Solving Fredholm Integral Equations with Physics-Informed Neural Networks: Single-and double-network approaches
Abstract
We investigate the use of Physics-Informed Neural Networks (PINNs) for the numerical solving of Fredholm integral equations of the first and second kind, proposing both single-network and double-network formulations. Rigorous error estimates are derived for both approaches, establishing theoretical convergence guarantees in terms of training residuals. Numerical experiments demonstrate convergence across a range of benchmark problems. A systematic comparison with the classical Nyström method reveals that, for well-posed, well-conditioned problems with clean data, the Nyström method remains the solver of choice. However, the PINN framework demonstrates a marked advantage in challenging settings: when the input data are corrupted by noise or the problem is ill-conditioned, PINNs exhibit significantly greater robustness. This inherent stability, attributed to the implicit regularization of neural network training, positions PINNs as a compelling alternative for the solution of Fredholm integral equations arising in ill-posed inverse problems and data-corrupted regimes.
Related
- Solving the scattering problem for open wave-guide networks, I: Fundamental solutions and integral equations
- CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations
- AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
- Boosted Physics-Informed Neural Networks for Solving Homogeneous and Non-Homogeneous Lane-Emden Type Equations
- Direct and Indirect Physics-Informed Neural Networks for Dirichlet Boundary Control of Semilinear Parabolic Equations: A Conditional Error Analysis
BibTeX
@misc{fabre2026solving,
title = {{Solving Fredholm Integral Equations with Physics-Informed Neural Networks: Single-and double-network approaches}},
author = {Mathieu Fabre and Laurence Grammont and Youcef Mammeri},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05799002v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Solving Fredholm Integral Equations with Physics-Informed Neural Networks: Single-and double-network approaches
AU - Mathieu Fabre
AU - Laurence Grammont
AU - Youcef Mammeri
PY - 2026
DA - 2026/10/14
PB - HAL
UR - https://hal.science/hal-05799002v1
AB - We investigate the use of Physics-Informed Neural Networks (PINNs) for the numerical solving of Fredholm integral equations of the first and second kind, proposing both single-network and double-network formulations. Rigorous error estimates are derived for both approaches, establishing theoretical convergence guarantees in terms of training residuals. Numerical experiments demonstrate convergence across a range of benchmark problems. A systematic comparison with the classical Nyström method reveals that, for well-posed, well-conditioned problems with clean data, the Nyström method remains the solver of choice. However, the PINN framework demonstrates a marked advantage in challenging settings: when the input data are corrupted by noise or the problem is ill-conditioned, PINNs exhibit significantly greater robustness. This inherent stability, attributed to the implicit regularization of neural network training, positions PINNs as a compelling alternative for the solution of Fredholm integral equations arising in ill-posed inverse problems and data-corrupted regimes.
ER -