用物理约束神经网络,从有限数据中恢复高精度电导率分布。
Recovering Sharp Conductivity Features in the Finite-Data Calderón Problem with Physics-Informed Neural Networks

- 分网络建模电导率与电势,结合边界激励进行物理约束
- 对包含物和尖锐界面的结构,重建误差低至3%-12%
- 傅里叶特征编码显著提升局部尖锐特征恢复能力
物理信息神经网络(PINNs)为从有限边界数据解决Calderón逆问题提供了新思路。本文通过引入基于随机小波函数的多尺度边界激励,并研究傅里叶特征编码(FFE)在表征高梯度电导率变化中的作用,提出一种物理信息重建框架。该框架使用独立的神经网络分别表示未知电导率及其对应的电势族,且条件依赖于施加的边界激励。通过物理残差强制满足椭圆型偏微分方程,同时利用有限的狄利克雷到诺伊曼(DtN)数据通过边界损失项进行约束。基于有限差分正演求解器生成的合成数据,我们在包含夹杂物、尖锐界面、光滑分布及非均匀介质的电导率场中评估了该方法。结果表明,该框架可从有限边界测量中恢复主导电导率结构,相对误差约为3%–12%。我们发现,傅里叶特征编码显著提升了对局域尖锐特征(尤其是夹杂物和界面)的重建效果,但并非普遍最优;对于平滑场,原始坐标网络表现相当。这些结果凸显了坐标表示方式与边界激励设计在神经型Calderón反演中的关键作用。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calderón inverse problem from limited boundary data. In this work, we revisit neural Calderón inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calderón inversion.
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