Potential identification via Tikhonov-PINNs

In this article, we introduce Tikhonov-physics informed neural networks (PINNs), a novel neural network-driven approach designed for tackling inverse potential problems. Through the combining of Tikhonov regularization with PINNs, we establish a stability estimate for the potential reconstruction. Additionally, leveraging learning theory and approximation theory of neural networks, we demonstrate the stochastic convergence of nonlinear potential identification problems, extending the analysis beyond linear settings and bounded noise constraints. A series of numerical illustrations are provided to showcase the efficacy and superiority of our method, contrasting it with both the traditional finite element approach and basic PINNs.