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

TL;DR

This paper introduces PCFM, a novel framework for enforcing hard physical constraints in flow-based generative models, improving simulation accuracy for PDE-governed systems with complex features.

Contribution

We propose a zero-shot physics-constrained sampling method that guarantees arbitrary nonlinear constraints in pretrained flow models, addressing limitations of soft penalty approaches.

Findings

Outperforms baseline methods on PDEs with shocks and discontinuities

Ensures exact satisfaction of physical constraints at the final solution

Effective in both scientific and general-purpose generative modeling

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 flexible framework for enforcing hard constraints in both scientific and general-purpose generative models, especially in applications where constraint satisfaction is essential.