Error Bound Analysis of Physics-Informed Neural Networks for Solving Nonlinear Projection Equations

Dawen Wu () and Abdel Lisser ()
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Dawen Wu: CNRS@CREATE
Abdel Lisser: Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes

Journal of Optimization Theory and Applications, 2025, vol. 207, issue 3, No 10, 20 pages

Abstract: Abstract This study presents an in-depth theoretical analysis for the physics-informed neural networks (PINNs) approach studied in (Wu, D., Lisser, A.: Neuro-PINN: A hybrid framework for efficient nonlinear projection equation solutions. Int. J. Numer. Meth. Eng. 125, e7377 (2024)) for solving nonlinear projection equations (NPEs). The NPE is first modeled by a system of ordinary differential equations (ODE system) and then solved by PINNs. We focus on establishing error bounds for both the neural network (NN) state solution, which solves the ODE system, and the NN terminal state, which solves the NPE. The proposed bounds are based on the use of the maximum loss notation $$\ell _\text {max}$$ ℓ max , which encapsulates the degree of training. Our theoretical results are validated by experiments on a 50-dimensional and a 500-dimensional NPE problem.

Keywords: Nonlinear projection equations; Physics-informed neural networks; Ordinary differential equations; Error bounds; 65L70; 68T07; 41A25 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02793-3

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