Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints

TL;DR

This paper introduces Physics-Based Flow Matching (PBFM), a novel training method that enforces physical constraints in generative models without sacrificing distributional accuracy, achieving Pareto-optimal trade-offs in physics-constrained generative tasks.

Contribution

PBFM is the first method to explicitly address the conflicting objectives of physical and distributional accuracy by enforcing constraints during training with conflict-free updates and unrolling.

Findings

Achieves Pareto-optimal trade-offs between physical and distributional accuracy.

Maintains inference speed while enforcing physical constraints.

Generalizes well across various physics-constrained generative tasks.

Abstract

Physics-constrained generative modeling aims to produce high-dimensional samples that are both physically consistent and distributionally accurate, a task that remains challenging due to often conflicting optimization objectives. Recent advances in flow matching and diffusion models have enabled efficient generative modeling, but integrating physical constraints often degrades generative fidelity or requires costly inference-time corrections. Our work is the first to recognize the trade-off between distributional and physical accuracy. Based on the insight of inherently conflicting objectives, we introduce Physics-Based Flow Matching (PBFM) a method that enforces physical constraints at training time using conflict-free gradient updates and unrolling to mitigate Jensen's gap. Our approach avoids manual loss balancing and enables simultaneous optimization of generative and physical objectives. As a consequence, physics constraints do not impede inference performance. We benchmark our method across three representative PDE benchmarks. PBFM achieves a Pareto-optimal trade-off, competitive inference speed, and generalizes to a wide range of physics-constrained generative tasks, providing a practical tool for scientific machine learning. Code and datasets available at https://github.com/tum-pbs/PBFM.