Ambient Physics: Training Neural PDE Solvers with Partial Observations

Abstract

In many scientific contexts, obtaining complete observations of PDE coefficients and solutions proves expensive, hazardous, or unfeasible. Recent diffusion-based approaches can reconstruct fields from incomplete data, yet require fully observed examples during training. We present Ambient Physics, a framework enabling models to learn joint distributions of coefficient-solution pairs using only partial observations, without needing any complete training examples. Our key insight involves randomly masking already-observed measurements during training and supervising predictions on them. This prevents the model from distinguishing between genuinely unobserved and artificially masked locations, forcing it to generate physically plausible predictions across entire domains. Results demonstrate state-of-the-art performance: a 62.51% reduction in average overall error while using 125x fewer function evaluations compared to prior diffusion approaches, despite training exclusively on partial observations. The framework is architecture-agnostic, and we identify a one-point transition where masking a single observed point substantially improves learning from partial observations.

Please cite this paper as:

@article{majid2026ambient,
  title={Ambient Physics: Training Neural PDE Solvers with Partial Observations},
  author={Abdul Majid, Harris and Daras, Giannis and Tudisco, Francesco and McDonagh, Steven},
  journal={arXiv preprint arXiv:2602.13873},
  year={2026}
}

Keywords: neural PDE solvers partial observations diffusion models AI4Science deep learning