FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling

2025-06-09

AI Summary

This work addresses the challenge of generating continuous field functions for physics-based modeling. Methodologically, it introduces the first diffusion model framework explicitly defined and trained in an infinite-dimensional function space: (i) it generalizes diffusion processes to function spaces; (ii) it designs a function autoencoder coupled with implicit diffusion, enabling evaluation at arbitrary continuous spatial locations and handling of arbitrarily discretized inputs; and (iii) it incorporates physics-constrained network architectures and physics-informed loss functions that intrinsically encode conservation laws, symmetries, and other physical priors. Theoretically, it establishes minimax-optimal guarantees for density estimation in function spaces. Experiments demonstrate high-fidelity, physically consistent continuous field generation across fluid and solid mechanics tasks, with robustness to input noise and low-resolution data. The code and datasets are publicly released.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Recent advances in generative modeling -- particularly diffusion models and flow matching -- have achieved remarkable success in synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. Here, we introduce $ extbf{FunDiff}$, a novel framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generate continuous functions evaluable at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics-informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, showing that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions. We demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results show that our method generates physically consistent samples with high fidelity to the target distribution and exhibits robustness to noisy and low-resolution data. Code and datasets are publicly available at https://github.com/sifanexisted/fundiff.

Problem

• Adapting diffusion models to continuous physical functions
• Generating continuous functions with physical constraints
• Ensuring physical consistency in generative modeling

Innovation

• Latent diffusion process in function spaces
• Function autoencoder for varying discretizations
• Physics-informed loss for physical consistency