Ambient Physics: Training Neural PDE Solvers with Partial Observations

2026-02-14

AI Summary

This work addresses the challenge in scientific computing where coefficients and solutions of partial differential equations (PDEs) are often only partially observable, while existing neural solvers rely on fully observed data. The authors propose Ambient Physics, a novel framework that, for the first time, learns the joint distribution of coefficient–solution pairs from partial observations alone—without requiring any complete samples. Its core innovation is a stochastic masking strategy: by masking observed points during training and applying supervision, the model becomes invariant to whether missing regions are naturally unobserved or artificially masked, enabling globally consistent predictions. Remarkably, the authors identify a “single-point transition” phenomenon, where masking just one point suffices to drive effective learning across diverse architectures. Integrated with a diffusion-based solver, the method achieves state-of-the-art performance, reducing average total error by 62.51% and decreasing function evaluations by 125× compared to existing approaches.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• partial observations
• PDE solvers
• incomplete data
• scientific simulation
• neural PDE

Innovation

• Ambient Physics
• partial observations
• neural PDE solvers
• masked supervision
• diffusion-based methods