Fredholm Neural Networks for inverse problems in elliptic PDEs
Building on our previous work on Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to inverse problems for linear and nonlinear elliptic partial differential equations. The proposed scheme consists of a custom-designed deep neural network (DNN) in which the number of layers, weights, biases and hyperparameters are computed in an explainable manner based on a fixed-point scheme, and we therefore refer to this as the Potential Fredholm Neural Network (PFNN). We first build the PFNN as a method for solving the forward problem, showing that this approach ensures both a high accuracy and explainability, achieving small errors in the interior of the domain, and near machine-precision on the boundary. We then use this approach to solve inverse problems for elliptic PDEs, and provide a rigorous proof for the consistency of the scheme and error bounds for both the interior and boundary of the domain, tied directly to the architecture of the PFNN. In particular, we show that these error bounds depend on the approximation of the boundary function and the integral discretization scheme, both of which directly correspond to components of the Fredholm NN architecture. In this way, we construct an explainable scheme that provides accurate solutions to the inverse problems, whilst still explicitly respecting the boundary conditions, due to the architecture of the PFNN. We assess the performance of the proposed scheme for linear and semi-linear elliptic PDEs in two and three dimensions.