结合已知物理的流匹配能量：PDE场的生成、分布外检测与反演
Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields
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中文总结 AI 辅助
本研究提出带势诱导速度的流匹配模型，生成显式标量能量，实现能量校正生成、OOD检测及反问题后验采样，可降低PDE残差、提升OOD检测精度。
中文摘要 AI 辅助
物理场的概率建模受益于数据驱动先验与控制方程等已知物理结构的结合。基于能量的模型（EBM）因能量可加性，能在推理阶段融入物理信息，是合适的选择；但由于配分函数难以处理，EBM的训练与采样一直存在困难。本研究表明，带势诱导速度的流匹配模型在所有传输时刻都会产生显式标量能量，其梯度恰好是转换后的学习得分，且在总体最优时可恢复边际负对数密度。该时变能量函数完全通过独立线性高斯插值的匹配回归目标获得，无需变分形式或额外MCMC步骤，采样过程保留流常微分方程（ODE）。训练后模型的能量函数可实现三类功能：能量校正的数据生成、作为分布外（OOD）检测评分函数的能量、用于反问题的能量组合后验采样。具体而言，显式能量允许预测-校正采样框架中使用通用MCMC采样器，与流ODE基线相比，可降低PDE残差和谱距离；同时，利用数据能量与基于物理的能量（如PDE残差）作为互补机制，可提升OOD任务的检测精度；此外，通过将能量与二次观测似然组合生成后验能量，将其作为推理时目标的显式选择族，探究其与基于MCMC的反问题推理的关联。
英文摘要
Probabilistic modeling of physical fields benefits from both a data-driven prior and known physical structure such as the governing equations. Energy-based models (EBMs) are a natural fit since energies compose additively, which enables augmenting physics information during inference. However, EBMs have been difficult to train and sample from due to the intractable partition function. We show in this work that flow matching models with a potential-induced velocity yield an explicit scalar energy at all transport times, whose gradient is exactly the converted learned score and which recovers the marginal negative log-density at the population optimum. The time-dependent energy functions are obtained purely from the matching regression objective on an independent linear Gaussian interpolation, without a variational form or additional MCMC steps, and the sampling retains the flow ODE. Access to the energy function from a trained model serves three roles: energy-corrected data generation, energy as a scoring function for out-of-distribution (OOD) detection, and energy compositional posterior sampling for inverse problems. In particular, we show the explicit energy permits general MCMC samplers in the predictor-corrector sampling framework, reducing PDE residual and spectral distance compared to the flow ODE baseline. Furthermore, we demonstrate utilizing the data energy and physics-based energy (e.g., PDE residuals) as complementary mechanisms to improve detection accuracy for OOD tasks. In addition, we explore the connection to MCMC-based inference for inverse problems by composing the energy with a quadratic observational likelihood that yields a posterior energy, used as an explicitly chosen family of inference-time targets.
