Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems

TL;DR

This paper introduces a method to fine-tune flow-matching generative models using physical constraints from PDEs, enabling accurate inverse problem solving and physical consistency in scientific modeling.

Contribution

It proposes a differentiable post-training procedure that enforces physical laws in generative models and jointly infers unknown parameters, enhancing scientific inference capabilities.

Findings

Improved satisfaction of PDE constraints in generated solutions

Accurate recovery of hidden physical parameters

Effective handling of ill-posed inverse problems

Abstract

We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems. Starting from a model trained on low-fidelity or observational data, we apply a differentiable post-training procedure that minimizes weak-form residuals of governing partial differential equations (PDEs), promoting physical consistency and adherence to boundary conditions without distorting the underlying learned distribution. To infer unknown physical inputs, such as source terms, material parameters, or boundary data, we augment the generative process with a learnable latent parameter predictor and propose a joint optimization strategy. The resulting model produces physically valid field solutions alongside plausible estimates of hidden parameters, effectively addressing ill-posed inverse problems in a data-driven yet physicsaware manner. We validate our method on canonical PDE benchmarks, demonstrating improved satisfaction of PDE constraints and accurate recovery of latent coefficients. Our approach bridges generative modelling and scientific inference, opening new avenues for simulation-augmented discovery and data-efficient modelling of physical systems.