Solving an inverse source problem by deep neural network method with convergence and error analysis

For the inverse source problem of an elliptic system using noisy internal measurement as inversion input, we approximate its solution by neural network function, which is obtained by optimizing an empirical loss function with appropriate regularizing terms. We analyze the convergence of the general loss from noisy inversion input data in deep Galerkin method by the regularizing empirical loss function. Based on the upper bound of the expected loss function by its regularizing empirical form, we establish the upper bound of the expected loss function at the minimizer of the regularizing empirical noisy loss function in terms of the number of sampling points as well as the noise level quantitatively, for suitably chosen regularizing parameters and regularizing terms. Then, by specifying the number of sampling points in terms of noise level of inversion input data, we establish the error orders representing the difference between the neural network solution and the exact one, under some a-priori restrictions on the source. Finally, we give numerical implementations for several examples to verify our theoretical results.