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.

翻译：本文提出了一种微调流匹配生成模型的框架，旨在强化物理约束并解决科学系统中的反问题。该方法从基于低精度或观测数据训练的模型出发，应用一种可微分的训练后处理程序，通过最小化控制偏微分方程（PDE）的弱形式残差，在避免扭曲已学习底层分布的前提下，提升物理一致性并满足边界条件。为推断未知物理输入（如源项、材料参数或边界数据），我们在生成过程中引入可学习的隐式参数预测器，并提出一种联合优化策略。所得模型能够生成物理有效的场解，同时提供隐藏参数的合理估计，从而以数据驱动且物理感知的方式有效处理不适定反问题。我们在经典PDE基准测试中验证了所提方法，证明了其在PDE约束满足度与隐式系数准确恢复方面的改进。本工作搭建了生成建模与科学推断之间的桥梁，为仿真增强的物理系统发现与数据高效建模开辟了新途径。