Valerie Tsao | Searcharxiv

arXiv2026

Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity

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.

cs.LG↗

arXiv2025

Probabilistic Spatial Interpolation of Sparse Data using Diffusion Models

The large underlying assumption of climate models today relies on the basis of a "confident" initial condition, a reasonably plausible snapshot of the Earth for which all future predictions depend on. However, given the inherently chaotic nature of our system, this assumption is complicated by sensitive dependence, where small uncertainties in initial conditions can lead to exponentially diverging outcomes over time. This challenge is particularly salient at global spatial scales and over centennial timescales, where data gaps are not just common but expected. The source of uncertainty is two-fold: (1) sparse, noisy observations from satellites and ground stations, and (2) internal variability stemming from the simplifying approximations within the models themselves. In practice, data assimilation methods are used to reconcile this missing information by conditioning model states on partial observations. Our work builds on this idea but operates at the extreme end of sparsity. We propose a conditional data imputation framework that reconstructs full temperature fields from as little as 1% observational coverage. The method leverages a diffusion model guided by a prekriged mask, effectively inferring the full-state fields from minimal data points. We validate our framework over the Southern Great Plains, focusing on afternoon (12:00-6:00 PM) temperature fields during the summer months of 2018-2020. Across varying observational densities--from swath data to isolated in-situ sensors--our model achieves strong reconstruction accuracy, highlighting its potential to fill in critical data gaps in both historical reanalysis and real-time forecasting pipelines.

stat.AP↗