用物理方程约束生成模型，让AI更准地解科学问题。
Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems
- 训练后微调生成模型，使其满足偏微分方程和边界条件。
- 能同时恢复隐藏参数（如源项、材料属性）和生成物理合理解。
- 适合需要数据少且依赖物理规律的科学建模任务。
我们提出一种框架，用于对流匹配生成模型进行物理约束微调，以解决科学系统中的正演与逆问题。从低保真度或观测数据训练的模型出发，采用可微分的后训练过程，最小化控制偏微分方程（PDE）的弱形式残差，从而在不扭曲原始学习分布的前提下，提升物理一致性与边界条件遵守能力。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变量预测器，并提出联合优化策略。所获模型不仅能生成物理合理的场解，还能给出隐藏参数的合理估计，以数据驱动且物理感知的方式有效解决病态逆问题。我们在典型PDE基准上验证了该方法，结果表明其显著提升了对PDE约束的满足程度，并准确恢复了潜在系数。本方法连接生成建模与科学推断，为模拟增强发现和高效物理系统建模开辟新路径。
原文摘要 · Abstract (English)
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.
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