Particle-Guided Diffusion Models for Partial Differential Equations

Generative AI & LLMs

Published: arXiv: 2601.23262v1

Authors

Andrew Millard Fredrik Lindsten Zheng Zhao

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.

Paper Summary

Problem

Partial differential equations (PDEs) are a fundamental tool in science and engineering, but solving them can be computationally expensive and challenging, especially when dealing with large parameter spaces or complex dynamics. Current numerical solvers are often slow and impractical for real-time or uncertainty-aware applications.

Key Innovation

The paper presents a new approach to solving PDEs using a guided stochastic sampling method, which combines the strengths of diffusion models and physics-based guidance. This method, called Particle-Guided Diffusion Models, uses a new Sequential Monte Carlo (SMC) framework to generate solution fields that are physically admissible and accurate.

Practical Impact

The proposed method has the potential to revolutionize the way PDEs are solved, enabling faster and more accurate simulations in various fields, such as fluid dynamics, heat transport, and electromagnetics. This could lead to breakthroughs in fields like climate modeling, materials science, and medical imaging.

Analogy / Intuitive Explanation

Imagine trying to solve a puzzle with millions of pieces, where each piece represents a tiny part of a complex system. Traditional numerical solvers are like trying to solve the puzzle by looking at each piece individually, which can be slow and error-prone. The Particle-Guided Diffusion Models approach is like using a powerful AI assistant that can look at the entire puzzle at once, making educated guesses about the missing pieces and filling them in with high accuracy.

Paper Information

Categories:

cs.LG

Published Date:

arXiv ID:

2601.23262v1

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