用图神经网络增强偏微分方程反问题求解，提升重建精度。
Graph Neural Regularizers for PDE Inverse Problems
- 结合有限元法与图神经网络，交替迭代优化系数估计。
- 在高度病态场景下仍能实现高精度重构，优于传统方法。
- 适用于复杂几何与多种PDE，兼具可解释性与泛化能力。
我们提出一种求解由偏微分方程（PDE）支配的广泛病态反问题的框架，通过交替使用基于有限元法（FEM）的反演与学习型图神经网络正则化，恢复前向算子的目标系数。利用FEM离散化中固有的图结构，采用物理启发的图神经网络作为学习型正则器，提供了一种稳健、可解释且可推广的替代方案。数值实验表明，该框架优于经典正则化技术，在高度病态情形下仍能实现精确重构。
原文摘要 · Abstract (English)
We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs), where the target coefficients of the forward operator are recovered through an iterative regularization scheme that alternates between FEM-based inversion and learned graph neural regularization. The forward problem is numerically solved using the finite element method (FEM), enabling applicability to a wide range of geometries and PDEs. By leveraging the graph structure inherent to FEM discretizations, we employ physics-inspired graph neural networks as learned regularizers, providing a robust, interpretable, and generalizable alternative to standard approaches. Numerical experiments demonstrate that our framework outperforms classical regularization techniques and achieves accurate reconstructions even in highly ill-posed scenarios.
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