Fredholm Neural Networks for forward and inverse problems in elliptic PDEs
Journal:
arXiv
Published Date:
Jul 8, 2025
Abstract
Building on our previous work introducing Fredholm Neural Networks (Fredholm
NNs/ FNNs) for solving integral equations, we extend the framework to tackle
forward and inverse problems for linear and semi-linear elliptic partial
differential equations. The proposed scheme consists of a deep neural network
(DNN) which is designed to represent the iterative process of fixed-point
iterations for the solution of elliptic PDEs using the boundary integral method
within the framework of potential theory. The number of layers, weights, biases
and hyperparameters are computed in an explainable manner based on the
iterative scheme, and we therefore refer to this as the Potential Fredholm
Neural Network (PFNN). We show that this approach ensures both accuracy and
explainability, achieving small errors in the interior of the domain, and near
machine-precision on the boundary. We provide a constructive proof for the
consistency of the scheme and provide explicit error bounds for both the
interior and boundary of the domain, reflected in the layers of the PFNN. 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 provide an explainable scheme
that explicitly respects the boundary conditions. We assess the performance of
the proposed scheme for the solution of both the forward and inverse problem
for linear and semi-linear elliptic PDEs in two dimensions.
