Potential coefficient identification problem in parabolic equation with deep neural networks

Abstract

Abstract The inverse problem of identifying an unknown space-dependent potential coefficient in the parabolic equation is considered from the additional observation at the terminal time in this work. A novel conditional stability estimate is established for a large terminal time T with suitable assumptions on the input data. Then the potential coefficient and solution of the parabolic equation are parameterized by separate deep neural networks (DNNs), and a new loss function is proposed to reconstruct the unknown potential coefficient. The DNN approximations of the potential coefficient for both continuous and empirical loss functions are analyzed rigorously via utilizing analogous arguments for the conditional stability. Meanwhile, the error estimates are expressed explicitly by the noise level and neural network architectural parameters, which yields a prior rule for determining the number of observations and choosing the size of neural networks. Some numerical experiments are provided to illustrate the robustness of the approach against various noise levels of measured observation and the accuracy of the numerical solutions.