On stability and regularization for data-driven solution of parabolic inverse source problems

The diffusion process from some internal source arising in engineering situations can be mathematically described by a parabolic system. We consider an inverse source problem for parabolic system using parametric approximations, where deep neural networks (DNNs) are used to approximate the solution of the inverse problem. First, we prove generalization error estimates depending on training errors and data noise levels by establishing conditional stability of the inverse problem. Following our analysis, we propose a new loss function involving the derivative of the residuals for PDE and measurement data. These extra terms effectively induce higher regularity in solution to deal with the ill-posedness of the inverse problem. Using these regularization terms, we develop reconstruction schemes and demonstrate the effectiveness of our proposed methodology on a number of test problems.