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Fredholm Neural Networks for forward and inverse problems in elliptic PDEs

Kyriakos Georgiou, Constantinos Siettos, Athanasios N. Yannacopoulos

TL;DR

The paper introduces Potential Fredholm Neural Networks (PFNNs) and Recurrent PFNNs that fuse fixed-point iteration schemes with boundary-integral formulations to solve forward and inverse problems for linear and semi-linear elliptic PDEs. By embedding potential theory into a DNN architecture, PFNNs deliver boundary-consistent solutions with explicit error bounds tied to boundary density approximation and integral discretization, enabling explainability. The authors derive constructive convergence proofs, present detailed error analyses for interior and boundary regions, and demonstrate high-accuracy results on Poisson, Helmholtz, and Liouville-type semi-linear PDEs in 2D, including inverse-source problems. They also discuss explainability, computational trade-offs, and potential extensions to high-dimensional, time-dependent problems and uncertainty quantification. Overall, PFNNs offer a physics-informed, transparent alternative to black-box ML approaches for complex PDE-based tasks.

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.

Fredholm Neural Networks for forward and inverse problems in elliptic PDEs

TL;DR

The paper introduces Potential Fredholm Neural Networks (PFNNs) and Recurrent PFNNs that fuse fixed-point iteration schemes with boundary-integral formulations to solve forward and inverse problems for linear and semi-linear elliptic PDEs. By embedding potential theory into a DNN architecture, PFNNs deliver boundary-consistent solutions with explicit error bounds tied to boundary density approximation and integral discretization, enabling explainability. The authors derive constructive convergence proofs, present detailed error analyses for interior and boundary regions, and demonstrate high-accuracy results on Poisson, Helmholtz, and Liouville-type semi-linear PDEs in 2D, including inverse-source problems. They also discuss explainability, computational trade-offs, and potential extensions to high-dimensional, time-dependent problems and uncertainty quantification. Overall, PFNNs offer a physics-informed, transparent alternative to black-box ML approaches for complex PDE-based tasks.

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.

Paper Structure

This paper contains 16 sections, 6 theorems, 56 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Potential Theory and Fredholm Neural Networks
  4. The Potential Fredholm NN for the Poisson PDE
  5. The Potential Fredholm NN for the Helmholtz PDE
  6. Consistency and error analysis
  7. Recurrent PFNN construction for semi-linear elliptic PDEs
  8. The Potential Fredholm Neural Network for Inverse Problems
  9. Numerical Implementation
  10. Solution of the Forward Problem
  11. The Poisson PDE
  12. The Helmholtz PDE
  13. Semi-linear elliptic PDEs
  14. Explainability
  15. Solution of Inverse problems
  16. ...and 1 more sections

Key Result

Theorem 2.1

Consider the two-dimensional linear Poisson equation for $u(x)$: Its solution can be written via the double layer boundary integral given by: where $\Phi(x,y)$ is the fundamental solution of the Laplace equation, $n_y$ is the outward pointing normal vector to $y$, $\sigma_y$ is the surface element at point $y\in \partial \Omega$, and $\frac{\partial \Phi}{\partial n_{y}} = n_y \cdot \nabla \, _{

Figures (7)

  • Figure 1: Schematic of the PFNN construction. The first component is a Fredholm Neural Network and the second encapsulates the representation of the double layer potential given in (\ref{['potential-form-1']}), decomposed into a the final hidden layer.
  • Figure 2: Schematic of the Recurrent Potential Fredholm NN to solve semi-linear elliptic PDEs.
  • Figure 3: Schematic representation of Algorithm \ref{['alg:inverse']} to solve the inverse problem using the Potential FNN framework.
  • Figure 4: Results of the PFNN for the solution of the linear Poisson PDE \ref{['ex1']}.
  • Figure 5: Results of the PFNN for the linear Helmholtz PDE (\ref{['ex-helmholtz']}).
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 2.1: folland2020introduction
  • Definition 2.2: Fredholm neural networks
  • Proposition 2.3: PFNN construction for the Poisson PDE
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • ...and 2 more