Ambient Physics: Training Neural PDE Solvers with Partial Observations

TL;DR

Ambient Physics introduces a novel framework for training neural PDE solvers using only partial observations, enabling accurate reconstructions without complete data and significantly reducing computational costs.

Contribution

It presents a new method that learns from partial observations by masking measurements, outperforming previous diffusion-based approaches in PDE reconstruction tasks.

Findings

Achieves 62.51% reduction in average error compared to prior methods.

Uses 125 times fewer function evaluations than existing diffusion-based models.

Enables learning from partial observations across different architectures and measurement patterns.

Abstract

In many scientific settings, acquiring complete observations of PDE coefficients and solutions can be expensive, hazardous, or impossible. Recent diffusion-based methods can reconstruct fields given partial observations, but require complete observations for training. We introduce Ambient Physics, a framework for learning the joint distribution of coefficient-solution pairs directly from partial observations, without requiring a single complete observation. The key idea is to randomly mask a subset of already-observed measurements and supervise on them, so the model cannot distinguish "truly unobserved" from "artificially unobserved", and must produce plausible predictions everywhere. Ambient Physics achieves state-of-the-art reconstruction performance. Compared with prior diffusion-based methods, it achieves a 62.51$\%$ reduction in average overall error while using 125$\times$ fewer function evaluations. We also identify a "one-point transition": masking a single already-observed point enables learning from partial observations across architectures and measurement patterns. Ambient Physics thus enables scientific progress in settings where complete observations are unavailable.