PILD: Physics-Informed Learning via Diffusion

TL;DR

PILD introduces a physics-informed diffusion framework that integrates physical laws into generative models, enhancing accuracy and stability in scientific and engineering applications.

Contribution

It unifies diffusion modeling with physical constraints using residual supervision and a conditional embedding module, enabling broad applicability to various physics-based problems.

Findings

Significantly improves accuracy over baselines

Enhances stability and generalization

Applicable to diverse scientific tasks

Abstract

Diffusion models have emerged as powerful generative tools for modeling complex data distributions, yet their purely data-driven nature limits applicability in practical engineering and scientific problems where physical laws need to be followed. This paper proposes Physics-Informed Learning via Diffusion (PILD), a framework that unifies diffusion modeling and first-principles physical constraints by introducing a virtual residual observation sampled from a Laplace distribution to supervise generation during training. To further integrate physical laws, a conditional embedding module is incorporated to inject physical information into the denoising network at multiple layers, ensuring consistent guidance throughout the diffusion process. The proposed PILD framework is concise, modular, and broadly applicable to problems governed by ordinary differential equations, partial differential equations, as well as algebraic equations or inequality constraints. Extensive experiments across engineering and scientific tasks including estimating vehicle trajectories, tire forces, Darcy flow and plasma dynamics, demonstrate that our PILD substantially improves accuracy, stability, and generalization over existing physics-informed and diffusion-based baselines.