# Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity

arXiv: https://arxiv.org/abs/2604.23867 · HTML: https://mlsanity.com/paper/2604.23867/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 15 of its 41 references that are in the map.
- First posted on arXiv 26 Apr 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 15 of 15)

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- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
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- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [Robust Neural Posterior Estimation and Statistical Model Criticism](https://mlsanity.com/paper/2210.06564/) (2022)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [The Hidden Language of Diffusion Models](https://mlsanity.com/paper/2306.00966/) (2023)
- [An operator learning perspective on parameter-to-observable maps](https://mlsanity.com/paper/2402.06031/) (2024)
- [HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions](https://mlsanity.com/paper/2304.02811/) (2023)
- [Inferencing Based on Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1803.02627/) (2018)

## Abstract (arXiv)

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.