PILD: Physics-Informed Learning via Diffusion

2026-01-29

AI Summary

Purely data-driven diffusion models often fail to satisfy the physical consistency required in scientific and engineering applications. To address this limitation, this work proposes the Physics-Informed Latent Diffusion (PILD) framework, which uniquely incorporates physical constraints into diffusion training by treating them as virtual residual observations drawn from a Laplace distribution. Furthermore, a multi-level conditional embedding module is introduced to deeply inject physical priors into every layer of the denoising network, enabling end-to-end physics-guided generation. The resulting approach is concise, modular, and highly generalizable. Extensive experiments across diverse tasks—including vehicle trajectory prediction, tire force modeling, Darcy flow, and plasma dynamics—demonstrate that PILD significantly outperforms existing physics-informed and diffusion-based baselines, achieving notable improvements in accuracy, stability, and generalization.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Learning to Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• diffusion models
• physics-informed learning
• physical constraints
• generative modeling
• scientific computing

Innovation

• Physics-Informed Learning
• Diffusion Models
• Virtual Residual Observation
• Conditional Embedding
• Physical Constraints