

    This implementation has the same computational structure as the exact Gaussian likelihood correction:<\/p>\n\n\n\n

      \n
    1. estimate the clean state from the current noisy state<\/li>\n\n\n\n
    2. apply a measurement operator to the estimate<\/li>\n\n\n\n
    3. compare the predicted measurement with the observed field<\/li>\n\n\n\n
    4. differentiate the discrepancy with respect to the current state<\/li>\n\n\n\n
    5. add the resulting correction to the unconditional score<\/li>\n<\/ol>\n\n\n\n

      The principal difference is the likelihood weighting. In the exact Gaussian model, the residual is weighted by the full effective covariance<\/p>\n\n\n\n

      \u0393<\/mi><\/mrow>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>=<\/mo>(<\/mo>R<\/mi>+<\/mo>H<\/mi>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>t<\/mi><\/mrow><\/msub>H<\/mi>\u22a4<\/mi><\/msup>)<\/mo><\/mrow>\u2212<\/mo>1<\/mn><\/mrow><\/msup>.<\/mi><\/mrow>\\Gamma_t^{-1}\n=\n\\left(\nR+H\\Sigma_{0\\mid t}H^\\top\n\\right)^{-1}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The implementation instead uses the scalar factor<\/p>\n\n\n\n

      1<\/mn>\u03c3<\/mi><\/mtext>scaled<\/mi><\/mrow><\/msub><\/mfrac>\\frac{1}{\\sigma_{\\mathrm{scaled}}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Thus, the Gaussian calculation identifies the matrix-valued uncertainty correction that is approximated in practice by a time-dependent scalar weighting. It also clarifies how the observation operator’s spectral bandwidth and spatial mask determine which state-space directions receive guidance.<\/p>\n\n\n\n

      6. Flow-Matching Extension<\/h2>\n\n\n\n

      The linear Gaussian model provides an exact setting in which the posterior distribution and conditional likelihood gradient can be computed analytically. The clean-state estimator is linear, and the noisy likelihood gradient takes the form<\/p>\n\n\n\n

      \u2207<\/mo>u<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>t<\/mi><\/msub>(<\/mo>y<\/mi>|<\/mo>u<\/mi>)<\/mo>=<\/mo>B<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>\u0393<\/mi><\/mrow>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>(<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>B<\/mi>t<\/mi><\/msub>u<\/mi>)<\/mo><\/mrow>,<\/mo><\/mrow>\\nabla_u \\log p_t(y\\mid u)\n=\nB_t^\\top H^\\top \\Gamma_t^{-1}\n\\left(y-HB_tu\\right),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where B<\/mi>t<\/mi><\/msub>u<\/mi><\/mrow>B_tu<\/annotation><\/semantics><\/math> is the conditional estimate of the clean state and \u0393<\/mi><\/mrow>t<\/mi><\/msub>\\Gamma_t<\/annotation><\/semantics><\/math> accounts for measurement noise and the remaining uncertainty in that estimate.<\/p>\n\n\n\n

      The same structure can be extended to a nonlinear flow-matching model by replacing the linear clean-state estimator with a time dependent denoising map.<\/p>\n\n\n\n

      6.1 Denoising Map<\/h4>\n\n\n\n

      Let U<\/mi>t<\/mi><\/msub>\u2208<\/mo>\u211d<\/mi>n<\/mi><\/msup><\/mrow>U_t\\in\\mathbb{R}^n<\/annotation><\/semantics><\/math> denote the state at time t<\/mi>t<\/annotation><\/semantics><\/math>. The unconditional flow is controlled by<\/p>\n\n\n\n
      d<\/mi>U<\/mi>t<\/mi><\/msub><\/mrow>d<\/mi>t<\/mi><\/mrow><\/mfrac>=<\/mo>v<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>,<\/mo><\/mrow>\\frac{dU_t}{dt}=v_t(U_t),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where v<\/mi>t<\/mi><\/msub>:<\/mo>\u211d<\/mi>n<\/mi><\/msup>\u2192<\/mo>\u211d<\/mi>n<\/mi><\/msup><\/mrow>v_t:\\mathbb{R}^n\\rightarrow\\mathbb{R}^n<\/annotation><\/semantics><\/math> is the unconditional velocity field.<\/p>\n\n\n\n

      Assume that a time-dependent map F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math> predicts the clean state U<\/mi>0<\/mn><\/msub>U_0<\/annotation><\/semantics><\/math> from the current state<\/p>\n\n\n\n
      U<\/mi>^<\/mo><\/mover>0<\/mn><\/msub>=<\/mo>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>\\widehat{U}_0=F_t(U_t).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The map F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math> is treated as given. No separate differential equation for F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math> is introduced.<\/p>\n\n\n\n

      The observation model is <\/p>\n\n\n\n

      Y<\/mi>=<\/mo>H<\/mi>U<\/mi>0<\/mn><\/msub>+<\/mo>N<\/mi>,<\/mo><\/mspace>N<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>0<\/mn>,<\/mo>R<\/mi>)<\/mo>,<\/mo><\/mrow>Y=HU_0+N,\n\\qquad\nN\\sim\\mathcal{N}(0,R),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where H<\/mi>H<\/annotation><\/semantics><\/math> is the observation operator and R<\/mi>\u227b<\/mo>0<\/mn><\/mrow>R\\succ0<\/annotation><\/semantics><\/math> is the measurement-noise covariance.<\/p>\n\n\n\n

      Applying the observation operator to the predicted clean state gives the predicted measurement<\/p>\n\n\n\n

      Y<\/mi>^<\/mo><\/mover>t<\/mi><\/msub>=<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>\\widehat{Y}_t=HF_t(U_t).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The corresponding measurement residual is<\/p>\n\n\n\n

      r<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>r_t(U_t)=y-HF_t(U_t).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      This residual measures the disagreement between the observed data and the observation predicted from the current state.<\/p>\n\n\n\n

      6.2 DPS Guidance<\/h4>\n\n\n\n

      Diffusion posterior sampling introduces an observation-dependent correction through the gradient of the likelihood. Using F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow>F_t(U_t)<\/annotation><\/semantics><\/math> as a point estimate of the clean state, the Gaussian likelihood is<\/p>\n\n\n\n
      p<\/mi>(<\/mo>y<\/mi>|<\/mo>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>)<\/mo>\u221d<\/mo>exp<\/mi>\u2061<\/mo><\/mrow>[<\/mo>\u2212<\/mo>1<\/mn>2<\/mn><\/mfrac>(<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>)<\/mo><\/mrow>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>)<\/mo><\/mrow>]<\/mo><\/mrow><\/mrow>p\\bigl(y\\mid F_t(U_t)\\bigr)\n\\propto\n\\exp\\left[\n-\\frac{1}{2}\n\\left(y-HF_t(U_t)\\right)^\\top\nR^{-1}\n\\left(y-HF_t(U_t)\\right)\n\\right]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The DPS guidance term is defined as<\/p>\n\n\n\n

      g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u03bb<\/mi>t<\/mi><\/msub>\u2207<\/mo>U<\/mi>t<\/mi><\/msub><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>(<\/mo>y<\/mi>|<\/mo>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>)<\/mo>,<\/mo><\/mrow>g_t(U_t,y)\n=\n\\lambda_t\n\\nabla_{U_t}\n\\log p\\bigl(y\\mid F_t(U_t)\\bigr),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where \u03bb<\/mi>t<\/mi><\/msub>\\lambda_t<\/annotation><\/semantics><\/math> controls the time-dependent strength and sign convention of the guidance.<\/p>\n\n\n\n

      Define the Jacobian of F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math> by<\/p>\n\n\n\n
      J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>\u2202<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow>\u2202<\/mi>U<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac>.<\/mi><\/mrow>J_{F_t}(U_t)\n=\n\\frac{\\partial F_t(U_t)}{\\partial U_t}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Applying the chain rule gives<\/p>\n\n\n\n

      g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u03bb<\/mi>t<\/mi><\/msub>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>\u22a4<\/mi><\/msup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>g_t(U_t,y)\n=\n\\lambda_t\nJ_{F_t}(U_t)^\\top\nH^\\top R^{-1}\n\\left[\ny-HF_t(U_t)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Equivalently,<\/p>\n\n\n\n

      g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>\u03bb<\/mi>t<\/mi><\/msub>2<\/mn><\/mfrac>\u2207<\/mo>U<\/mi>t<\/mi><\/msub><\/msub>\u2016<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>\u2016<\/mo><\/mrow>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>2<\/mn><\/msubsup>,<\/mo><\/mrow>g_t(U_t,y)\n=\n-\\frac{\\lambda_t}{2}\n\\nabla_{U_t}\n\\left\\|\ny-HF_t(U_t)\n\\right\\|_{R^{-1}}^2,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where<\/p>\n\n\n\n

      \u2016<\/mi>z<\/mi>\u2016<\/mi>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>2<\/mn><\/msubsup>=<\/mo>z<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>z<\/mi>.<\/mi><\/mrow>\\|z\\|_{R^{-1}}^2\n=\nz^\\top R^{-1}z.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The residual is evaluated in observation space. Multiplication by H<\/mi>\u22a4<\/mi><\/msup>H^\\top<\/annotation><\/semantics><\/math>maps it to the clean-state space, while J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>\u22a4<\/mi><\/msup><\/mrow>J_{F_t}(U_t)^\\top<\/annotation><\/semantics><\/math>propogates the correction from the predicted clean state back to the current state.<\/p>\n\n\n\n

      Although the measurement is defined on U<\/mi>0<\/mn><\/msub>U_0<\/annotation><\/semantics><\/math>, the likelihood gradient can guide intermediate states throughout the flow.<\/p>\n\n\n\n

      6.3 Guided Flow<\/h4>\n\n\n\n

      After adding the DPS correction, the state evolves according to<\/p>\n\n\n\n

      d<\/mi>U<\/mi>t<\/mi><\/msub><\/mrow>d<\/mi>t<\/mi><\/mrow><\/mfrac>=<\/mo>v<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>+<\/mo>g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>\\frac{dU_t}{dt}\n=\nv_t(U_t)+g_t(U_t,y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Define the full guided velocity as<\/p>\n\n\n\n

      f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>v<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>)<\/mo>+<\/mo>g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>,<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>f_t(U,y)=v_t(U)+g_t(U,y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The guided dynamics are therefore<\/p>\n\n\n\n

      d<\/mi>U<\/mi>t<\/mi><\/msub><\/mrow>d<\/mi>t<\/mi><\/mrow><\/mfrac>=<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>\\frac{dU_t}{dt}=f_t(U_t,y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The complete observation-guided update follows the sequence<\/p>\n\n\n\n

      U<\/mi>t<\/mi><\/msub>\u27f6<\/mo>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>\u27f6<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>\u27f6<\/mo>g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>U_t\n\\longrightarrow\nF_t(U_t)\n\\longrightarrow\nHF_t(U_t)\n\\longrightarrow\ng_t(U_t,y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The current state is first mapped to a clean-state prediction. This prediction is passed through the observation operator, and the resulting residual is propagated back through the denoising map.<\/p>\n\n\n\n

      7. Moment Evolution<\/h2>\n\n\n\n

      The DPS equation describes the evolution of an individual trajectory. To characterize the conditional distribution of trajectories given Y<\/mi>=<\/mo>y<\/mi><\/mrow>Y=y<\/annotation><\/semantics><\/math>, consider the conditional mean and covariance.<\/p>\n\n\n\n

      7.1 Conditional Mean<\/h4>\n\n\n\n

      Define the conditional mean as<\/p>\n\n\n\n

      m<\/mi>t<\/mi><\/msub>=<\/mo>\ud835\udd3c<\/mi>[<\/mo>U<\/mi>t<\/mi><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>]<\/mo>.<\/mi><\/mrow>m_t\n=\n\\mathbb{E}[U_t\\mid Y=y].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Since each trajectory satisfies<\/p>\n\n\n\n

      U<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>=<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>,<\/mo><\/mrow>\\dot{U}_t=f_t(U_t,y),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      differentiating the conditional expectation gives<\/p>\n\n\n\n

      m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>=<\/mo>\ud835\udd3c<\/mi>[<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>]<\/mo><\/mrow>.<\/mi><\/mrow>\\dot{m}_t\n=\n\\mathbb{E}\n\\left[\nf_t(U_t,y)\\mid Y=y\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Substituting the definition of the guided velocity yields<\/p>\n\n\n\n

      m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>=<\/mo>\ud835\udd3c<\/mi>[<\/mo><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>v<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow><\/mtd><\/mtr><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>+<\/mo>\u03bb<\/mi>t<\/mi><\/msub>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>\u22a4<\/mi><\/msup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>)<\/mo><\/mrow><\/mspace>|<\/mo><\/mspace>Y<\/mi>=<\/mo>y<\/mi>]<\/mo>.<\/mi><\/mrow><\/mtd><\/mtr><\/mtable>\\begin{aligned}\n\\dot{m}_t\n=\n\\mathbb{E}\\Big[\n&amp;\nv_t(U_t)\\\\\n&amp;+\n\\lambda_t\nJ_{F_t}(U_t)^\\top\nH^\\top R^{-1}\n\\left(\ny-HF_t(U_t)\n\\right)\n\\;\\Big|\\;Y=y\n\\Big].\n\\end{aligned}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      When either v<\/mi>t<\/mi><\/msub>v_t<\/annotation><\/semantics><\/math> or F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math> is nonlinear, the expectation depends on the full conditional distribution of U<\/mi>t<\/mi><\/msub>U_t<\/annotation><\/semantics><\/math>, rather than only on m<\/mi>t<\/mi><\/msub>m_t<\/annotation><\/semantics><\/math>. The mean equation is there not closed in general.<\/p>\n\n\n\n

      7.2 Conditional Covariance<\/h4>\n\n\n\n

      Define the conditional covariance by<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>t<\/mi><\/msub>=<\/mo>Cov<\/mi>\u2061<\/mo>(<\/mo>U<\/mi>t<\/mi><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>\\Sigma_t\n=\n\\operatorname{Cov}(U_t\\mid Y=y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Equivalently,<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>t<\/mi><\/msub>=<\/mo>\ud835\udd3c<\/mi>[<\/mo>(<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>(<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>\u22a4<\/mi><\/msup>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>]<\/mo><\/mrow>.<\/mi><\/mrow>\\Sigma_t\n=\n\\mathbb{E}\n\\left[\n(U_t-m_t)(U_t-m_t)^\\top\n\\mid Y=y\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Let<\/p>\n\n\n\n

      \u03b4<\/mi>U<\/mi>t<\/mi><\/msub>=<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>.<\/mi><\/mrow>\\delta U_t=U_t-m_t.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Then<\/p>\n\n\n\n

      \u03b4<\/mi>U<\/mi><\/mrow>\u02d9<\/mo><\/mover>t<\/mi><\/msub>=<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>\u2212<\/mo>m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>.<\/mi><\/mrow>\\dot{\\delta U}_t\n=\nf_t(U_t,y)-\\dot{m}_t.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Differentiating<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>t<\/mi><\/msub>=<\/mo>\ud835\udd3c<\/mi>[<\/mo>\u03b4<\/mi>U<\/mi>t<\/mi><\/msub>\u03b4<\/mi>U<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>]<\/mo><\/mrow><\/mrow>\\Sigma_t\n=\n\\mathbb{E}\n\\left[\n\\delta U_t\\delta U_t^\\top\n\\mid Y=y\n\\right]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      gives<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>t<\/mi><\/msub>=<\/mo>\ud835\udd3c<\/mi>[<\/mo><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>(<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>\u2212<\/mo>m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>)<\/mo><\/mrow>(<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>\u22a4<\/mi><\/msup><\/mrow><\/mtd><\/mtr><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>+<\/mo>(<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>(<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>\u2212<\/mo>m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>)<\/mo><\/mrow>\u22a4<\/mi><\/msup>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>]<\/mo>.<\/mi><\/mrow><\/mtd><\/mtr><\/mtable>\\begin{aligned}\n\\dot{\\Sigma}_t\n=\n\\mathbb{E}\\Big[\n&amp;\n\\left(\nf_t(U_t,y)-\\dot{m}_t\n\\right)\n(U_t-m_t)^\\top\\\\\n&amp;+\n(U_t-m_t)\n\\left(\nf_t(U_t,y)-\\dot{m}_t\n\\right)^\\top\n\\mid Y=y\n\\Big].\n\\end{aligned}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      This covariance equation is also exact, but it is not closed for a nonlinear guided velocity field.<\/p>\n\n\n\n

      The flow-matching dynamics considered here are deterministic. Consequently, there is no separate process-noise covariance term. The conditional covariance instead comes from the distribution of the initial state and from conditioning on the observation.<\/p>\n\n\n\n

      A local approximation is therefore required to obtain closed evolution equations for m<\/mi>t<\/mi><\/msub>m_t<\/annotation><\/semantics><\/math> and \u03a3<\/mi><\/mrow>t<\/mi><\/msub>\\Sigma_t<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

      8. Local Linearization<\/h2>\n\n\n\n

      8.1 Local Approximation<\/h4>\n\n\n\n

      Locally linearize the full guided velocity around the conditional mean:<\/p>\n\n\n\n

      f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>,<\/mo>y<\/mi>)<\/mo>\u2248<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>+<\/mo>J<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>,<\/mo><\/mrow>f_t(U,y)\n\\approx\nf_t(m_t,y)+J_t(U-m_t),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where<\/p>\n\n\n\n

      J<\/mi>t<\/mi><\/msub>=<\/mo><\/mo>\u2202<\/mi>f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>\u2202<\/mi>U<\/mi><\/mrow><\/mfrac>|<\/mo><\/mrow>U<\/mi>=<\/mo>m<\/mi>t<\/mi><\/msub><\/mrow><\/msub>.<\/mi><\/mrow>J_t\n=\n\\left.\n\\frac{\\partial f_t(U,y)}{\\partial U}\n\\right|_{U=m_t}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Because<\/p>\n\n\n\n

      f<\/mi>t<\/mi><\/msub>=<\/mo>v<\/mi>t<\/mi><\/msub>+<\/mo>g<\/mi>t<\/mi><\/msub>,<\/mo><\/mrow>f_t=v_t+g_t,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      the Jacobian can be decomposed as<\/p>\n\n\n\n

      J<\/mi>t<\/mi><\/msub>=<\/mo>J<\/mi>t<\/mi>v<\/mi><\/msubsup>+<\/mo>J<\/mi>t<\/mi>g<\/mi><\/msubsup>,<\/mo><\/mrow>J_t=J_t^v+J_t^g,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where<\/p>\n\n\n\n

      J<\/mi>t<\/mi>v<\/mi><\/msubsup>=<\/mo><\/mo>\u2202<\/mi>v<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>)<\/mo><\/mrow>\u2202<\/mi>U<\/mi><\/mrow><\/mfrac>|<\/mo><\/mrow>U<\/mi>=<\/mo>m<\/mi>t<\/mi><\/msub><\/mrow><\/msub><\/mrow>J_t^v\n=\n\\left.\n\\frac{\\partial v_t(U)}{\\partial U}\n\\right|_{U=m_t}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      and<\/p>\n\n\n\n

      J<\/mi>t<\/mi>g<\/mi><\/msubsup>=<\/mo><\/mo>\u2202<\/mi>g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>\u2202<\/mi>U<\/mi><\/mrow><\/mfrac>|<\/mo><\/mrow>U<\/mi>=<\/mo>m<\/mi>t<\/mi><\/msub><\/mrow><\/msub>.<\/mi><\/mrow>J_t^g\n=\n\\left.\n\\frac{\\partial g_t(U,y)}{\\partial U}\n\\right|_{U=m_t}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Since<\/p>\n\n\n\n

      \ud835\udd3c<\/mi>[<\/mo>U<\/mi>t<\/mi><\/msub>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>]<\/mo>=<\/mo>0<\/mn>,<\/mo><\/mrow>\\mathbb{E}[U_t-m_t\\mid Y=y]=0,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      the locally closed mean equation becomes<\/p>\n\n\n\n

      m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>\\dot{m}_t\n\\approx\nf_t(m_t,y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Expanding the guidance term gives<\/p>\n\n\n\n

      m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo>v<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>+<\/mo>\u03bb<\/mi>t<\/mi><\/msub>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>\u22a4<\/mi><\/msup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>\\dot{m}_t\n\\approx\nv_t(m_t)\n+\n\\lambda_t\nJ_{F_t}(m_t)^\\top\nH^\\top R^{-1}\n\\left[\ny-HF_t(m_t)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The covariance equation becomes<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo>J<\/mi>t<\/mi><\/msub>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>+<\/mo>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>J<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>.<\/mi><\/mrow>\\dot{\\Sigma}_t\n\\approx\nJ_t\\Sigma_t+\\Sigma_tJ_t^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      These equations have the same structure as the moment equations for a locally linear deterministic system. The effective Jacobian additionally includes an observation-dependent DPS contribution.<\/p>\n\n\n\n

      8.2 DPS Jacobian<\/h4>\n\n\n\n

      To identify this contribution, locally linearize the denoising map:<\/p>\n\n\n\n

      F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>)<\/mo>\u2248<\/mo>F<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>+<\/mo>G<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>,<\/mo><\/mrow>F_t(U)\n\\approx\nF_t(m_t)+G_t(U-m_t),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      where<\/p>\n\n\n\n

      G<\/mi>t<\/mi><\/msub>=<\/mo>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>G_t=J_{F_t}(m_t).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The guidance term is approximated by<\/p>\n\n\n\n

      g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>,<\/mo>y<\/mi>)<\/mo>\u2248<\/mo>\u03bb<\/mi>t<\/mi><\/msub>G<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>\u2212<\/mo>H<\/mi>G<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>\u2212<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>g_t(U,y)\n\\approx\n\\lambda_t\nG_t^\\top H^\\top R^{-1}\n\\left[\ny-HF_t(m_t)-HG_t(U-m_t)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Therefore,<\/p>\n\n\n\n

      J<\/mi>t<\/mi>g<\/mi><\/msubsup>\u2248<\/mo>\u2212<\/mo>\u03bb<\/mi>t<\/mi><\/msub>G<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>H<\/mi>G<\/mi>t<\/mi><\/msub>.<\/mi><\/mrow>J_t^g\n\\approx\n-\\lambda_t\nG_t^\\top H^\\top R^{-1}HG_t.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The complete local Jacobian becomes<\/p>\n\n\n\n

      J<\/mi>t<\/mi><\/msub>\u2248<\/mo>J<\/mi>t<\/mi>v<\/mi><\/msubsup>\u2212<\/mo>\u03bb<\/mi>t<\/mi><\/msub>G<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>H<\/mi>G<\/mi>t<\/mi><\/msub>.<\/mi><\/mrow>J_t\n\\approx\nJ_t^v\n–\n\\lambda_t\nG_t^\\top H^\\top R^{-1}HG_t.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Substituting this expression into the covariance equation gives<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo><\/mrow><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>(<\/mo>J<\/mi>t<\/mi>v<\/mi><\/msubsup>\u2212<\/mo>\u03bb<\/mi>t<\/mi><\/msub>G<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>H<\/mi>G<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow>\u03a3<\/mi><\/mrow>t<\/mi><\/msub><\/mrow><\/mtd><\/mtr><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>+<\/mo>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>(<\/mo>J<\/mi>t<\/mi>v<\/mi><\/msubsup>\u2212<\/mo>\u03bb<\/mi>t<\/mi><\/msub>G<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>H<\/mi>G<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow>\u22a4<\/mi><\/msup>.<\/mi><\/mrow><\/mtd><\/mtr><\/mtable>\\begin{aligned}\n\\dot{\\Sigma}_t\n\\approx{}&amp;\n\\left(\nJ_t^v\n–\n\\lambda_tG_t^\\top H^\\top R^{-1}HG_t\n\\right)\\Sigma_t\\\\\n&amp;+\n\\Sigma_t\n\\left(\nJ_t^v\n–\n\\lambda_tG_t^\\top H^\\top R^{-1}HG_t\n\\right)^\\top.\n\\end{aligned}\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Under a convention in which \u03bb<\/mi>t<\/mi><\/msub>\u2265<\/mo>0<\/mn><\/mrow>\\lambda_t\\geq0<\/annotation><\/semantics><\/math>, <\/p>\n\n\n\n
      \u2212<\/mo>\u03bb<\/mi>t<\/mi><\/msub>G<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>H<\/mi>G<\/mi>t<\/mi><\/msub><\/mrow>-\\lambda_t\nG_t^\\top H^\\top R^{-1}HG_t<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      is negative semidefinite. It therefore contributes local contraction in directions that affect the predicted measurement. This does not imply that the full covariance must decrease, since the unconditional dynamics and coupling between state components also contribute to its evolution.<\/p>\n\n\n\n

      For a nonlinear F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math>, the exact guidance Jacobian also contains second-order derivatives. Define<\/p>\n\n\n\n
      q<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>)<\/mo>=<\/mo>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>q_t(U)\n=\nH^\\top R^{-1}\n\\left[\ny-HF_t(U)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Since<\/p>\n\n\n\n

      g<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u03bb<\/mi>t<\/mi><\/msub>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>)<\/mo>\u22a4<\/mi><\/msup>q<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>)<\/mo>,<\/mo><\/mrow>g_t(U,y)\n=\n\\lambda_t\nJ_{F_t}(U)^\\top q_t(U),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      the exact derivative is<\/p>\n\n\n\n

      \u2202<\/mi>g<\/mi>t<\/mi><\/msub><\/mrow>\u2202<\/mi>U<\/mi><\/mrow><\/mfrac>=<\/mo>\u03bb<\/mi>t<\/mi><\/msub>[<\/mo>\u2211<\/mo>i<\/mi><\/munder><\/mrow>q<\/mi>t<\/mi>,<\/mo>i<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>)<\/mo>\u2207<\/mo>2<\/mn><\/msup>F<\/mi>t<\/mi>,<\/mo>i<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>)<\/mo>\u2212<\/mo>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>)<\/mo>\u22a4<\/mi><\/msup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>H<\/mi>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>\\frac{\\partial g_t}{\\partial U}\n=\n\\lambda_t\n\\left[\n\\sum_i\nq_{t,i}(U)\\nabla^2F_{t,i}(U)\n–\nJ_{F_t}(U)^\\top\nH^\\top R^{-1}H\nJ_{F_t}(U)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The first term contains the Hessians of the components of F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math>. The locally linear approximation neglects these Hessian terms and retains the first-order contribution.<\/p>\n\n\n\n

      9. Splitting C <\/em>and D<\/em><\/h2>\n\n\n\n

      9.1 State Decomposition<\/h4>\n\n\n\n

      The state is now partitioned into near and far components:<\/p>\n\n\n\n

      U<\/mi>t<\/mi><\/msub>=<\/mo>(<\/mo>C<\/mi>t<\/mi><\/msub><\/mtd><\/mtr>D<\/mi>t<\/mi><\/msub><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>.<\/mi><\/mrow>U_t=\n\\begin{pmatrix}\nC_t\\\\\nD_t\n\\end{pmatrix}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The conditional mean is partitioned as<\/p>\n\n\n\n

      m<\/mi>t<\/mi><\/msub>=<\/mo>(<\/mo>m<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mtd><\/mtr>m<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>,<\/mo><\/mrow>m_t=\n\\begin{pmatrix}\nm_{C,t}\\\\\nm_{D,t}\n\\end{pmatrix},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      and the conditional covariance is partitioned as<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>t<\/mi><\/msub>=<\/mo>(<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>.<\/mi><\/mrow>\\Sigma_t=\n\\begin{pmatrix}\n\\Sigma_{CC,t}&amp;\\Sigma_{CD,t}\\\\\n\\Sigma_{DC,t}&amp;\\Sigma_{DD,t}\n\\end{pmatrix}.\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Similarly, the guided velocity and its Jacobian are written as<\/p>\n\n\n\n

      f<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>=<\/mo>(<\/mo>f<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo><\/mrow><\/mtd><\/mtr>f<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow><\/mrow>f_t(U_t,y)\n=\n\\begin{pmatrix}\nf_{C,t}(U_t,y)\\\\\nf_{D,t}(U_t,y)\n\\end{pmatrix}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      and<\/p>\n\n\n\n

      J<\/mi>t<\/mi><\/msub>=<\/mo>(<\/mo>J<\/mi>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>J<\/mi>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr>J<\/mi>D<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>J<\/mi>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>.<\/mi><\/mrow>J_t=\n\\begin{pmatrix}\nJ_{CC,t}&amp;J_{CD,t}\\\\\nJ_{DC,t}&amp;J_{DD,t}\n\\end{pmatrix}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      9.2 Block Moment Equations<\/h4>\n\n\n\n

      The mean equations become<\/p>\n\n\n\n

      m<\/mi>\u02d9<\/mo><\/mover>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u2248<\/mo>f<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo><\/mrow>\\dot{m}_{C,t}\n\\approx\nf_{C,t}(m_t,y)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      and<\/p>\n\n\n\n

      m<\/mi>\u02d9<\/mo><\/mover>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u2248<\/mo>f<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>.<\/mi><\/mrow>\\dot{m}_{D,t}\n\\approx\nf_{D,t}(m_t,y).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Expanding<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>t<\/mi><\/msub>=<\/mo>J<\/mi>t<\/mi><\/msub>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>+<\/mo>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>J<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup><\/mrow>\\dot{\\Sigma}_t\n=\nJ_t\\Sigma_t+\\Sigma_tJ_t^\\top<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      block by block gives the near-state covariance equation<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>=<\/mo><\/mrow><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>J<\/mi>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u03a3<\/mi><\/mrow>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>+<\/mo>J<\/mi>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>+<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>J<\/mi>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>+<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>J<\/mi>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>,<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>\\begin{aligned}\n\\dot{\\Sigma}_{CC,t}\n={}&amp;\nJ_{CC,t}\\Sigma_{CC,t}\n+\nJ_{CD,t}\\Sigma_{DC,t}\\\\\n&amp;+\n\\Sigma_{CC,t}J_{CC,t}^\\top\n+\n\\Sigma_{CD,t}J_{CD,t}^\\top,\n\\end{aligned}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      the near-far cross covariance equation<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>=<\/mo><\/mrow><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>J<\/mi>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>+<\/mo>J<\/mi>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>+<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>J<\/mi>D<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>+<\/mo>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>J<\/mi>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>,<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>\\begin{aligned}\n\\dot{\\Sigma}_{CD,t}\n={}&amp;\nJ_{CC,t}\\Sigma_{CD,t}\n+\nJ_{CD,t}\\Sigma_{DD,t}\\\\\n&amp;+\n\\Sigma_{CC,t}J_{DC,t}^\\top\n+\n\\Sigma_{CD,t}J_{DD,t}^\\top,\n\\end{aligned}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      and the far-state covariance equation<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>=<\/mo><\/mrow><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>J<\/mi>D<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>+<\/mo>J<\/mi>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub><\/mrow><\/mtd><\/mtr><\/mrow><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>+<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>J<\/mi>D<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>+<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>J<\/mi>D<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow>\u22a4<\/mi><\/msubsup>.<\/mi><\/mrow><\/mtd><\/mtr><\/mtable>\\begin{aligned}\n\\dot{\\Sigma}_{DD,t}\n={}&amp;\nJ_{DC,t}\\Sigma_{CD,t}\n+\nJ_{DD,t}\\Sigma_{DD,t}\\\\\n&amp;+\n\\Sigma_{DC,t}J_{DC,t}^\\top\n+\n\\Sigma_{DD,t}J_{DD,t}^\\top.\n\\end{aligned}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The far-state covariance therefore does not evolve independently. Its evolution depends on the far-state dynamics, the near-far cross-covariance, and the local coupling between the near and far components of the guided velocity.<\/p>\n\n\n\n

      9.3 Observation Influence on D<\/mi>t<\/mi><\/msub>D_t<\/annotation><\/semantics><\/math><\/h4>\n\n\n\n

      Partition the predicted clean state as<\/p>\n\n\n\n

      F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>(<\/mo>F<\/mi>t<\/mi>C<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow><\/mtd><\/mtr>F<\/mi>t<\/mi>D<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>.<\/mi><\/mrow>F_t(U_t)\n=\n\\begin{pmatrix}\nF_t^C(U_t)\\\\\nF_t^D(U_t)\n\\end{pmatrix}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Only the near component is directly observed, so<\/p>\n\n\n\n

      H<\/mi>=<\/mo>(<\/mo>A<\/mi><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>.<\/mi><\/mrow>H=\n\\begin{pmatrix}\nA&amp;0\n\\end{pmatrix}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The predicted observation is therefore<\/p>\n\n\n\n

      H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>A<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>,<\/mo><\/mrow>HF_t(U_t)=AF_t^C(U_t),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      and the measurement residual becomes<\/p>\n\n\n\n

      r<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>y<\/mi>\u2212<\/mo>A<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>r_t(U_t)\n=\ny-AF_t^C(U_t).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Although the observation depends only on F<\/mi>t<\/mi>C<\/mi><\/msubsup>F_t^C<\/annotation><\/semantics><\/math>, the predicted near component may depend on both C<\/mi>t<\/mi><\/msub>C_t<\/annotation><\/semantics><\/math> and D<\/mi>t<\/mi><\/msub>D_t<\/annotation><\/semantics><\/math>. Its Jacobian can be partitioned as<\/p>\n\n\n\n
      J<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>=<\/mo>(<\/mo>\u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>C<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac><\/mstyle><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>\u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>D<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac><\/mstyle><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>.<\/mi><\/mrow>J_{F_t^C}(U_t)\n=\n\\begin{pmatrix}\n\\dfrac{\\partial F_t^C}{\\partial C_t}\n&amp;\n\\dfrac{\\partial F_t^C}{\\partial D_t}\n\\end{pmatrix}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The DPS correction applied to the current near component is<\/p>\n\n\n\n

      g<\/mi>C<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>=<\/mo>\u03bb<\/mi>t<\/mi><\/msub>(<\/mo>\u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>C<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mrow>\u22a4<\/mi><\/msup>A<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>A<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>g_{C,t}\n=\n\\lambda_t\n\\left(\n\\frac{\\partial F_t^C}{\\partial C_t}\n\\right)^\\top\nA^\\top R^{-1}\n\\left[\ny-AF_t^C(U_t)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      The correction applied to the current far component is<\/p>\n\n\n\n

      g<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>=<\/mo>\u03bb<\/mi>t<\/mi><\/msub>(<\/mo>\u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>D<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mrow>\u22a4<\/mi><\/msup>A<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>A<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>g_{D,t}\n=\n\\lambda_t\n\\left(\n\\frac{\\partial F_t^C}{\\partial D_t}\n\\right)^\\top\nA^\\top R^{-1}\n\\left[\ny-AF_t^C(U_t)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      This expression identifies the mechanism through which a partial observation can influence the unobserved component. Although D is not measured directly, the DPS correction can modify D<\/mi>t<\/mi><\/msub>D_t<\/annotation><\/semantics><\/math> whenever perturbations in D<\/mi>t<\/mi><\/msub>D_t<\/annotation><\/semantics><\/math> affect the predicted near state F<\/mi>t<\/mi>C<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>F_t^C(U_t).<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

      In the linear Gaussian model, the analogous information pathway is determined by<\/p>\n\n\n\n

      \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>.<\/mi><\/mrow>\\Sigma_{DC}A^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      Both the conditional mean of D<\/mi>D<\/annotation><\/semantics><\/math> and the reduction in its covariance depend on this cross-covariance term. If the observed directions of C<\/mi>C<\/annotation><\/semantics><\/math> are uncorrelated with D<\/mi>D<\/annotation><\/semantics><\/math>, then the measurement provides no information about D<\/mi>D<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

      Furthermore, the mean update and covariance reduction are restricted to far-state directions selected by \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup><\/mrow>\\Sigma_{DC}A^\\top<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

      In the nonlinear flow-matching setting, the local Jacobian<\/p>\n\n\n\n

      \u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>D<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac>\\frac{\\partial F_t^C}{\\partial D_t}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      plays an analogous role. It identifies the directions in the current far state that influence the predicted observation and are therefore accessible to measurement-based guidance.<\/p>\n\n\n\n

      The Gaussian cross-covariance and nonlinear denoiser Jacobian are distinct mathematical objects, but both characterize the coupling required for information to propagate from the observed component to the hidden component.<\/p>\n\n\n\n

      The preceding sections characterize how partial observations influence posterior guidance and hidden-state uncertainty. The following section documents a complementary fluid-specific evaluation framework for comparing reconstructed quasi-geostrophic fields.<\/p>\n\n\n\n

      Within this section, C<\/mi>C<\/annotation><\/semantics><\/math> in the feature-loss definitions denotes the number of feature channels, rather than the near-state component introduced earlier.<\/p>\n\n\n\n

      10. QG-SSL Evaluation Framework<\/h2>\n\n\n\n

      10.1 Objective<\/h4>\n\n\n\n

      QG-SSL is a self-supervised encoder for comparing two-dimensional quasi-geostrophic (QG) vorticity fields. It learns spatial structure and short-term dynamics from real trajectories, without quality labels or generated samples.<\/p>\n\n\n\n

      10.2 Data and Preprocessing<\/h4>\n\n\n\n

      We use the original paper\u2019s released 64<\/mn>64<\/annotation><\/semantics><\/math> \u00d7 64<\/mn>64<\/annotation><\/semantics><\/math> filtered vorticity fields. These were obtained by spectrally filtering 512<\/mn>512<\/annotation><\/semantics><\/math> \u00d7 512<\/mn>512<\/annotation><\/semantics><\/math> QG simulations, as described in the paper. We use eddy and jet flows at Reynolds numbers 10<\/mn>3<\/mn><\/msup>10^3<\/annotation><\/semantics><\/math> and 10<\/mn>4<\/mn><\/msup>10^4<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

      Each combination contains 500<\/mn>500<\/annotation><\/semantics><\/math> trajectories with 196<\/mn>196<\/annotation><\/semantics><\/math> saved fields. We discard the spin-up portion and use frames 101<\/mn>101<\/annotation><\/semantics><\/math>\u2013195<\/mn>195<\/annotation><\/semantics><\/math>. The random initial conditions produce an early transient; the paper reports that the energy spectrum becomes self-similar only after t<\/mi>=<\/mo>50<\/mn><\/mrow>t=50<\/annotation><\/semantics><\/math>. Restricting training to this later regime avoids learning initialization artifacts. Each example is a pair (<\/mo>x<\/mi>t<\/mi><\/msub>,<\/mo>x<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msub>)<\/mo><\/mrow>(x_t,x_{t+1})<\/annotation><\/semantics><\/math>; one step is approximately 0.5<\/mn>0.5<\/annotation><\/semantics><\/math> non-dimensional time units.<\/p>\n\n\n\n

      We train only on these 64<\/mn>64<\/annotation><\/semantics><\/math> \u00d7 64<\/mn>64<\/annotation><\/semantics><\/math> target fields, not on the paper\u2019s 16<\/mn>16<\/annotation><\/semantics><\/math> \u00d7 16<\/mn>16<\/annotation><\/semantics><\/math> observations or on outputs from a generative model. Trajectories 0<\/mn>0<\/annotation><\/semantics><\/math>\u2013399<\/mn>399<\/annotation><\/semantics><\/math> are used for training and 400<\/mn>400<\/annotation><\/semantics><\/math>\u2013446<\/mn>446<\/annotation><\/semantics><\/math> for validation and for setting feature scales. All remaining trajectories are held out. For each physical configuration, we compute one scalar mean and standard deviation from the training fields and use them to standardize its inputs. Flow regime and Reynolds number are not given to the network.<\/p>\n\n\n\n

      The simulated square is periodic: opposite edges are connected, so a field leaving one edge re-enters from the other. We use this by cyclically rolling x<\/mi>t<\/mi><\/msub>x_t<\/annotation><\/semantics><\/math> and x<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msub>x_{t+1}<\/annotation><\/semantics><\/math> by the same random multiples of 8 pixels. This changes the origin without changing their relative alignment. We then hide 50<\/mn>50<\/annotation><\/semantics><\/math>% of x<\/mi>t<\/mi><\/msub>x_t<\/annotation><\/semantics><\/math> in random 8 \u00d7 8 blocks and create a second view by rolling both the masked field and its mask again. Matching their global embeddings discourages dependence on absolute position.<\/p>\n\n\n\n

      The primary spatial distance nevertheless remains location-sensitive and penalizes translating only one of the two compared fields. The encoder receives two channels: the masked vorticity field and a binary visibility mask. At inference time the mask is entirely visible.<\/p>\n\n\n\n

      10.3 Encoder Architecture<\/h4>\n\n\n\n

      The encoder is a compact, approximately 9<\/mn>9<\/annotation><\/semantics><\/math>-million-parameter hierarchical transformer designed for a periodic domain. A stage is a group of transformer blocks operating at one fixed spatial resolution:<\/p>\n\n\n\n
        \n
      1. A 4<\/mn>4<\/annotation><\/semantics><\/math> \u00d7 4<\/mn>4<\/annotation><\/semantics><\/math>, stride-4 convolution converts the input into a 16<\/mn>16<\/annotation><\/semantics><\/math> \u00d7 16<\/mn>16<\/annotation><\/semantics><\/math> grid with 96<\/mn>96<\/annotation><\/semantics><\/math> channels.<\/li>\n\n\n\n
      2. Three transformer stages operate at resolutions A 4<\/mn>4<\/annotation><\/semantics><\/math> \u00d7 4<\/mn>4<\/annotation><\/semantics><\/math>, stride-4<\/mn>4<\/annotation><\/semantics><\/math> convolution converts the input into a 16<\/mn>16<\/annotation><\/semantics><\/math> \u00d7 16<\/mn>16<\/annotation><\/semantics><\/math> grid with 96<\/mn>96<\/annotation><\/semantics><\/math> channels., 8 \u00d7 8, and 4<\/mn>4<\/annotation><\/semantics><\/math> \u00d7 4<\/mn>4<\/annotation><\/semantics><\/math>, with 96<\/mn>96<\/annotation><\/semantics><\/math>, 192<\/mn>192<\/annotation><\/semantics><\/math>, and 384<\/mn>384<\/annotation><\/semantics><\/math> channels.<\/li>\n\n\n\n
      3. The stages contain 2<\/mn>2<\/annotation><\/semantics><\/math>, 2<\/mn>2<\/annotation><\/semantics><\/math>, and 4<\/mn>4<\/annotation><\/semantics><\/math> shifted-window attention blocks, respectively. Attention windows are 4<\/mn>4<\/annotation><\/semantics><\/math> \u00d7 4<\/mn>4<\/annotation><\/semantics><\/math>, and cyclic shiftswrap across the domain boundary.<\/li>\n\n\n\n
      4. Each stage is projected to a 32<\/mn>32<\/annotation><\/semantics><\/math>-channel spatial feature map. These three aligned maps form the primary representation.<\/li>\n\n\n\n
      5. The mean and standard deviation of every stage are concatenated and passed through an MLP to produce an optional 128<\/mn>128<\/annotation><\/semantics><\/math>-dimensional global embedding.<\/li>\n<\/ol>\n\n\n\n

        Training uses a student encoder and an exponential-moving-average (EMA) teacher with the same architecture. The teacher sees complete fields; the student sees the masked current field. At each stage, separate spatial heads predict the teacher\u2019s current features at hidden locations and its future features everywhere. Two MLP heads predict the corresponding global embeddings. The heads predict features, not pixels, and are discarded after training.<\/p>\n\n\n\n

        10.4 Training Losses<\/h4>\n\n\n\n

        Setup<\/h5>\n\n\n\n

        A training example is a pair of consecutive saved fields, (<\/mo>x<\/mi>t<\/mi><\/msub>,<\/mo>x<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msub>)<\/mo><\/mrow>(x_t,x_{t+1})<\/annotation><\/semantics><\/math>. Let \u03a9<\/mi><\/mrow>\\Omega<\/annotation><\/semantics><\/math> be the field\u2019s spatial domain, let \u210b<\/mi>\u2282<\/mo>\u03a9<\/mi><\/mrow><\/mrow>\\mathcal{H}\\subset\\Omega<\/annotation><\/semantics><\/math> be the randomly hidden region, and let \ud835\udcb1<\/mi>=<\/mo>\u03a9<\/mi><\/mrow>\u2216<\/mo>\u210b<\/mi><\/mrow>\\mathcal{V}=\\Omega\\setminus\\mathcal{H}<\/annotation><\/semantics><\/math> be the visible region. The student encodes x<\/mi>t<\/mi><\/msub>x_t<\/annotation><\/semantics><\/math> using only \ud835\udcb1<\/mi>\\mathcal{V}<\/annotation><\/semantics><\/math>; the EMA teacher encodes the complete x<\/mi>t<\/mi><\/msub>x_t<\/annotation><\/semantics><\/math> and x<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msub>x_{t+1}<\/annotation><\/semantics><\/math>. Teacher outputs are treated as fixed targets.<\/p>\n\n\n\n

        At scale s<\/mi>\u2208<\/mo>{<\/mo>1,2,3<\/mn>}<\/mo><\/mrow>s\\in\\{1,2,3\\}<\/annotation><\/semantics><\/math>, S<\/mi>s<\/mi><\/msub>S_s<\/annotation><\/semantics><\/math> is the student\u2019s spatial feature map, T<\/mi>s<\/mi>t<\/mi><\/msubsup>T_s^t<\/annotation><\/semantics><\/math> and T<\/mi>s<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msubsup>T_s^{t+1}<\/annotation><\/semantics><\/math> are the teacher\u2019s current and future maps, \u03a9<\/mi><\/mrow>s<\/mi><\/msub>\\Omega_s<\/annotation><\/semantics><\/math> is the set of all spatial positions, and \u210b<\/mi>s<\/mi><\/msub>\u2282<\/mo>\u03a9<\/mi><\/mrow>s<\/mi><\/msub><\/mrow>\\mathcal{H}_s\\subset\\Omega_s<\/annotation><\/semantics><\/math> is the hidden region at that scale. The student global embedding is g<\/mi>g<\/annotation><\/semantics><\/math>, while the teacher embeddings are u<\/mi>t<\/mi><\/msup>u^t<\/annotation><\/semantics><\/math> and u<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msup>u^{t+1}<\/annotation><\/semantics><\/math>. A second student view, obtained by periodically translating both the field and its visible region by the same displacement, has embedding g<\/mi>~<\/mo><\/mover>\\widetilde{g}<\/annotation><\/semantics><\/math>.\u0303<\/p>\n\n\n\n

        All feature errors are normalized coordinate by coordinate. For spatial maps F<\/mi>^<\/mo><\/mover>\\widehat{F}<\/annotation><\/semantics><\/math> and F<\/mi>F<\/annotation><\/semantics><\/math> at scale s<\/mi>s<\/annotation><\/semantics><\/math>, each with \ud835\udc36 channels, and any set \ud835\udcab<\/mi>\u2286<\/mo>\u03a9<\/mi><\/mrow>s<\/mi><\/msub><\/mrow>\\mathcal{P}\\subseteq\\Omega_s<\/annotation><\/semantics><\/math> of evaluated positions, define<\/p>\n\n\n\n
        d<\/mi><\/mtext>sp<\/mi><\/mrow><\/msub>(<\/mo>F<\/mi>^<\/mo><\/mover>,<\/mo>F<\/mi>;<\/mo>\ud835\udcab<\/mi>)<\/mo><\/mrow>=<\/mo>1<\/mn>|<\/mi>\ud835\udcab<\/mi>|<\/mi>C<\/mi><\/mrow><\/mfrac>\u2211<\/mo>p<\/mi>\u2208<\/mo>\ud835\udcab<\/mi><\/mrow><\/munder><\/mrow>\u2211<\/mo>c<\/mi>=<\/mo>1<\/mn><\/mrow>C<\/mi><\/munderover><\/mrow>(<\/mo>F<\/mi>^<\/mo><\/mover>c<\/mi><\/msub>(<\/mo>p<\/mi>)<\/mo>\u2212<\/mo>F<\/mi>c<\/mi><\/msub>(<\/mo>p<\/mi>)<\/mo><\/mrow>\u03c3<\/mi>c<\/mi><\/msub>(<\/mo>F<\/mi>)<\/mo><\/mrow><\/mfrac>)<\/mo><\/mrow>2<\/mn><\/msup>,<\/mo><\/mrow>d_{\\mathrm{sp}}\n\\left(\n\\widehat{F},F;\\mathcal{P}\n\\right)\n=\n\\frac{1}{|\\mathcal{P}|C}\n\\sum_{p\\in\\mathcal{P}}\n\\sum_{c=1}^{C}\n\\left(\n\\frac{\n\\widehat{F}_c(p)-F_c(p)\n}{\n\\sigma_c(F)\n}\n\\right)^2,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        \u03c3<\/mi>c<\/mi><\/msub>(<\/mo>F<\/mi>)<\/mo>=<\/mo>Var<\/mi><\/mtext>examples<\/mi><\/mrow>,<\/mo><\/mspace>p<\/mi>\u2208<\/mo>\u03a9<\/mi><\/mrow>s<\/mi><\/msub><\/mrow><\/msub>\u2061<\/mo>[<\/mo>F<\/mi>c<\/mi><\/msub>(<\/mo>p<\/mi>)<\/mo>]<\/mo><\/mrow>+<\/mo>\u03b5<\/mi><\/mrow><\/msqrt>.<\/mi><\/mrow>\\sigma_c(F)\n=\n\\sqrt{\n\\operatorname{Var}_{\\mathrm{examples},\\,p\\in\\Omega_s}\n\\left[\nF_c(p)\n\\right]\n+\\varepsilon\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        For global vectors v<\/mi>^<\/mo><\/mover>,<\/mo>v<\/mi>\u2208<\/mo>\u211d<\/mi>D<\/mi><\/msup><\/mrow>\\widehat{v},v\\in\\mathbb{R}^{D}<\/annotation><\/semantics><\/math>, define<\/p>\n\n\n\n
        d<\/mi><\/mtext>vec<\/mi><\/mrow><\/msub>(<\/mo>v<\/mi>^<\/mo><\/mover>,<\/mo>v<\/mi>)<\/mo><\/mrow>=<\/mo>1<\/mn>D<\/mi><\/mfrac>\u2211<\/mo>k<\/mi>=<\/mo>1<\/mn><\/mrow>D<\/mi><\/munderover><\/mrow>(<\/mo>v<\/mi>^<\/mo><\/mover>k<\/mi><\/msub>\u2212<\/mo>v<\/mi>k<\/mi><\/msub><\/mrow>\u03c1<\/mi>k<\/mi><\/msub>(<\/mo>v<\/mi>)<\/mo><\/mrow><\/mfrac>)<\/mo><\/mrow>2<\/mn><\/msup>,<\/mo><\/mrow>d_{\\mathrm{vec}}\n\\left(\n\\widehat{v},v\n\\right)\n=\n\\frac{1}{D}\n\\sum_{k=1}^{D}\n\\left(\n\\frac{\n\\widehat{v}_k-v_k\n}{\n\\rho_k(v)\n}\n\\right)^2,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        \u03c1<\/mi>k<\/mi><\/msub>(<\/mo>v<\/mi>)<\/mo>=<\/mo>Var<\/mi><\/mtext>examples<\/mi><\/mrow><\/msub>\u2061<\/mo>[<\/mo>v<\/mi>k<\/mi><\/msub>]<\/mo><\/mrow>+<\/mo>\u03b5<\/mi><\/mrow><\/msqrt>.<\/mi><\/mrow>\\rho_k(v)\n=\n\\sqrt{\n\\operatorname{Var}_{\\mathrm{examples}}\n\\left[\nv_k\n\\right]\n+\\varepsilon\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        Here p<\/mi>p<\/annotation><\/semantics><\/math> indexes spatial positions, c<\/mi>c<\/annotation><\/semantics><\/math> indexes spatial feature channels, j<\/mi>j<\/annotation><\/semantics><\/math> and k<\/mi>k<\/annotation><\/semantics><\/math> index global coordinates, D<\/mi>=<\/mo>128<\/mn><\/mrow>D=128<\/annotation><\/semantics><\/math>, and \u03b5<\/mi>=<\/mo>10<\/mn>\u2212<\/mo>4<\/mn><\/mrow><\/msup><\/mrow>\\varepsilon=10^{-4}<\/annotation><\/semantics><\/math>. Each variance is taken over the training examples used to evaluate the loss; spatial variances also include all positions at the corresponding scale. Below, \ud835\udd3c<\/mi>\\mathbb{E}<\/annotation><\/semantics><\/math> averages over training pairs, hidden regions, and translations.<\/p>\n\n\n\n
        Masked Current-Feature Prediction<\/h5>\n\n\n\n

        The head P<\/mi>s<\/mi>t<\/mi><\/msubsup>P_s^t<\/annotation><\/semantics><\/math> predicts the teacher’s current map from the student’s map. Only hidden positions are scored:<\/p>\n\n\n\n
        \u2112<\/mi><\/mtext>masked<\/mi><\/mrow><\/msub>=<\/mo>1<\/mn>3<\/mn><\/mfrac>\u2211<\/mo>s<\/mi>=<\/mo>1<\/mn><\/mrow>3<\/mn><\/munderover><\/mrow>\ud835\udd3c<\/mi>[<\/mo>d<\/mi><\/mtext>sp<\/mi><\/mrow><\/msub>(<\/mo>P<\/mi>s<\/mi>t<\/mi><\/msubsup>(<\/mo>S<\/mi>s<\/mi><\/msub>)<\/mo>,<\/mo>T<\/mi>s<\/mi>t<\/mi><\/msubsup>;<\/mo>\u210b<\/mi>s<\/mi><\/msub>)<\/mo><\/mrow>]<\/mo><\/mrow>.<\/mi><\/mrow>\\mathcal{L}_{\\mathrm{masked}}\n=\n\\frac{1}{3}\n\\sum_{s=1}^{3}\n\\mathbb{E}\n\\left[\nd_{\\mathrm{sp}}\n\\left(\nP_s^t(S_s),\nT_s^t;\n\\mathcal{H}_s\n\\right)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        This forces the student to infer unobserved spatial structure from its visible context.<\/p>\n\n\n\n

        Future-Feature Prediction<\/h5>\n\n\n\n

        A separate head P<\/mi>s<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msubsup>P_s^{t+1}<\/annotation><\/semantics><\/math> predicts the teacher’s next-field map. The loss uses every position because the task is to predict the complete future:<\/p>\n\n\n\n
        \u2112<\/mi><\/mtext>future<\/mi><\/mrow><\/msub>=<\/mo>1<\/mn>3<\/mn><\/mfrac>\u2211<\/mo>s<\/mi>=<\/mo>1<\/mn><\/mrow>3<\/mn><\/munderover><\/mrow>\ud835\udd3c<\/mi>[<\/mo>d<\/mi><\/mtext>sp<\/mi><\/mrow><\/msub>(<\/mo>P<\/mi>s<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msubsup>(<\/mo>S<\/mi>s<\/mi><\/msub>)<\/mo>,<\/mo>T<\/mi>s<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msubsup>;<\/mo>\u03a9<\/mi><\/mrow>s<\/mi><\/msub>)<\/mo><\/mrow>]<\/mo><\/mrow>.<\/mi><\/mrow>\\mathcal{L}_{\\mathrm{future}}\n=\n\\frac{1}{3}\n\\sum_{s=1}^{3}\n\\mathbb{E}\n\\left[\nd_{\\mathrm{sp}}\n\\left(\nP_s^{t+1}(S_s),\nT_s^{t+1};\n\\Omega_s\n\\right)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        Global-Feature Prediction<\/h5>\n\n\n\n

        The heads Q<\/mi>t<\/mi><\/msup>Q^t<\/annotation><\/semantics><\/math> and Q<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msup>Q^{t+1}<\/annotation><\/semantics><\/math> predict the teacher’s current and future global embeddings from the student’s current embedding:<\/p>\n\n\n\n
        \u2112<\/mi><\/mtext>global<\/mi><\/mrow><\/msub>=<\/mo>1<\/mn>2<\/mn><\/mfrac>\ud835\udd3c<\/mi>[<\/mo>d<\/mi><\/mtext>vec<\/mi><\/mrow><\/msub>(<\/mo>Q<\/mi>t<\/mi><\/msup>(<\/mo>g<\/mi>)<\/mo>,<\/mo>u<\/mi>t<\/mi><\/msup>)<\/mo><\/mrow>+<\/mo>d<\/mi><\/mtext>vec<\/mi><\/mrow><\/msub>(<\/mo>Q<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>g<\/mi>)<\/mo>,<\/mo>u<\/mi>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msup>)<\/mo><\/mrow>]<\/mo><\/mrow>.<\/mi><\/mrow>\\mathcal{L}_{\\mathrm{global}}\n=\n\\frac{1}{2}\n\\mathbb{E}\n\\left[\nd_{\\mathrm{vec}}\n\\left(\nQ^t(g),u^t\n\\right)\n+\nd_{\\mathrm{vec}}\n\\left(\nQ^{t+1}(g),u^{t+1}\n\\right)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        This is the global counterpart of the two spatial prediction losses.<\/p>\n\n\n\n

        Periodic-Translation Consistency<\/h5>\n\n\n\n

        The original and translated views should describe the same physical field, so their global embeddings are matched:<\/p>\n\n\n\n

        \u2112<\/mi><\/mtext>shift<\/mi><\/mrow><\/msub>=<\/mo>\ud835\udd3c<\/mi>[<\/mo>d<\/mi><\/mtext>vec<\/mi><\/mrow><\/msub>(<\/mo>g<\/mi>,<\/mo>g<\/mi>~<\/mo><\/mover>)<\/mo><\/mrow>]<\/mo><\/mrow>.<\/mi><\/mrow>\\mathcal{L}_{\\mathrm{shift}}\n=\n\\mathbb{E}\n\\left[\nd_{\\mathrm{vec}}\n\\left(\ng,\\widetilde{g}\n\\right)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        Variance Regularization<\/h5>\n\n\n\n

        Let \ud835\udca2<\/mi>\\mathcal{G}<\/annotation><\/semantics><\/math> be the set containing the embeddings \ud835\udc54 and g<\/mi>~<\/mo><\/mover>\\widetilde{g}<\/annotation><\/semantics><\/math> from all training examples used to evaluate the loss. The variance penalty is<\/p>\n\n\n\n
        \u2112<\/mi><\/mtext>variance<\/mi><\/mrow><\/msub>=<\/mo>1<\/mn>D<\/mi><\/mfrac>\u2211<\/mo>k<\/mi>=<\/mo>1<\/mn><\/mrow>D<\/mi><\/munderover><\/mrow>max<\/mi>\u2061<\/mo><\/mrow>(<\/mo>0<\/mn>,<\/mo>1<\/mn>\u2212<\/mo>Var<\/mi>v<\/mi>\u2208<\/mo>\ud835\udca2<\/mi><\/mrow><\/msub>\u2061<\/mo>[<\/mo>v<\/mi>k<\/mi><\/msub>]<\/mo><\/mrow>+<\/mo>\u03b5<\/mi><\/mrow><\/msqrt>)<\/mo><\/mrow>.<\/mi><\/mrow>\\mathcal{L}_{\\mathrm{variance}}\n=\n\\frac{1}{D}\n\\sum_{k=1}^{D}\n\\max\n\\left(\n0,\n1-\n\\sqrt{\n\\operatorname{Var}_{v\\in\\mathcal{G}}\n\\left[\nv_k\n\\right]\n+\\varepsilon\n}\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        It prevents collapse to a constant embedding by requiring every coordinate to vary across examples.<\/p>\n\n\n\n

        Covariance Regularization<\/h5>\n\n\n\n

        Let<\/p>\n\n\n\n

        \u03bc<\/mi>=<\/mo>1<\/mn>|<\/mi>\ud835\udca2<\/mi>|<\/mi><\/mrow><\/mfrac>\u2211<\/mo>v<\/mi>\u2208<\/mo>\ud835\udca2<\/mi><\/mrow><\/munder><\/mrow>v<\/mi>,<\/mo><\/mrow>\\mu\n=\n\\frac{1}{|\\mathcal{G}|}\n\\sum_{v\\in\\mathcal{G}}\nv,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        C<\/mi>j<\/mi>k<\/mi><\/mrow><\/msub>=<\/mo>1<\/mn>|<\/mi>\ud835\udca2<\/mi>|<\/mi><\/mrow><\/mfrac>\u2211<\/mo>v<\/mi>\u2208<\/mo>\ud835\udca2<\/mi><\/mrow><\/munder><\/mrow>(<\/mo>v<\/mi>j<\/mi><\/msub>\u2212<\/mo>\u03bc<\/mi>j<\/mi><\/msub>)<\/mo><\/mrow>(<\/mo>v<\/mi>k<\/mi><\/msub>\u2212<\/mo>\u03bc<\/mi>k<\/mi><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>C_{jk}\n=\n\\frac{1}{|\\mathcal{G}|}\n\\sum_{v\\in\\mathcal{G}}\n\\left(\nv_j-\\mu_j\n\\right)\n\\left(\nv_k-\\mu_k\n\\right).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        The covariance penalty suppresses redundant correlations between distinct global coordinates:<\/p>\n\n\n\n

        \u2112<\/mi><\/mtext>covariance<\/mi><\/mrow><\/msub>=<\/mo>1<\/mn>D<\/mi><\/mfrac>\u2211<\/mo>j<\/mi>,<\/mo>k<\/mi>=<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr>j<\/mi>\u2260<\/mo>k<\/mi><\/mrow><\/mtd><\/mtr><\/mtable><\/mstyle>D<\/mi><\/munderover><\/mrow>C<\/mi>j<\/mi>k<\/mi><\/mrow>2<\/mn><\/msubsup>.<\/mi><\/mrow>\\mathcal{L}_{\\mathrm{covariance}}\n=\n\\frac{1}{D}\n\\sum_{\\substack{j,k=1\\\\j\\neq k}}^{D}\nC_{jk}^{2}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        Complete Objective<\/h5>\n\n\n\n
        <\/mspace>\u2112<\/mi>=<\/mo>\u2112<\/mi><\/mtext>masked<\/mi><\/mrow><\/msub>+<\/mo>\u2112<\/mi><\/mtext>future<\/mi><\/mrow><\/msub>+<\/mo>0.25<\/mn>\u2112<\/mi><\/mtext>global<\/mi><\/mrow><\/msub>+<\/mo>\u2112<\/mi><\/mtext>shift<\/mi><\/mrow><\/msub>+<\/mo>0.1<\/mn>\u2112<\/mi><\/mtext>variance<\/mi><\/mrow><\/msub>+<\/mo>0.01<\/mn>\u2112<\/mi><\/mtext>covariance<\/mi><\/mrow><\/msub><\/mrow><\/mspace><\/mrow>\\boxed{\n\\mathcal{L}\n=\n\\mathcal{L}_{\\mathrm{masked}}\n+\n\\mathcal{L}_{\\mathrm{future}}\n+\n0.25\\mathcal{L}_{\\mathrm{global}}\n+\n\\mathcal{L}_{\\mathrm{shift}}\n+\n0.1\\mathcal{L}_{\\mathrm{variance}}\n+\n0.01\\mathcal{L}_{\\mathrm{covariance}}\n}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        Optimization and Final Metric<\/h5>\n\n\n\n

        We train with AdamW, batch size 128, learning rate 3<\/mn>\u00d7<\/mo>10<\/mn>\u2212<\/mo>4<\/mn><\/mrow><\/msup><\/mrow>3\\times10^{-4}<\/annotation><\/semantics><\/math>, weight decay 10<\/mn>\u2212<\/mo>4<\/mn><\/mrow><\/msup>10^{-4}<\/annotation><\/semantics><\/math>, a cosine learning-rate schedule, bfloat16 arithmetic, and gradient clipping at 5. The teacher is updated after every batch with EMA decay 0.996. The current model is the teacher checkpoint after 10 epochs.<\/p>\n\n\n\n

        After training, we run the frozen encoder on the validation\/calibration fields and compute the standard deviation of each feature channel across fields and spatial positions. When comparing two fields, their channel-wise feature differences are divided by these standard deviations. The primary distance is the resulting root-mean-square difference, averaged over the three spatial scales. This stops channels with naturally large numerical ranges from dominating.<\/p>\n\n\n\n

        For an ensemble of \ud835\udc5a generated fields x<\/mi>1<\/mn><\/msub>,<\/mo>\u2026<\/mo>,<\/mo>x<\/mi>m<\/mi><\/msub><\/mrow>x_1,\\ldots,x_m<\/annotation><\/semantics><\/math>, a target field \ud835\udc66, and a field distance \ud835\udc51, we compute the energy score<\/p>\n\n\n\n
        ES<\/mi>d<\/mi><\/msub>\u2061<\/mo>(<\/mo>{<\/mo>x<\/mi>i<\/mi><\/msub>}<\/mo>i<\/mi>=<\/mo>1<\/mn><\/mrow>m<\/mi><\/msubsup>,<\/mo>y<\/mi>)<\/mo><\/mrow>=<\/mo>1<\/mn>m<\/mi><\/mfrac>\u2211<\/mo>i<\/mi>=<\/mo>1<\/mn><\/mrow>m<\/mi><\/munderover><\/mrow>d<\/mi>(<\/mo>x<\/mi>i<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>\u2212<\/mo>1<\/mn>2<\/mn>m<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>\u2211<\/mo>i<\/mi>=<\/mo>1<\/mn><\/mrow>m<\/mi><\/munderover><\/mrow>\u2211<\/mo>j<\/mi>=<\/mo>1<\/mn><\/mrow>m<\/mi><\/munderover><\/mrow>d<\/mi>(<\/mo>x<\/mi>i<\/mi><\/msub>,<\/mo>x<\/mi>j<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>\\operatorname{ES}_{d}\n\\left(\n\\{x_i\\}_{i=1}^{m},y\n\\right)\n=\n\\frac{1}{m}\n\\sum_{i=1}^{m}\nd(x_i,y)\n–\n\\frac{1}{2m^2}\n\\sum_{i=1}^{m}\n\\sum_{j=1}^{m}\nd(x_i,x_j).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        The first term measures accuracy against the target. The subtracted pairwise term rewards ensemble diversity and therefore penalizes collapse. Lower is better. In later tables, \u201cES\u201d means this score with \ud835\udc51 replaced by the named distance.’<\/p>\n\n\n\n

        10.5 Compared Metrics and Baselines<\/h4>\n\n\n\n

        Most metrics below define a distance d<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>d(x,y)<\/annotation><\/semantics><\/math> between two 64<\/mn>64<\/annotation><\/semantics><\/math> x 64<\/mn>64<\/annotation><\/semantics><\/math> fields. Pairwise experiments use that distance directly; a table entry ending in \u201cES\u201d substitutes it into the energy score above. The paper\u2019s full-cycle statistic instead compares a generated field with its given observation, while its ensemble-spread statistic has no target. Lower is better within any one metric, but absolute values cannot be compared between metrics because their scales differ.<\/p>\n\n\n\n
        QG-SSL Aligned<\/h5>\n\n\n\n

        This is our primary distance. At each of the three spatial scales, corresponding feature vectors at the same location are compared after dividing each channel by its calibration standard deviation. We take the root-mean-square difference over channels and locations, then average the three scales. It therefore measures learned structure while retaining relative spatial alignment.<\/p>\n\n\n\n

        QG-SSL Global<\/h5>\n\n\n\n

        This uses the optional 128-dimensional global embedding instead of the spatial maps. If g<\/mi>k<\/mi><\/msub>(<\/mo>x<\/mi>)<\/mo><\/mrow>g_k(x)<\/annotation><\/semantics><\/math> is coordinate \ud835\udc58 and \u03c4<\/mi>k<\/mi><\/msub>\\tau_k<\/annotation><\/semantics><\/math> is its standard deviation on held-out real fields, then<\/p>\n\n\n\n
        d<\/mi><\/mtext>global<\/mi><\/mrow><\/msub>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>1<\/mn>128<\/mn><\/mfrac>\u2211<\/mo>k<\/mi>=<\/mo>1<\/mn><\/mrow>128<\/mn><\/munderover><\/mrow>(<\/mo>g<\/mi>k<\/mi><\/msub>(<\/mo>x<\/mi>)<\/mo>\u2212<\/mo>g<\/mi>k<\/mi><\/msub>(<\/mo>y<\/mi>)<\/mo><\/mrow>\u03c4<\/mi>k<\/mi><\/msub><\/mfrac>)<\/mo><\/mrow>2<\/mn><\/msup><\/mrow><\/msqrt>.<\/mi><\/mrow>d_{\\mathrm{global}}(x,y)\n=\n\\sqrt{\n\\frac{1}{128}\n\\sum_{k=1}^{128}\n\\left(\n\\frac{\ng_k(x)-g_k(y)\n}{\n\\tau_k\n}\n\\right)^2\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        The translation-consistency loss makes this representation approximately insensitive to the choice of spatial origin.<\/p>\n\n\n\n

        Pixel<\/h5>\n\n\n\n

        The fields are flattened and corresponding grid values are compared directly. With \u03c3<\/mi>p<\/mi><\/msub>\\sigma_p<\/annotation><\/semantics><\/math> denoting the calibration standard deviation at grid position \ud835\udc5d,<\/p>\n\n\n\n
        d<\/mi><\/mtext>pixel<\/mi><\/mrow><\/msub>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>1<\/mn>|<\/mi>\u03a9<\/mi><\/mrow>|<\/mi><\/mrow><\/mfrac>\u2211<\/mo>p<\/mi>\u2208<\/mo>\u03a9<\/mi><\/mrow><\/mrow><\/munder><\/mrow>(<\/mo>x<\/mi>(<\/mo>p<\/mi>)<\/mo>\u2212<\/mo>y<\/mi>(<\/mo>p<\/mi>)<\/mo><\/mrow>\u03c3<\/mi>p<\/mi><\/msub><\/mfrac>)<\/mo><\/mrow>2<\/mn><\/msup><\/mrow><\/msqrt>,<\/mo><\/mrow>d_{\\mathrm{pixel}}(x,y)\n=\n\\sqrt{\n\\frac{1}{|\\Omega|}\n\\sum_{p\\in\\Omega}\n\\left(\n\\frac{\nx(p)-y(p)\n}{\n\\sigma_p\n}\n\\right)^2\n},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        where \u03a9<\/mi><\/mrow>\\Omega<\/annotation><\/semantics><\/math> is the set of all 64<\/mn>64<\/annotation><\/semantics><\/math> x 64<\/mn>64<\/annotation><\/semantics><\/math> positions. This baseline retains exact location and fine detail but has no learned notion of structure.<\/p>\n\n\n\n
        Spectrum<\/h5>\n\n\n\n

        For each field we compute energy and enstrophy in 30 radial Fourier-wavenumber shells, take their logarithms, and concatenate them into a 60-dimensional vector. Each coordinate is standardized on held-out real fields and the distance is the root-mean-square vector difference. Because Fourier phase is discarded, this metric cannot locate structures in space.<\/p>\n\n\n\n

        DINOv2<\/h5>\n\n\n\n

        DINOv2 is a self-supervised vision transformer pre-trained on natural images. We use its ViT-S\/14 model without fine-tuning. A standardized vorticity field is clipped to three standard deviations, mapped to a grayscale image, resized to 224<\/mn>224<\/annotation><\/semantics><\/math> \u00d7 224<\/mn>224<\/annotation><\/semantics><\/math>, and copied into three color channels. The model reduces this image to one 384-dimensional global embedding. We standardize each embedding coordinate on held-out QG fields and use the root-mean-square distance between embeddings.<\/p>\n\n\n\n
        LSiM<\/h5>\n\n\n\n

        LSiM is a pretrained learned similarity metric for simulation fields. Each field is resized to 224<\/mn>224<\/annotation><\/semantics><\/math> \u00d7 224<\/mn>224<\/annotation><\/semantics><\/math>, copied into three channels, and linearly mapped to [<\/mo>0,225<\/mn>]<\/mo><\/mrow>[0,225]<\/annotation><\/semantics><\/math> using the joint minimum and maximum of the fields being compared. A five-scale convolutional network compares normalized feature maps using learned nonnegative channel weights; the square root of the summed multiscale error is the distance. The released model is used without training on our QG data.<\/p>\n\n\n\n
        Paper Reconstruction<\/h5>\n\n\n\n

        This is the paper\u2019s relative vorticity error. For a generated field \ud835\udc65 and target \ud835\udc66,<\/p>\n\n\n\n

        d<\/mi><\/mtext>reconstruction<\/mi><\/mrow><\/msub>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2016<\/mi>x<\/mi>\u2212<\/mo>y<\/mi>\u2016<\/mi>2<\/mn><\/msub><\/mrow>\u2016<\/mi>y<\/mi>\u2016<\/mi>2<\/mn><\/msub><\/mrow><\/mfrac>.<\/mi><\/mrow>d_{\\mathrm{reconstruction}}(x,y)\n=\n\\frac{\n\\|x-y\\|_2\n}{\n\\|y\\|_2\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        For an ensemble, the paper reports the mean of this error over members. Unlike an energy score, it contains no reward for ensemble diversity.<\/p>\n\n\n\n

        Paper Full-Cycle Consistency<\/h5>\n\n\n\n

        Let \ud835\udc34 be the paper\u2019s observation operator, which filters a 64<\/mn>64<\/annotation><\/semantics><\/math> \u00d7 64<\/mn>64<\/annotation><\/semantics><\/math> field to the resolved 16<\/mn>16<\/annotation><\/semantics><\/math> \u00d7 16<\/mn>16<\/annotation><\/semantics><\/math> information, and let \ud835\udc5c be the given observation. The cycle error of a generated field is<\/p>\n\n\n\n
        d<\/mi><\/mtext>cycle<\/mi><\/mrow><\/msub>(<\/mo>x<\/mi>,<\/mo>o<\/mi>)<\/mo>=<\/mo>\u2016<\/mi>A<\/mi>(<\/mo>x<\/mi>)<\/mo>\u2212<\/mo>o<\/mi>\u2016<\/mi>2<\/mn><\/msub><\/mrow>\u2016<\/mi>o<\/mi>\u2016<\/mi>2<\/mn><\/msub><\/mrow><\/mfrac>.<\/mi><\/mrow>d_{\\mathrm{cycle}}(x,o)\n=\n\\frac{\n\\|A(x)-o\\|_2\n}{\n\\|o\\|_2\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        \u201cFull-cycle\u201d uses observations covering the full periodic domain; the partial version additionally restricts the comparison to observed regions. We retain this diagnostic only when \ud835\udc5c is the actual observation used to condition the generated sample; it is not used as a distance between arbitrary pairs of fields. Ensemble results average the member-wise errors.<\/p>\n\n\n\n

        Paper Log-Energy<\/h5>\n\n\n\n

        Let \ud835\udc38(\ud835\udc65) be the 30-shell kinetic-energy spectrum of field \ud835\udc65. The paper\u2019s metric is<\/p>\n\n\n\n

        d<\/mi>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>E<\/mi><\/mrow><\/msub>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2016<\/mi><\/mspace>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>E<\/mi>(<\/mo>x<\/mi>)<\/mo>\u2212<\/mo>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>E<\/mi>(<\/mo>y<\/mi>)<\/mo>\u2016<\/mi>2<\/mn><\/msub><\/mrow>\u2016<\/mi><\/mspace>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>E<\/mi>(<\/mo>y<\/mi>)<\/mo>\u2016<\/mi>2<\/mn><\/msub><\/mrow><\/mfrac>.<\/mi><\/mrow>d_{\\log E}(x,y)\n=\n\\frac{\n\\|\\log E(x)-\\log E(y)\\|_2\n}{\n\\|\\log E(y)\\|_2\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        The paper averages this member-wise error for an ensemble.<\/p>\n\n\n\n

        Paper Log-Enstrophy<\/h5>\n\n\n\n

        With \ud835\udc4d(\ud835\udc65) denoting the corresponding 30-shell enstrophy spectrum, this metric is<\/p>\n\n\n\n

        d<\/mi>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>Z<\/mi><\/mrow><\/msub>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2016<\/mi><\/mspace>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>Z<\/mi>(<\/mo>x<\/mi>)<\/mo>\u2212<\/mo>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>Z<\/mi>(<\/mo>y<\/mi>)<\/mo>\u2016<\/mi>2<\/mn><\/msub><\/mrow>\u2016<\/mi><\/mspace>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>Z<\/mi>(<\/mo>y<\/mi>)<\/mo>\u2016<\/mi>2<\/mn><\/msub><\/mrow><\/mfrac>.<\/mi><\/mrow>d_{\\log Z}(x,y)\n=\n\\frac{\n\\|\\log Z(x)-\\log Z(y)\\|_2\n}{\n\\|\\log Z(y)\\|_2\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        It is also averaged member-wise and, like log-energy, ignores Fourier phase.<\/p>\n\n\n\n

        Paper Ensemble Standard Deviation<\/h5>\n\n\n\n

        For ensemble X<\/mi>=<\/mo>{<\/mo>x<\/mi>1<\/mn><\/msub>,<\/mo>\u2026<\/mo>,<\/mo>x<\/mi>m<\/mi><\/msub>}<\/mo><\/mrow>X=\\{x_1,\\ldots,x_m\\}<\/annotation><\/semantics><\/math>, the paper reports the mean pointwise spread<\/p>\n\n\n\n
        s<\/mi>(<\/mo>X<\/mi>)<\/mo>=<\/mo>1<\/mn>|<\/mi>\u03a9<\/mi><\/mrow>|<\/mi><\/mrow><\/mfrac>\u2211<\/mo>p<\/mi>\u2208<\/mo>\u03a9<\/mi><\/mrow><\/mrow><\/munder><\/mrow>1<\/mn>m<\/mi><\/mfrac>\u2211<\/mo>i<\/mi>=<\/mo>1<\/mn><\/mrow>m<\/mi><\/munderover><\/mrow>(<\/mo>x<\/mi>i<\/mi><\/msub>(<\/mo>p<\/mi>)<\/mo>\u2212<\/mo>1<\/mn>m<\/mi><\/mfrac>\u2211<\/mo>j<\/mi>=<\/mo>1<\/mn><\/mrow>m<\/mi><\/munderover><\/mrow>x<\/mi>j<\/mi><\/msub>(<\/mo>p<\/mi>)<\/mo>)<\/mo><\/mrow>2<\/mn><\/msup><\/mrow><\/msqrt>.<\/mi><\/mrow>s(X)\n=\n\\frac{1}{|\\Omega|}\n\\sum_{p\\in\\Omega}\n\\sqrt{\n\\frac{1}{m}\n\\sum_{i=1}^{m}\n\\left(\nx_i(p)\n–\n\\frac{1}{m}\n\\sum_{j=1}^{m}\nx_j(p)\n\\right)^2\n}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        This has no target and is not a quality score by itself: either too little or too much spread can be wrong. In the controlled diversity benchmark, \u201censemble-std discrepancy\u201d is the relative difference between \ud835\udc60(\ud835\udc4b) and the spread of a clean reference ensemble.<\/p>\n\n\n\n

        10.5 Initial Metric Benchmarks<\/h4>\n\n\n\n

        Temporal Neighborhood Self-Consistency<\/h5>\n\n\n\n

        Each metric ranks seven real candidate trajectories relative to a query, then ranks the same trajectories h frames later. Spearman correlation measures preservation of the complete ranking. Each early, middle, or late origin panel contains 192 rankings across the four physical cases; the table averages available panels. Shuffling future identities gives correlations near zero.<\/p>\n\n\n\n\n \n \n \n \n \n \n \n \n \n
        h<\/em><\/th>\n Time<\/th>\n QG-SSL aligned<\/th>\n Paper reconstruction<\/th>\n Pixel<\/th>\n LSiM<\/th>\n DINOv2<\/th>\n <\/tr>\n <\/thead>\n
        1<\/td>\n 0.5<\/td>\n 0.968<\/td>\n 0.970<\/td>\n 0.971<\/strong><\/td>\n 0.906<\/td>\n 0.605<\/td>\n <\/tr>\n
        4<\/td>\n 2<\/td>\n 0.899<\/strong><\/td>\n 0.884<\/td>\n 0.875<\/td>\n 0.715<\/td>\n 0.332<\/td>\n <\/tr>\n
        8<\/td>\n 4<\/td>\n 0.847<\/strong><\/td>\n 0.782<\/td>\n 0.764<\/td>\n 0.595<\/td>\n 0.285<\/td>\n <\/tr>\n
        16<\/td>\n 8<\/td>\n 0.783<\/strong><\/td>\n 0.598<\/td>\n 0.554<\/td>\n 0.387<\/td>\n 0.156<\/td>\n <\/tr>\n
        32<\/td>\n 16<\/td>\n 0.681<\/strong><\/td>\n 0.461<\/td>\n 0.376<\/td>\n 0.267<\/td>\n 0.182<\/td>\n <\/tr>\n
        80*<\/td>\n 40<\/td>\n 0.428<\/strong><\/td>\n 0.209<\/td>\n 0.087<\/td>\n -0.015<\/td>\n 0.096<\/td>\n <\/tr>\n <\/tbody>\n<\/table>\n\n\n\n
        Controlled Phase and Diversity Failures<\/h5>\n\n\n\n

        We randomize Fourier phases while preserving magnitudes, destroying spatial structure without changing the spectrum. We divide this distance by that caused by a periodic translation. A high ratio means phase sensitivity with translation insensitivity; absolute values cannot be compared between metrics.<\/p>\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n
        Metric<\/th>\n Phase-randomized<\/th>\n Translated<\/th>\n Ratio<\/th>\n <\/tr>\n <\/thead>\n
        QG-SSL aligned<\/td>\n 1.161<\/td>\n 1.294<\/td>\n 0.90<\/td>\n <\/tr>\n
        QG-SSL global<\/td>\n 0.877<\/td>\n 0.035<\/td>\n 24.78<\/strong><\/td>\n <\/tr>\n
        DINOv2<\/td>\n 3.249<\/td>\n 0.639<\/td>\n 5.09<\/td>\n <\/tr>\n
        LSiM<\/td>\n 0.623<\/td>\n 0.644<\/td>\n 0.97<\/td>\n <\/tr>\n
        Pixel<\/td>\n 1.597<\/td>\n 1.668<\/td>\n 0.96<\/td>\n <\/tr>\n
        Spectrum<\/td>\n 2.56 \u00d7 10-6<\/sup><\/td>\n 7.24 \u00d7 10-7<\/sup><\/td>\n 3.54<\/td>\n <\/tr>\n
        Paper reconstruction<\/td>\n 1.381<\/td>\n 1.432<\/td>\n 0.96<\/td>\n <\/tr>\n
        Paper log-energy<\/td>\n 2.24 \u00d7 10-7<\/sup><\/td>\n 6.22 \u00d7 10-8<\/sup><\/td>\n 3.60<\/td>\n <\/tr>\n
        Paper log-enstrophy<\/td>\n 3.40 \u00d7 10-7<\/sup><\/td>\n 9.03 \u00d7 10-8<\/sup><\/td>\n 3.76<\/td>\n <\/tr>\n <\/tbody>\n<\/table>\n\n\n\n
        M1-M4 Selection by Future Physical Utility<\/h5>\n\n\n\n

        For each of the four physical cases, we take one 16-member ensemble from each of M1\u2013M4. We evaluate every metric on each ensemble at the current time. Lower is better, so these four scores produce a current-time ranking of M<\/mi>1<\/mn>\u2212<\/mo>M<\/mi>4<\/mn><\/mrow>M1-M4<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

        Independently, we lift every generated field to 512<\/mn>512<\/annotation><\/semantics><\/math> \u00d7 512<\/mn>512<\/annotation><\/semantics><\/math>, evolve it with the matching QG solver, filter it back, and compute the ensemble\u2019s future pixel-space ES against the ground-truth future field from the same QG trajectory that supplied its conditioning observation. This produces a reference ranking of M<\/mi>1<\/mn>\u2212<\/mo>M<\/mi>4<\/mn><\/mrow>M1-M4<\/annotation><\/semantics><\/math> by future physical utility.<\/p>\n\n\n\n

        There are six unordered pairs among four methods. \u201cPairwise order agreement\u201d is the fraction of the 4<\/mn>4 <\/annotation><\/semantics><\/math> x 6<\/mn>6<\/annotation><\/semantics><\/math> = 24<\/mn>24<\/annotation><\/semantics><\/math> method pairs for which the current-time ranking and the future ranking choose the same method as better; ties are excluded. \u201cCorrect winner\u201d counts the physical cases in which the method with the lowest current-time score is also the method with the lowest future score.<\/p>\n\n\n\n

        Solver error on held-out real states is 0.003<\/mn>\u2212<\/mo>0.010<\/mn><\/mrow>0.003-0.010<\/annotation><\/semantics><\/math> at h<\/mi>=<\/mo>1<\/mn><\/mrow>h=1<\/annotation><\/semantics><\/math> and 0.017<\/mn>\u2212<\/mo>0.064<\/mn><\/mrow>0.017-0.064<\/annotation><\/semantics><\/math> at h = 8. The future M<\/mi>1<\/mn>\u2212<\/mo>M<\/mi>4<\/mn><\/mrow>M1-M4<\/annotation><\/semantics><\/math> order is identical at h<\/mi>=<\/mo>1<\/mn>,<\/mo>4<\/mn>,<\/mo>8<\/mn><\/mrow>h=1, 4, 8<\/annotation><\/semantics><\/math>, so the table applies to each horizon.<\/p>\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n
        Metric evaluated before rollout<\/th>\n Pairwise order agreement<\/th>\n Correct winner<\/th>\n <\/tr>\n <\/thead>\n
        QG-SSL aligned ES<\/td>\n 1.000<\/strong><\/td>\n 4\/4<\/strong><\/td>\n <\/tr>\n
        Pixel ES<\/td>\n 1.000<\/strong><\/td>\n 4\/4<\/strong><\/td>\n <\/tr>\n
        Paper reconstruction<\/td>\n 1.000<\/strong><\/td>\n 4\/4<\/strong><\/td>\n <\/tr>\n
        LSiM ES<\/td>\n 0.917<\/td>\n 2\/4<\/td>\n <\/tr>\n
        QG-SSL global ES<\/td>\n 0.875<\/td>\n 1\/4<\/td>\n <\/tr>\n
        Paper cycle-consistency<\/td>\n 0.875<\/td>\n 1\/4<\/td>\n <\/tr>\n
        Spectrum ES<\/td>\n 0.750<\/td>\n 1\/4<\/td>\n <\/tr>\n
        Paper log-energy<\/td>\n 0.708<\/td>\n 0\/4<\/td>\n <\/tr>\n
        Paper log-enstrophy<\/td>\n 0.708<\/td>\n 0\/4<\/td>\n <\/tr>\n
        DINOv2 ES<\/td>\n 0.625<\/td>\n 0\/4<\/td>\n <\/tr>\n <\/tbody>\n<\/table>\n\n\n\n

        11. Conclusion<\/h2>\n\n\n\n

        This analysis examined how partial observations influence both observed and unobserved components of a high-dimensional physical state. The clean state was decomposed into a near component C<\/mi>C<\/annotation><\/semantics><\/math>, which is directly measured, and a far component D<\/mi>D<\/annotation><\/semantics><\/math>, which is not directly observed. The observation model was<\/p>\n\n\n\n
        Y<\/mi>=<\/mo>A<\/mi>C<\/mi>+<\/mo>N<\/mi>=<\/mo>H<\/mi>U<\/mi>0<\/mn><\/msub>+<\/mo>N<\/mi>,<\/mo><\/mspace>H<\/mi>=<\/mo>(<\/mo>A<\/mi><\/mtd>a<\/mi>m<\/mi>p<\/mi>;<\/mo>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>.<\/mi><\/mrow>Y\n=\nAC+N\n=\nHU_0+N,\n\\qquad\nH\n=\n\\begin{pmatrix}\nA&amp;0\n\\end{pmatrix}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        The linear Gaussian model provides an exact description of this inverse problem. The posterior distribution of the far state is<\/p>\n\n\n\n

        D<\/mi>|<\/mo>Y<\/mi>=<\/mo>y<\/mi>\u223c<\/mo>\ud835\udca9<\/mi>(<\/mo>m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>,<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>,<\/mo><\/mrow>D\\mid Y=y\n\\sim\n\\mathcal{N}\n\\left(\nm_{D\\mid y},\n\\Sigma_{D\\mid Y}\n\\right),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        with<\/p>\n\n\n\n

        m<\/mi>D<\/mi>|<\/mo>y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>y<\/mi><\/mrow>m_{D\\mid y}\n=\n\\Sigma_{DC}A^\\top S_Y^{-1}y<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        and<\/p>\n\n\n\n

        \u03a3<\/mi><\/mrow>D<\/mi>|<\/mo>Y<\/mi><\/mrow><\/msub>=<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>D<\/mi><\/mrow><\/msub>\u2212<\/mo>\u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>S<\/mi>Y<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>A<\/mi>\u03a3<\/mi><\/mrow>C<\/mi>D<\/mi><\/mrow><\/msub>.<\/mi><\/mrow>\\Sigma_{D\\mid Y}\n=\n\\Sigma_{DD}\n–\n\\Sigma_{DC}A^\\top\nS_Y^{-1}\nA\\Sigma_{CD}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        These expressions show that information reaches the hidden component only through directions that are both visible to the observation operator A<\/mi>A<\/annotation><\/semantics><\/math> and correlated with the far state through \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>\\Sigma_{DC}<\/annotation><\/semantics><\/math>. If<\/p>\n\n\n\n
        \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup>=<\/mo>0<\/mn>,<\/mo><\/mrow>\\Sigma_{DC}A^\\top\n=\n0,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        then the observation does not change either the posterior mean or covariance of D<\/mi>D<\/annotation><\/semantics><\/math>. More generally, the number of far-state directions in which uncertainty can be reduced is limited by the rank of the observation and cross-covariance operators.<\/p>\n\n\n\n

        The Gaussian setting also yields an exact conditional score. The likelihood correction is<\/p>\n\n\n\n

        \u2207<\/mo>u<\/mi><\/msub>log<\/mi>\u2061<\/mo><\/mspace><\/mrow>p<\/mi>t<\/mi><\/msub>(<\/mo>y<\/mi>|<\/mo>u<\/mi>)<\/mo>=<\/mo>B<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>\u0393<\/mi><\/mrow>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>(<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>B<\/mi>t<\/mi><\/msub>u<\/mi>)<\/mo><\/mrow>,<\/mo><\/mrow>\\nabla_u\\log p_t(y\\mid u)\n=\nB_t^\\top H^\\top\n\\Gamma_t^{-1}\n\\left(\ny-HB_tu\n\\right),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        where B<\/mi>t<\/mi><\/msub>u<\/mi><\/mrow>B_tu<\/annotation><\/semantics><\/math> is the exact clean-state estimate and \u0393<\/mi><\/mrow>t<\/mi><\/msub>\\Gamma_t<\/annotation><\/semantics><\/math> combines measurement noise with the remaining uncertainty in that estimate. This result provides an exact benchmark for diffusion posterior sampling.<\/p>\n\n\n\n

        The flow-matching analysis extends the same structure to a nonlinear denoising map F<\/mi>t<\/mi><\/msub>F_t<\/annotation><\/semantics><\/math>. The DPS-guided dynamics are<\/p>\n\n\n\n
        d<\/mi>U<\/mi>t<\/mi><\/msub><\/mrow>d<\/mi>t<\/mi><\/mrow><\/mfrac>=<\/mo>v<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>+<\/mo>\u03bb<\/mi>t<\/mi><\/msub>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>\u22a4<\/mi><\/msup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>H<\/mi>F<\/mi>t<\/mi><\/msub>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>\\frac{dU_t}{dt}\n=\nv_t(U_t)\n+\n\\lambda_t\nJ_{F_t}(U_t)^\\top\nH^\\top R^{-1}\n\\left[\ny-HF_t(U_t)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        After local linearization around the conditional mean, the first two moments evolve approximately as<\/p>\n\n\n\n

        m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo><\/mrow>\\dot{m}_t\n\\approx\nf_t(m_t,y)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        and<\/p>\n\n\n\n

        \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo>J<\/mi>t<\/mi><\/msub>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>+<\/mo>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>J<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>.<\/mi><\/mrow>\\dot{\\Sigma}_t\n\\approx\nJ_t\\Sigma_t+\\Sigma_tJ_t^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        The local DPS contribution to the Jacobian is<\/p>\n\n\n\n

        J<\/mi>t<\/mi>g<\/mi><\/msubsup>\u2248<\/mo>\u2212<\/mo>\u03bb<\/mi>t<\/mi><\/msub>G<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>H<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>H<\/mi>G<\/mi>t<\/mi><\/msub>,<\/mo><\/mrow>J_t^g\n\\approx\n-\\lambda_t\nG_t^\\top H^\\top R^{-1}HG_t,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        where<\/p>\n\n\n\n

        G<\/mi>t<\/mi><\/msub>=<\/mo>J<\/mi>F<\/mi>t<\/mi><\/msub><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>G_t\n=\nJ_{F_t}(m_t).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        After splitting the flow state into near and far components, the correction acting on the far coordinates is<\/p>\n\n\n\n

        g<\/mi>D<\/mi>,<\/mo>t<\/mi><\/mrow><\/msub>=<\/mo>\u03bb<\/mi>t<\/mi><\/msub>(<\/mo>\u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>D<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mrow>\u22a4<\/mi><\/msup>A<\/mi>\u22a4<\/mi><\/msup>R<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>[<\/mo>y<\/mi>\u2212<\/mo>A<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup>(<\/mo>U<\/mi>t<\/mi><\/msub>)<\/mo>]<\/mo><\/mrow>.<\/mi><\/mrow>g_{D,t}\n=\n\\lambda_t\n\\left(\n\\frac{\\partial F_t^C}{\\partial D_t}\n\\right)^\\top\nA^\\top R^{-1}\n\\left[\ny-AF_t^C(U_t)\n\\right].<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        Thus, a measurement applied only to the near component can influence the current far component whenever the predicted near state depends on the far coordinates. In the linear Gaussian model, this coupling is represented by \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup><\/mrow>\\Sigma_{DC}A^\\top<\/annotation><\/semantics><\/math>. In the nonlinear flow-matching model, it is represented locally by<\/p>\n\n\n\n
        \u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>D<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac>\\frac{\\partial F_t^C}{\\partial D_t}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        These are distinct mathematical objects, but they express the same central principle: information can propagate from an observed region to an unobserved region only through statistical or dynamical coupling between the two.<\/p>\n\n\n\n

        The analysis therefore provides a framework for interpreting the spatial reach of posterior guidance. It separates direct observational support from indirect information transfer and helps explain why reconstruction quality can deteriorate outside the observed region when the relevant coupling is weak.<\/p>\n\n\n\n

        12. Future Work<\/h3>\n\n\n\n

        Future work should focus on testing how observation design, nonlinear coupling, and physical dynamics affect information transfer into unobserved regions.<\/p>\n\n\n\n

        12.1 Observation Bandwidth<\/h4>\n\n\n\n

        Different observation operators A<\/mi>A<\/annotation><\/semantics><\/math> measure different spatial scales and regions. Their effect can be studied through the Gaussian coupling<\/p>\n\n\n\n
        \u03a3<\/mi><\/mrow>D<\/mi>C<\/mi><\/mrow><\/msub>A<\/mi>\u22a4<\/mi><\/msup><\/mrow>\\Sigma_{DC}A^\\top<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        and the nonlinear sensitivity<\/p>\n\n\n\n

        \u2202<\/mi>F<\/mi>t<\/mi>C<\/mi><\/msubsup><\/mrow>\u2202<\/mi>D<\/mi>t<\/mi><\/msub><\/mrow><\/mfrac>.<\/mi><\/mrow>\\frac{\\partial F_t^C}{\\partial D_t}.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        Comparing these quantities with far-region reconstruction error could show how observation bandwidth and placement determine the spatial reach of guidance.<\/p>\n\n\n\n

        12.2 Numerical Validation<\/h4>\n\n\n\n

        The locally linear moment equations should be compared with ensemble estimates from a trained model:<\/p>\n\n\n\n

        m<\/mi>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo>f<\/mi>t<\/mi><\/msub>(<\/mo>m<\/mi>t<\/mi><\/msub>,<\/mo>y<\/mi>)<\/mo>,<\/mo><\/mrow>\\dot{m}_t\n\\approx\nf_t(m_t,y),<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        \u03a3<\/mi><\/mrow>\u02d9<\/mo><\/mover>t<\/mi><\/msub>\u2248<\/mo>J<\/mi>t<\/mi><\/msub>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>+<\/mo>\u03a3<\/mi><\/mrow>t<\/mi><\/msub>J<\/mi>t<\/mi>\u22a4<\/mi><\/msubsup>.<\/mi><\/mrow>\\dot{\\Sigma}_t\n\\approx\nJ_t\\Sigma_t+\\Sigma_tJ_t^\\top.<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        This would determine when the first-order approximation is accurate and when nonlinear Hessian terms become important.<\/p>\n\n\n\n

        12.3 Irregular Domains and Stability<\/h4>\n\n\n\n

        The analysis should also be extended to irregular masks, such as land-ocean boundaries, while avoiding numerical artifacts near mask edges. Reconstructed states should then be evolved with the governing fluid solver to test long-time stability, forecast skill, energy and enstrophy behavior, and ensemble calibration.<\/p>\n\n\n\n

        12.4 Improved Posterior Guidance<\/h4>\n\n\n\n

        The exact Gaussian model weights the residual using<\/p>\n\n\n\n

        \u0393<\/mi><\/mrow>t<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msubsup>=<\/mo>(<\/mo>R<\/mi>+<\/mo>H<\/mi>\u03a3<\/mi><\/mrow>0<\/mn>|<\/mo>t<\/mi><\/mrow><\/msub>H<\/mi>\u22a4<\/mi><\/msup>)<\/mo><\/mrow>\u2212<\/mo>1<\/mn><\/mrow><\/msup>,<\/mo><\/mrow>\\Gamma_t^{-1}\n=\n\\left(\nR+H\\Sigma_{0\\mid t}H^\\top\n\\right)^{-1},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

        whereas practical DPS often uses a simpler scalar weighting. Future work could develop ensemble-based or low-rank approximations to this effective covariance and compare them with standard DPS, conditional generative models, and PDE-constrained approaches.<\/p>\n\n\n\n

        13. References<\/h2>\n\n\n\n

        [1<\/mn>1<\/annotation><\/semantics><\/math>] A. N. Suresh Babu, A. Sadam, and P. F. J. Lermusiaux, “Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence,” Journal of Advances in Modeling Earth Systems, vol. 18, no. 3, e<\/mi>2025<\/mn>M<\/mi>S<\/mi>005324<\/mn><\/mrow>e2025MS005324<\/annotation><\/semantics><\/math>, 2026. DOI: 10.1029<\/mn>\/<\/mi>2025<\/mn>M<\/mi>S<\/mi>005324.<\/mn><\/mrow>10.1029\/2025MS005324.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

        [2<\/mn>2<\/annotation><\/semantics><\/math>] H. Chung, J. Kim, M. T. McCann, M. L. Klasky, and J. C. Ye, “Diffusion Posterior Sampling for General Noisy Inverse Problems,” International Conference on Learning Representations, 2023.<\/p>\n\n\n\n

        [3<\/mn>3<\/annotation><\/semantics><\/math>] A. N. Suresh Babu et al., “quasi-geostrophic-beta-plane-super-resolution,” GitHub repository, Models\/samplers.py. This implementation contains the VP-SDE reverse sampler, DPS correction, Fourier\/coarsening observation operator, and gappy-observation mask.<\/p>\n\n\n\n

        [4<\/mn>4<\/annotation><\/semantics><\/math>] R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 6th ed., Pearson, Result 4.6, p. 160.<\/p>\n\n\n\n

        [5<\/mn>5<\/annotation><\/semantics><\/math>] Y. Polyanskiy and Y. Wu, Information Theory: From Coding to Learning, Cambridge University Press, data-processing inequality, Theorem 3.7(c).<\/p>\n","protected":false},"excerpt":{"rendered":"

        SGI Mentor: Akhil Sadam SGI Fellows: Santoshi Yadagiri, Pietro Palombini 1. Introduction Many geophysical inverse problems require reconstruction of a high-dimensional physical state from observations that are incomplete, noisy, or available only over part of the spatial domain. In oceanic and atmospheric applications, observations may be coarse, sparse, or separated by large unmeasured regions. The […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[110],"tags":[],"ppma_author":[44,48],"class_list":["post-347","post","type-post","status-publish","format-standard","hentry","category-research"],"authors":[{"term_id":44,"user_id":0,"is_guest":1,"slug":"cap-pietropalombini","display_name":"pietropalombini","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g","author_category":"","first_name":"","last_name":"","user_url":"","job_title":"","description":""},{"term_id":48,"user_id":0,"is_guest":1,"slug":"cap-sayadagiri3","display_name":"sayadagiri3","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g","author_category":"","first_name":"","last_name":"","user_url":"","job_title":"","description":""}],"_links":{"self":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts\/347","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/comments?post=347"}],"version-history":[{"count":9,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts\/347\/revisions"}],"predecessor-version":[{"id":363,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts\/347\/revisions\/363"}],"wp:attachment":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/media?parent=347"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/categories?post=347"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/tags?post=347"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/ppma_author?post=347"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}