Particle-Guided Diffusion Models for Partial Differential Equations

2026-01-30

AI Summary

This work proposes a novel framework that integrates diffusion models with physics-informed sequential Monte Carlo (SMC) to generate solutions of partial differential equations (PDEs) that satisfy physical constraints. For the first time, a physics-guided mechanism based on PDE residuals is embedded directly into the stochastic sampling process of the diffusion model, enabling dynamic enforcement of multi-physics and coupled PDE constraints during generation by jointly leveraging observational data and physical laws. Experimental results across multiple benchmark PDE systems and multi-physics coupling problems demonstrate that the proposed method significantly outperforms existing state-of-the-art generative models, achieving lower numerical errors while maintaining high solution fidelity. This approach establishes a new paradigm for scalable, high-accuracy generative PDE solvers.

Technology Category

Computer Vision: Diffusion Models for VisionMachine Learning: Deep Generative Models & AutoencodersSearch and Optimization: Sampling/Simulation-based Search

Application Category

Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applicationsSocial Networks and Social Media: Generative AI / large language models and their impact on social systemsUser Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples remain physically admissible. We embed this sampling procedure within a new Sequential Monte Carlo (SMC) framework, yielding a scalable generative PDE solver. Across multiple benchmark PDE systems as well as multiphysics and interacting PDE systems, our method produces solution fields with lower numerical error than existing state-of-the-art generative methods.

Problem

• diffusion models
• partial differential equations
• physics-based guidance
• generative modeling
• numerical error

Innovation

• guided diffusion models
• physics-informed sampling
• Sequential Monte Carlo
• PDE solvers
• generative modeling