PhyRes-MDNF：用于多级离散神经场反演的物理耦合残差GNN校正
ResiPhy-MDNF: A Residual-Based Physics-Aware Multilevel Discrete Neural Field Framework for PDE-Constrained Inverse Problems
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中文总结 AI 辅助
研究针对偏微分方程约束下细网格系数反演病态问题，提出PhyRes-MDNF框架，通过固定物理的多级离散神经场优化，在全空间达西实现中联合优化状态场与系数场，实验表明该方法提升了精度与迭代效率。
中文摘要 AI 辅助
在偏微分方程约束下，细网格上的系数反演是病态的：稀疏观测对细尺度参数的约束较弱，直接的单分辨率优化必须同时恢复所有尺度上的状态和系数场。这导致收敛缓慢且对初始化敏感；学习到的传递模型需要离线数据，并且可能会在数值物理中引入近似误差。我们提出了PhyRes-MDNF，一种固定物理的多级离散神经场框架。在每个级别上，单级DNF将反演未知数直接表示为可训练场，并优化离散目标。在全空间达西实现中，状态场\(U\)及其共享系数场\(K\)在一个固定物理反演过程中联合优化。在级别之间，一个零初始化的PhyRes-GNN联合执行固定模板延拓和有界残差校正，以构建输入目标表示，一个固定初始化映射将其转换为下一个DNF变量。它从观测和不变的数值模型中重新拟合，无需离线预训练或细网格真值。因此，粗级别在引入精细自由度之前解析大尺度结构，缩短了细网格优化路径，同时保留了原始离散算子。在相同的最终网格更新预算下，多级达西实现分别将系数和状态误差降低了约\(85\%\)和\(90\%\)，证明了精度和最终网格迭代效率的提高。在实测的KTC2023 EIT数据上，全\(W\)管道比官方线性化CEM重建平均Otsu mIoU提高了约\(3.4\%\)，比直接单级DNF提高了\(16.9\%\)。
英文摘要
Inverse problems governed by partial differential equations are difficult when observations are sparse and the unknown coefficient field contains both large- and small-scale structures. We introduce a residual-based, physics-aware multilevel discrete neural field framework, ResiPhy-MDNF, for such problems. The method couples a coarse-to-fine discrete neural field (DNF) optimizer with a residual-based graph neural network (GNN) transfer operator, called ResiPhy-GNN. At each level, the DNF directly optimizes trainable grid- or mesh-based state and coefficient arrays using the prescribed numerical model. Between levels, ResiPhy-GNN maps the coarse representation to the fine representation through a learned prolongation based on graph connectivity, spatial features, and residual information. The method requires neither surrogate models nor offline pretraining. We evaluate the framework on coefficient inversion in Darcy flow for subsurface modeling and on electrical impedance tomography (EIT). In the controlled Darcy test case, the \(64^2\!\to128^2\) multilevel path uses \(1.25\times\) the cumulative grid-work proxy of the direct single-level \(128^2\) solve, while achieving \(6.76\times\) lower permeability error and \(10.4\times\) lower state error. On measured Kuopio Tomography Challenge 2023 EIT data, the framework improves the mean intersection-over-union after Otsu thresholding by roughly \(3.4\%\) over the official linearized complete electrode model reconstruction and by \(16.9\%\) over direct single-level discrete-field optimization. These results indicate that the same multilevel construction can be used across different coefficient structures, discretizations, and observation geometries.
