SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1443-1464, June 2024.
Abstract. In this work we connect machine learning techniques, in particular kernel machine learning, to inverse source and scattering problems. We show the proposed kernel machine learning has demonstrated generalization capability and has a rigorous mathematical foundation. The proposed learning is based on the Mercer kernel, the reproducing kernel Hilbert space, the kernel trick, as well as the mathematical theory of inverse source and scattering theory, and the restricted Fourier integral operator. The kernel machine learns a multilayer neural network which outputs an [math]-neighborhood average of the unknown or its nonlinear transformation. We then apply the general architecture to the multifrequency inverse source problem for a fixed observation direction and the Born inverse medium scattering problem. We establish a mathematically justified kernel machine indicator with demonstrated capability in both shape identification and parameter identification, under very general assumptions on the physical unknowns. More importantly, stability estimates are established in the case of both noiseless and noisy measurement data. Of central importance is the interplay between a restricted Fourier integral operator and a corresponding Sturm–Liouville differential operator. Several numerical examples are presented to demonstrate the capability of the proposed kernel machine learning.

中文翻译：

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用于逆源和散射问题的核机器学习


《SIAM 数值分析杂志》，第 62 卷，第 3 期，第 1443-1464 页，2024 年 6 月。

抽象的。在这项工作中，我们将机器学习技术（特别是内核机器学习）与逆源和散射问题联系起来。我们证明了所提出的内核机器学习已经展示了泛化能力并且具有严格的数学基础。所提出的学习基于 Mercer 核、再现核 Hilbert 空间、核技巧以及逆源和散射理论的数学理论以及受限傅里叶积分算子。内核机器学习一个多层神经网络，该网络输出未知数或其非线性变换的数学邻域平均值。然后，我们将通用架构应用于固定观测方向的多频逆源问题和玻恩逆介质散射问题。我们建立了一个数学上合理的内核机器指标，在对物理未知数的非常普遍的假设下，具有形状识别和参数识别的能力。更重要的是，稳定性估计是在无噪声和噪声测量数据的情况下建立的。最重要的是受限傅里叶积分算子和相应的 Sturm-Liouville 微分算子之间的相互作用。提出了几个数值示例来证明所提出的内核机器学习的能力。