Training Neural PDE Solvers with Partial Observations

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. Unlike self-supervised methods, which create supervision by masking complete observations and then predicting the held-out content; Ambient Physics randomly masks a subset of already-observed measurements and supervises only where measurements exist, so the model cannot distinguish "truly unobserved" from "artificially unobserved", and must produce plausible predictions everywhere. Compared with prior diffusion-based methods, Ambient Physics achieves state-of-the-art reconstruction performance: a 61.82% reduction in average overall error while using 500x fewer function evaluations. We provide theoretical justification for learning directly from partial observations and identify a "one-measurement transition": masking a single already-observed measurement enables learning from partial observations across architectures and measurement patterns. We also extend our empirical evaluation to super-resolution tasks and real-world ERA5 weather fields. Ambient Physics thus enables scientific progress in settings where complete observations are unavailable.