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arxiv: 2604.23867 · v1 · submitted 2026-04-26 · 💻 cs.LG

Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity

Valerie Tsao , Nathaniel Chaney , Manolis Veveakis This is my paper

Pith reviewed 2026-05-08 06:14 UTC · model grok-4.3

classification 💻 cs.LG
keywords latent diffusionPDE parameterizationsparse reconstructionsuper-resolutioninterpretable latent spacephysics-guided generative modelsstructured sparsity
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The pith

Directly setting latent variables to PDE coefficients and source terms lets a diffusion model reconstruct physical fields from sparse or low-resolution data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents LatentPDE, a diffusion framework for recovering scientific fields when observations suffer from noise, incomplete coverage, or limited resolution. It achieves this by constructing the latent space so that each variable directly equals the coefficients and source terms of an assumed governing partial differential equation. The diffusion process then generates fields that automatically obey the chosen physics without added penalty terms. A sympathetic reader would care because the resulting reconstructions remain reliable across widely varying patterns of missing data and can be produced at any target resolution.

Core claim

LatentPDE constructs an inherently interpretable latent space by directly parameterizing the latent variables as the coefficients and source terms of an assumed governing PDE. This allows the diffusion process to produce physically compliant reconstructions across highly disparate and structured data gaps, achieving high-fidelity recovery at any desired resolution while tracking the underlying predictive uncertainty.

What carries the argument

The direct parameterization of latent variables as the coefficients and source terms of a governing PDE, which enforces physical compliance inside the generative diffusion process.

If this is right

  • Reconstructions stay physically valid even when observations contain highly structured gaps.
  • The same model produces high-fidelity fields at any chosen resolution.
  • Predictive uncertainty is obtained as a direct output of the diffusion process.
  • The method works across diverse data configurations without relying on soft loss penalties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the chosen PDE form is incorrect for the true system, the model could still output smooth fields that appear physical but systematically deviate from reality.
  • The learned coefficients could be inspected after training to recover estimates of physical parameters from the data.
  • The same latent-parameterization idea might apply to other inverse problems where a governing equation is known but observations are incomplete.
  • Testing the framework on nonlinear or time-dependent PDEs would reveal how far the current linear-assumption construction generalizes.

Load-bearing premise

An appropriate governing PDE exists whose coefficients and source terms can be directly used to parameterize the latent space so that the resulting diffusion produces physically compliant reconstructions for arbitrary structured data gaps.

What would settle it

Reconstructed fields that violate the assumed governing PDE by amounts larger than numerical discretization error on held-out test cases with large structured gaps would show that physical compliance is not reliably enforced.

Figures

Figures reproduced from arXiv: 2604.23867 by Manolis Veveakis, Nathaniel Chaney, Valerie Tsao.

Figure 1
Figure 1. Figure 1: Schematic of LatentPDE framework. Sparse, noised, low-resolution observations and initial conditions (u lr 0 ) are encoded (via MAP or MLP) into a latent space of unknown PDE coefficients. A noising-denoising algorithm refines these parameters before a spectral solver decodes them, recovering full and high-resolution fields that strictly enforce governing physics. state-of-the-art methods on full reconstru… view at source ↗
Figure 2
Figure 2. Figure 2: Full-field reconstruction of PDEs from sparse (s = 5.0%) and noisy (σ = 0.15) observations. Note that LatentPDE is conditioned here on the low-resolution initial state (u0) shown, demonstrating simultaneous super-resolution and reconstruction functionality, whereas other baselines require a high-resolution u0 (not pictured). Despite this, LatentPDE consistently achieves the highest fidelity, reducing RMSE … view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of LatentPDE and FunDPS reconstructions. FunDPS was trained on the same 20000 samples of advection–diffusion data as our model. Error distributions and spectral alignment curves over 500 samples are shown on the left. Top and bottom row show different individual reconstructions of radial and grid mask respectively. EnKF methods, then report the one yielding the lowest Root Mean Square Error (RMS… view at source ↗
Figure 4
Figure 4. Figure 4: Radially averaged PSD across advection–diffusion sub￾regimes and mask configurations. LatentPDE matches the ground truth energy even at the sharp high-frequency cutoff, whereas the baselines suffer from spectral bias. To visually illustrate the spectral alignment with ground truth, view at source ↗
Figure 5
Figure 5. Figure 5: Ensemble mean and uncertainty quantification for Latent￾PDE across our three governing equations. The rightmost columns demonstrate that the generated uncertainty standard deviation maps spatially correlate to regions of elevated absolute error view at source ↗
Figure 6
Figure 6. Figure 6: Metric distributions for RMSE and PSD for all sub￾regimes of the advection–diffusion equation. These high-uncertainty regions coincide qualitatively with larger absolute errors, indicating that the posterior ensemble captures meaningful localized failure modes under sparse conditioning. Beyond these spatial insights, encoding physi￾cal coefficients into our latents provides a restriction upon the solution … view at source ↗
Figure 7
Figure 7. Figure 7: Representative initial conditions used in our study. These were chosen to provide challenging initial states with diverse spatial regularity and frequency content. Initial conditions. Initial conditions are generated in the following four ways: broadband random Fourier field, sharp transition fronts, localized vortex dipoles, and multiscale Gaussian-Fourier fields. All initial conditions are standardized t… view at source ↗
Figure 8
Figure 8. Figure 8: Sample masks of each type at a fixed 5.0% sparsity. The top row denotes the masks used for training and the bottom row denotes the masks used during inference. 15 view at source ↗
Figure 9
Figure 9. Figure 9: Monte Carlo convergence of the LatentPDE posterior mean as a function of the number of diffusion ensemble members K. Empirical errors are computed by subsampling posterior samples and comparing the resulting ensemble mean to the full pilot-ensemble mean, while theoretical curves use the estimate tr(Σ) b /(dK) from the posterior covariance, as described by Equation 56. F.2. Partitioning the Dataset For each… view at source ↗
Figure 10
Figure 10. Figure 10: Monte Carlo convergence of the EnKF analysis mean as a function of the number of ensemble members K. We choose K = 16 for where the error ϵ drops below 0.01. For a more detailed explanation of how this number was derived, refer to Section F.1. 19 view at source ↗
Figure 11
Figure 11. Figure 11: Reconstructions of the advection–diffusion equation under increasing observation noise (σ) at a fixed 5.0% sparsity. s = 5.0%, = 0.15 Sparse Observations u0 Ground Truth RMSE = 0.2231 LatentPDE RMSE = 0.2994 EnKF RMSE = 0.3155 3DVar RMSE = 0.5509 PINN s = 5.0%, = 0.30 RMSE = 0.1839 RMSE = 0.6082 RMSE = 0.6260 RMSE = 0.9689 s = 5.0%, = 0.45 RMSE = 0.7691 RMSE = 0.9091 RMSE = 0.9488 RMSE = 1.0006 1.5 1.0 0.… view at source ↗
Figure 12
Figure 12. Figure 12: Reconstructions of the Klein–Gordon equation under increasing observation noise (σ) at a fixed 5.0% sparsity 20 view at source ↗
Figure 13
Figure 13. Figure 13: Reconstructions of the Helmholtz equation under increasing observation noise (σ) at a fixed 5.0% sparsity 21 view at source ↗
Figure 14
Figure 14. Figure 14: Reconstructions of the advection–diffusion equation under diverse mask configurations (grid, boundary, radial, and single patch). For all realizations here, σ is fixed at 0.15 and sparsity at 5.0%. 22 view at source ↗
Figure 15
Figure 15. Figure 15: Reconstructions of the Klein–Gordon equation under diverse mask configurations (grid, boundary, radial, and single patch). For all realizations here, σ is fixed at 0.15 and sparsity at 5.0%. s = 5.0%, = 0.15 Sparse Observations u0 Ground Truth RMSE = 0.0075 LatentPDE RMSE = 0.0264 EnKF RMSE = 0.0109 3DVar RMSE = 0.0546 PINN s = 5.0%, = 0.15 RMSE = 0.0045 RMSE = 0.0289 RMSE = 0.0104 RMSE = 0.0178 s = 5.0%,… view at source ↗
Figure 16
Figure 16. Figure 16: Reconstructions of the Helmholtz equation under diverse mask configurations (grid, boundary, radial, and single patch). For all realizations here, σ is fixed at 0.15 and sparsity at 5.0%. 23 view at source ↗
Figure 17
Figure 17. Figure 17: Qualitative comparison of reconstructions for advection–diffusion at 5 levels of sparsity (from 1% to 5%) for the single-patch mask. Noise (σ) is fixed at 0.15. 24 view at source ↗
Figure 18
Figure 18. Figure 18: Qualitative comparison of reconstructions for Klein–Gordon at 5 levels of sparsity (from 1% to 5%) for the single-patch mask. Noise (σ) is fixed at 0.15. s = 1.0%, = 0.15 Sparse Observations u0 Ground Truth RMSE = 0.0245 LatentPDE RMSE = 0.0324 EnKF RMSE = 0.0317 3DVar RMSE = 0.1018 PINN s = 2.0%, = 0.15 RMSE = 0.0133 RMSE = 0.0504 RMSE = 0.0477 RMSE = 0.0691 s = 3.0%, = 0.15 RMSE = 0.0103 RMSE = 0.0195 R… view at source ↗
Figure 19
Figure 19. Figure 19: Qualitative comparison of reconstructions for Helmholtz at 5 levels of sparsity (from 1% to 5%) for the single-patch mask. Noise (σ) is fixed at 0.15. 25 view at source ↗
Figure 20
Figure 20. Figure 20: Random realizations generated by LatentPDE for the different equation scenarios at σ = 0.15 and s = 5.0%. This visualization is meant to show a subset of the ensemble average that we use to report our results. K. Ablation Study Our ablation study evaluates three inference-time modifications to the standard diffusion model setup. We follow standard practice and compare variants using RMSE under the same do… view at source ↗
read the original abstract

Scientific measurements are often bottlenecked by suboptimal conditions, whether that be noise, incomplete spatial coverage, or limited resolution, rendering accurate field reconstruction a difficult task. We introduce LatentPDE, a latent diffusion framework designed to simultaneously resolve sparse-observation reconstruction and super-resolution. While existing physics-guided diffusion models typically rely on soft loss penalties or uninterpretable representations, our approach enforces physical compliance by constructing an inherently interpretable latent space. Specifically, we parameterize the latent variables directly as the coefficients and source terms of an assumed governing PDE. In doing so, LatentPDE is able to reliably reconstruct dynamics across highly disparate and structured data gaps. Empirical results on diverse configurations demonstrate that our model achieves high-fidelity recovery at any desired resolution while also tracking the underlying predictive uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript introduces LatentPDE, a latent diffusion framework for simultaneous sparse-observation reconstruction and super-resolution of fields governed by PDEs. The core innovation is to parameterize the latent variables directly as the coefficients and source terms of an assumed governing PDE, thereby enforcing physical compliance in an interpretable manner rather than via soft penalties. The authors claim this enables reliable high-fidelity recovery across highly disparate structured data gaps at arbitrary resolutions while also tracking predictive uncertainty.

Significance. If the central claims can be substantiated with quantitative evidence, the work would represent a meaningful step toward inherently interpretable physics-informed generative models. Directly embedding PDE coefficients in the latent space could improve generalization and trustworthiness compared with penalty-based or black-box alternatives, particularly for scientific applications involving structured sparsity and uncertainty quantification.

major comments (3)
  1. [Abstract] Abstract: The assertions of 'high-fidelity recovery' and 'reliable reconstruction' across structured data gaps are made without any quantitative metrics, baselines, error bars, ablation studies, or comparisons to existing methods. The results section must supply these to support the central empirical claims.
  2. [Methods] Methods (latent parameterization): The approach assumes the data are generated by (or well-approximated by) the exact PDE structure chosen by the authors. If this form is misspecified (e.g., missing nonlinear terms), the diffusion over coefficients can still satisfy the training objective while producing non-compliant reconstructions in the gaps; a sensitivity analysis to PDE misspecification is required.
  3. [Methods] Methods (diffusion loop): No description is given of how forward PDE solutions are computed inside the diffusion process or how consistency between sampled coefficients and the sparse observations is enforced at every scale. Without this, it is unclear whether the generative model is constrained to physically valid parameter combinations.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it briefly named the specific PDEs and source terms used in the empirical demonstrations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight important areas for strengthening the empirical support and methodological clarity. We address each major comment below, indicating revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The assertions of 'high-fidelity recovery' and 'reliable reconstruction' across structured data gaps are made without any quantitative metrics, baselines, error bars, ablation studies, or comparisons to existing methods. The results section must supply these to support the central empirical claims.

    Authors: We agree that the abstract would be strengthened by including quantitative support for the claims. The results section (Sections 4 and 5) already provides comprehensive quantitative metrics, baselines, error bars, ablation studies, and comparisons to existing methods across multiple PDEs and sparsity patterns. To directly address the concern, we have revised the abstract to highlight key quantitative findings, such as average reconstruction errors and uncertainty calibration metrics from the experiments. revision: yes

  2. Referee: [Methods] Methods (latent parameterization): The approach assumes the data are generated by (or well-approximated by) the exact PDE structure chosen by the authors. If this form is misspecified (e.g., missing nonlinear terms), the diffusion over coefficients can still satisfy the training objective while producing non-compliant reconstructions in the gaps; a sensitivity analysis to PDE misspecification is required.

    Authors: This is a fair point regarding the core assumption. LatentPDE is explicitly designed for scenarios where the governing PDE structure is known and correctly specified, consistent with many physics-informed approaches. To demonstrate robustness, we have added a new sensitivity analysis subsection in the revised manuscript. This includes experiments where the assumed PDE omits small nonlinear terms or has perturbed coefficients, showing graceful degradation in reconstruction quality while maintaining advantages over non-physics baselines. revision: yes

  3. Referee: [Methods] Methods (diffusion loop): No description is given of how forward PDE solutions are computed inside the diffusion process or how consistency between sampled coefficients and the sparse observations is enforced at every scale. Without this, it is unclear whether the generative model is constrained to physically valid parameter combinations.

    Authors: We apologize for the insufficient detail in the original submission. The latent diffusion operates on the PDE coefficients and source terms. During each denoising step, the current coefficient estimates are used to numerically solve the forward PDE (via finite differences for regular domains or finite elements for irregular ones) to produce the field at the target resolution. Consistency with sparse observations is enforced by a scale-aware conditioning loss that penalizes mismatches between the PDE-solved field and the available measurements, integrated into the diffusion training objective at multiple noise levels. We have substantially expanded the Methods section with a new subsection, detailed equations, and an algorithm box with pseudocode to fully specify this process. revision: yes

Circularity Check

0 steps flagged

No circularity: parameterization is an explicit modeling assumption, not a self-referential derivation

full rationale

The abstract describes a deliberate design choice to represent latent variables as coefficients and source terms of an assumed governing PDE. This is presented as an input modeling decision that enables the diffusion process to operate in an interpretable space, rather than a result derived from the data or from prior steps within the paper. No equations, self-citations, or fitted-parameter renamings are provided in the available text that would reduce the claimed physical compliance to a tautology or to the training observations by construction. The approach remains self-contained as a generative framework whose validity rests on the external correctness of the assumed PDE form and the training procedure, neither of which is shown to collapse into the outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the existence of a suitable governing PDE for the target dynamics and on the premise that its coefficients and source terms can serve as an interpretable and sufficient latent representation.

free parameters (1)
  • PDE coefficients and source terms
    These become the latent variables and are therefore learned or optimized during training on the reconstruction task.
axioms (1)
  • domain assumption A governing PDE exists for the observed dynamics and can be assumed a priori
    The method requires that the latent space be defined in terms of this PDE; the abstract does not specify how the PDE form is chosen.

pith-pipeline@v0.9.0 · 5430 in / 1104 out tokens · 61089 ms · 2026-05-08T06:14:24.038688+00:00 · methodology

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Reference graph

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