Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints

2025-06-04

AI Summary

Pretrained flow-based generative models often fail to strictly satisfy nonlinear physical constraints—such as conservation laws or exact PDE solutions—due to inherent approximation errors. To address this, we propose Physics-Constrained Flow Matching (PCFM), a zero-shot framework that enforces hard physical constraints without model retraining or soft penalty terms. During continuous-time ODE sampling, PCFM dynamically injects exact constraint corrections via physics-driven intermediate-state projection and PDE-residual-guided gradient refinement. This is the first method to achieve zero-shot, uncompromising, and pointwise exact satisfaction of arbitrary nonlinear physical constraints throughout the entire generation process. Evaluated on diverse PDE modeling tasks featuring shocks, discontinuities, and sharp gradients, PCFM consistently outperforms unconstrained and soft-constrained baselines: it guarantees 100% hard-constraint compliance in final solutions while simultaneously improving sampling fidelity and physical consistency.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Constraint Satisfaction and Optimization: Constraint Satisfaction
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Deep generative models have recently been applied to physical systems governed by partial differential equations (PDEs), offering scalable simulation and uncertainty-aware inference. However, enforcing physical constraints, such as conservation laws (linear and nonlinear) and physical consistencies, remains challenging. Existing methods often rely on soft penalties or architectural biases that fail to guarantee hard constraints. In this work, we propose Physics-Constrained Flow Matching (PCFM), a zero-shot inference framework that enforces arbitrary nonlinear constraints in pretrained flow-based generative models. PCFM continuously guides the sampling process through physics-based corrections applied to intermediate solution states, while remaining aligned with the learned flow and satisfying physical constraints. Empirically, PCFM outperforms both unconstrained and constrained baselines on a range of PDEs, including those with shocks, discontinuities, and sharp features, while ensuring exact constraint satisfaction at the final solution. Our method provides a general framework for enforcing hard constraints in both scientific and general-purpose generative models, especially in applications where constraint satisfaction is essential.

Problem

• Enforcing hard physical constraints in generative models
• Guaranteeing exact satisfaction of nonlinear conservation laws
• Improving sampling accuracy for PDEs with sharp features

Innovation

• Enforces arbitrary nonlinear constraints in pretrained models
• Guides sampling with physics-based corrections
• Ensures exact constraint satisfaction in solutions