用神经网络学习边界距离，提升缺陷重建精度
The learned range test method for the inverse inclusion problem
- 将范围测试法转化为特定结构的神经网络
- 对多边形缺陷重建准确率超传统方法与纯数据驱动模型
- 结合预训练分类器，可按距边界远近区分缺陷
研究在有界区域Ω⊂ℝᵈ内包含一个夹杂物B的反问题，仅凭边界上的一组柯西数据（u|∂Ω, ∂_νu|∂Ω）重构B，其中Δu=0在Ω\overline B内，且u=0在∂B上。我们发现基于范围测试的重构算法可表示为具有特定结构的神经网络。提出在监督学习框架下学习该网络权重，并结合预训练分类器，以根据夹杂物到边界的距离进行区分。数值模拟表明，该学习型范围测试方法能实现对多边形夹杂物的精确且稳定的重建。结果优于标准范围测试法（无学习）和端到端全连接深度神经网络（纯数据驱动方法）。
原文摘要 · Abstract (English)
We consider the inverse problem consisting of the reconstruction of an inclusion $B$ contained in a bounded domain $Ω\subset\mathbb{R}^d$ from a single pair of Cauchy data $(u|_{\partialΩ},\partial_νu|_{\partialΩ})$, where $Δu=0$ in $Ω\setminus\overline B$ and $u=0$ on $\partial B$. We show that the reconstruction algorithm based on the range test, a domain sampling method, can be written as a neural network with a specific architecture. We propose to learn the weights of this network in the framework of supervised learning, and to combine it with a pre-trained classifier, with the purpose of distinguishing the inclusions based on their distance from the boundary. The numerical simulations show that this learned range test method provides accurate and stable reconstructions of polygonal inclusions. Furthermore, the results are superior to those obtained with the standard range test method (without learning) and with an end-to-end fully connected deep neural network, a purely data-driven method.
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