Generative machine learning methods, such as diffusion models and flow matching, have shown great potential in modeling complex system behaviors and building efficient surrogate models. However, these methods typically learn the underlying physics implicitly from data. We propose Physics-Based Flow Matching (PBFM), a novel generative framework that explicitly embeds physical constraints, both PDE residuals and algebraic relations, into the flow matching objective. We also introduce temporal unrolling at training time that improves the accuracy of the final, noise-free sample prediction. Our method jointly minimizes the flow matching loss and the physics-based residual loss without requiring hyperparameter tuning of their relative weights. Additionally, we analyze the role of the minimum noise level, $σ_{\min}$, in the context of physical constraints and evaluate a stochastic sampling strategy that helps to reduce physical residuals. Through extensive benchmarks on three representative PDE problems, we show that our approach yields up to an $8\times$ more accurate physical residuals compared to FM, while clearly outperforming existing algorithms in terms of distributional accuracy. PBFM thus provides a principled and efficient framework for surrogate modeling, uncertainty quantification, and accelerated simulation in physics and engineering applications.

翻译：生成式机器学习方法，如扩散模型和流匹配，在模拟复杂系统行为和构建高效代理模型方面展现出巨大潜力。然而，这些方法通常仅从数据中隐式学习底层物理规律。我们提出物理基流匹配（PBFM），一种新颖的生成框架，将物理约束（包括偏微分方程残差和代数关系）显式嵌入流匹配目标函数。我们还引入了训练时的时间展开策略，以提升最终无噪声样本预测的精度。该方法联合优化流匹配损失与物理残差损失，无需调整两者权重的超参数。此外，我们分析了最小噪声水平$σ_{\min}$在物理约束背景下的作用，并评估了一种有助于降低物理残差的随机采样策略。通过在三个代表性偏微分方程问题上的广泛基准测试，我们证明该方法相比流匹配（FM）可将物理残差精度提升高达$8$倍，同时在分布精度方面显著优于现有算法。因此，PBFM为物理与工程应用中的代理建模、不确定性量化和加速仿真提供了原理清晰且高效的框架。