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

2025-08-05

AI Summary

This work addresses the challenge of simultaneously satisfying physical constraints and preserving data distribution fidelity in flow-matching generative models for scientific computing. Methodologically, it introduces a physics-informed post-training fine-tuning framework that embeds the weak-form residual of partial differential equations (PDEs) into a differentiable optimization objective—enhancing physical consistency without distorting the original distribution—and couples a learnable latent variable predictor to jointly optimize field solutions and unknown physical parameters (e.g., source terms, material properties, or boundary conditions). It is the first approach to unify flow-matching generative modeling with latent-parameter joint inverse inference within a single paradigm, enabling injection of physics priors without retraining. Evaluated on canonical PDE benchmarks, the method significantly improves PDE residual accuracy and parameter inversion fidelity, achieving both high-fidelity generation and robustness to ill-posed inverse problems—thereby advancing data-efficient, interpretable scientific discovery.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Enforce physical constraints in flow-matching generative models
• Solve inverse problems with unknown physical inputs
• Improve PDE constraint satisfaction and latent coefficient recovery

Innovation

• Fine-tune flow-matching models with physics constraints
• Differentiable weak-form PDE residuals for physical consistency
• Latent parameter predictor for inverse problem solving