Ambient Physics: Training Neural PDE Solvers with Partial Observations

TL;DR

Ambient Physics is introduced, a framework for learning the joint distribution of coefficient-solution pairs directly from partial observations, without requiring a single complete observation, and identifies a "one-point transition": masking a single already-observed point enables learning from partial observations across 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.

Figures (13)

• Figure 1: Examples of Training Data used by Existing Methods vs Ambient Physics.Top: Existing diffusion-based methods require access to complete observations to learn the joint distribution of coefficient-solution pairs. Bottom: Ambient Physics learns the joint distribution directly from partial observations, without requiring a single complete observation. Left to Right: Darcy flow with point observations, Helmholtz with patch observations, Navier-Stokes with slice observations, and Poisson with window observations. Within each panel, the left field represents the coefficients (or initial condition) and the right field represents the solution (or final state).
• Figure 2: Examples of Ambient FNO. Reconstructions From Partial Observations.Top to Bottom: Ground truth coefficients and solution $(a, u)$, partial observations ($A_a a, A_u u$) with 3% uniformly random measurements, Ambient FNO reconstructed coefficients and solutions $(\hat{a}, \hat{u})$, Ambient FNO $+$Joint Dist. Modeling reconstructed coefficients and solutions $(\hat{a}, \hat{u})$. Left: Darcy flow. Right: Navier-Stokes. Within each panel, the left field represents the coefficients (or initial condition) and the right field represents the solution (or final state).
• Figure 3: Naive Training vs Ambient Physics Predictions from Partial Observations.Top: Naively training on partial observations yields a model that predicts accurately only at observed locations and arbitrarily elsewhere. Bottom: Ambient Physics, trained on the same partial observations, yields a model that predicts physically-plausible fields everywhere. Left to Right: Darcy flow with point observations, Helmholtz with patch observations, Navier-Stokes with slice observations, and Poisson with window observations. Within each panel, the left field represents the coefficients (or initial condition) and the right field represents the solution (or final state).
• Figure 4: One-Point Transition to Ambient Physics. Relative $L_2$ error (log scale) versus the number of additional masked already-observed points used during training, across Darcy flow, Helmholtz, Navier-Stokes, and Poisson. Left: Coefficient error. Right: Solution error. The point at 0 additional masked points corresponds to naive training ($B=I$); masking even a single already-observed point leads to a $\sim 10$--$100\times$ reduction in error.
• Figure 5: Example of "One-Point Transition" (Darcy Flow). Each subplot shows an Ambient Flow reconstruction from partial observations (3% uniformly random points), varying the number of additional masked (already-observed) points used during training (increasing across the grid; top-left: 0, which corresponds to naive training). Within Each Subplot:Top: Solution field. Bottom: Coefficient Field. Left to Right: Input Fields, Target Fields, Reconstructed Fields, Error (Log Scale). Within each input panel, gray pixels represent unobserved locations and green pixels represent masked measurements withheld from the model.