PILD: Physics-Informed Learning via Diffusion

TL;DR

Diffusion models excel at learning complex distributions but often violate governing physical laws in engineering and scientific problems. PILD combines diffusion-based generation with first-principles physics by introducing a virtual residual observation $\hat{\boldsymbol{r}} \sim \text{Laplace}(\hat{\boldsymbol{r}}; \mathcal{R}(\boldsymbol{x}_0), \sigma\mathbf{I})$ and a conditional embedding module to inject physical information across timesteps, yielding a joint objective that preserves data fidelity while enforcing physics. The approach demonstrates improved accuracy, stability, and generalization across vehicle tracking, tire forces, Darcy flow, and plasma dynamics, with ablations validating the benefits of Laplace residuals and deep conditioning. By enabling physically consistent, conditional generation and uncertainty quantification via DDIM sampling, PILD broadens diffusion-model applicability to physics-driven engineering and scientific domains. The framework offers a principled, modular path to integrating complex physical constraints into probabilistic generative modeling.

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.

Figures (14)

• Figure 1: Overview of our PILD (Physics-Informed Learning via Diffusion) framework: The current state $\boldsymbol{x_t}$ is fed into the network, while the current timestep $t$ and condition $O$ undergo conditional embedding via the U-FiLM/U-Att module. An estimation $\boldsymbol{x}_0^*$ of original value is obtained by DDIM denoising. Modeling the virtual residual observable $\boldsymbol{\hat{r}}$ with Laplace distribution, we train the network under a unified loss with physics information.
• Figure 2: U-FiLM and U-Att condition embedding structure.
• Figure 3: Experiment results on tracking tasks.
• Figure 4: Samples of tire force estimation experiments on PINN, DDPM and PILD (Ours).
• Figure 5: Evaluation of residual error and data loss of Darcy flow.

Theorems & Definitions (2)

• Proposition 1.1
• proof