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PILD: Physics-Informed Learning via Diffusion

Diffusion models have emerged as powerful generative tools for modeling complex data distributions, yet their purely data-driven nature limits applicability in engineering and scientific problems where physical laws must be respected. This paper proposes Physics-Informed Learning via Diffusion (PILD), a framework that unifies diffusion modeling and physical constraints through a probabilistic residual formulation with a virtual residual observation sampled from a Laplace distribution. To make this formulation practical under noisy diffusion states, we introduce a Jensen-gap-aware adaptive residual scale, which reduces the bias induced by residual likelihood marginalization. Additionally, we develop a physics-conditional alignment mechanism for conditional tasks that encourages intermediate latent representations to remain consistent with the observation conditions during denoising. The proposed 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 show that PILD improves physical fidelity and predictive accuracy over representative physics-informed and diffusion-based baselines.

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