A novel artificial neural network method is proposed for solving Cauchy inverse problems. It allows multiple hidden layers with arbitrary width and depth, which theoretically yields better approximations to the inverse problems. In this research, the existence and convergence are shown to establish the well-posedness of neural network method for Cauchy inverse problems, and various numerical examples are presented to illustrate its accuracy and stability. The numerical examples are from different points of view, including time-dependent and time-independent cases, high spatial dimension cases up to 8D, and cases with noisy boundary data and singular computational domain. Moreover, numerical results also show that neural networks with wider and deeper hidden layers could lead to better approximation for Cauchy inverse problems.
Diffuse Interface Methods for Inverse Problems: Case Study for an Elliptic Cauchy Problem
Institute for Computational and Applied Mathematics, University of Münster（明斯特大学计算与应用数学研究所）；Cells in Motion Cluster of Excellence, University of Münster（明斯特大学细胞运动卓越集群）；Dept. of Mathematical Sciences and Technology, Norwegian University of Life Sciences（挪威生命科学大学数学科学与技术系）
Data driven Koopman spectral analysis in Vandermonde-Cauchy form via the DFT: numerical method and theoretical insights
Faculty of Science, Department of Mathematics, University of Zagreb（萨格勒布大学理学院数学系）；Department of Mechanical Engineering and Mathematics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校机械工程和数学系）；AIMdyn, Inc.（AIMdyn公司）
Numerical computation of convolutions in free probability theory
School of Mathematics and Statistics, The University of Sydney（悉尼大学数学与统计学院）；Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气与计算机工程系）
