Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints
This addresses the challenge of guaranteeing physical constraints in generative models for scientific applications, though it is incremental as it builds on existing flow-based methods.
The paper tackled the problem of enforcing hard physical constraints in deep generative models for PDE-governed systems, proposing Physics-Constrained Flow Matching (PCFM) to guide sampling with physics-based corrections, which outperformed baselines and ensured exact constraint satisfaction.
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.