Numerical Optimization Method for Solving High-Dimensional Sideways Parabolic Problems

This paper is concerned with the sideways problem for parabolic equations in a bounded domain. The Dirichlet data on the remaining part of the boundary is reconstructed from the Dirichlet and Neumann measurements on a portion of the boundary. Based on existing theories, the uniqueness of the inverse problem can be established, and its ill-posedness is analyzed. Then, by introducing two auxiliary problems, the inverse problem is reformulated as an optimal control problem with Kohn-Vogelius regularization. We further prove the existence and stability of solutions to the optimal control problem. Finally, a physics-informed neural network framework based on the Kohn-Vogelius type functional is applied to several numerical examples.