Convergence analysis of a PINNs-based approach to the inverse source problem of the heat equation with local measurements

This paper investigates the problem of recovering the spatial distribution of the source term in a parabolic system from local measurement data. We propose a method based on Physics-Informed Neural Networks (PINNs) to approximate the solution of this inverse problem. Within this framework, we design a novel loss function that incorporates the residuals of the partial differential equation (PDE), measurement data residuals, as well as residuals from initial–boundary conditions and boundary derivatives. These additional terms enhance the regularity of the solution to address the ill-posedness of the inverse problem. Based on the conditional stability of the inverse problem, we demonstrate that the neural network can effectively approximate its solution, and we further establish generalization error estimates for the source term, which depend on the training error and data noise level, while also verifying the convergence of the network. A series of numerical experiments further validate the effectiveness and robustness of the proposed method, showing that it achieves high accuracy and efficiency under noisy data and varying measurement conditions.