Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity

TL;DR

LatentPDE is a physics-guided diffusion model that constructs an interpretable latent space based on PDE coefficients, enabling accurate, high-resolution reconstructions from sparse and incomplete scientific data.

Contribution

We introduce LatentPDE, a novel framework that enforces physical interpretability in diffusion models for improved reconstruction and super-resolution of scientific measurements.

Findings

Achieves high-fidelity reconstruction across diverse data gaps

Successfully tracks predictive uncertainty

Enforces physical compliance through PDE-based latent space

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.