面向鲁棒PDE反问题的物理引导条件流匹配与能量正则化

中文摘要 AI 辅助

我们考虑从稀疏、含噪声和损坏的观测中进行偏微分方程（PDE）反问题求解，目标是在无网格设置下恢复未知系数场和相关的状态变量。在如此稀疏和损坏的观测下，标准的物理信息和生成式方法通常不加区分地处理所有样本，因此缺乏一种原则性机制来调和物理定律与受污染数据。我们通过一个两阶段流匹配框架来解决这一难题。在第一阶段，我们开发了物理引导的条件流匹配（PG-CFM），该算法通过沿生成轨迹的残差正则化以及间歇性的全局配点约束来融入强形式PDE信息。在第二阶段，我们引入了能量正则化流匹配（ERFM），通过为每个观测分配来自冻结教师模型的物理-数据能量分数，并重新加权流匹配目标以减少高能量、与PDE不一致样本的影响，从而对第一阶段模型进行微调。我们表明，由此产生的第二阶段目标等价于在教师诱导的重新加权数据分布下的流匹配，这为鲁棒性机制提供了总体层面的解释。在多个反问题基准（包括泊松和纳维-斯托克斯问题）上的数值实验表明，所提出的框架比鲁棒PINN变体和竞争性生成基线实现了更准确的系数恢复。

英文摘要

We consider partial differential equation (PDE) inverse problems from sparse, noisy, and corrupted observations, with the aim of recovering unknown coefficient fields and associated state variables in a mesh-free setting. Under such sparse and corrupted observations, standard physics-informed and generative approaches typically treat all samples indiscriminately and therefore lack a principled mechanism for reconciling physical laws with contaminated data. We address this difficulty with a two-stage flow-matching framework. In the first stage, we develop physics-guided conditional flow matching (PG-CFM), which incorporates strong-form PDE information through residual regularization along the generative trajectories together with intermittent global collocation constraints. In the second stage, we introduce energy-regularized flow matching (ERFM), which fine-tunes the Stage-1 model by assigning each observation a physics--data energy score from a frozen teacher and reweighting the flow-matching objective to reduce the influence of high-energy, PDE-inconsistent samples. We show that the resulting Stage-2 objective is equivalent to flow matching under a teacher-induced reweighted data distribution, which gives a population-level interpretation of the robustness mechanism. Numerical experiments on several inverse benchmarks, including Poisson and Navier--Stokes problems, show that the proposed framework yields more accurate coefficient recovery than robust PINN variants and competing generative baselines.

发表机构

• Zhejiang University（浙江大学）
• Texas Tech University（德克萨斯理工大学）

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