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

Author

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  • Dawen Wu

    (CNRS@CREATE
    National University of Singapore)

  • Abdel Lisser

    (Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes)

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.

Suggested Citation

  • Dawen Wu & Abdel Lisser, 2025. "Error Bound Analysis of Physics-Informed Neural Networks for Solving Nonlinear Projection Equations," Journal of Optimization Theory and Applications, Springer, vol. 207(3), pages 1-20, December.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:3:d:10.1007_s10957-025-02793-3
    DOI: 10.1007/s10957-025-02793-3
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    References listed on IDEAS

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    1. Carlos J. García-Cervera & Mathieu Kessler & Francisco Periago, 2023. "Control of Partial Differential Equations via Physics-Informed Neural Networks," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 391-414, February.
    2. Wu, Dawen & Lisser, Abdel, 2024. "Solving Constrained Pseudoconvex Optimization Problems with deep learning-based neurodynamic optimization," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 424-434.
    3. Y. S. Xia & J. Wang, 2000. "On the Stability of Globally Projected Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 106(1), pages 129-150, July.
    4. Stephen M. Robinson, 1992. "Normal Maps Induced by Linear Transformations," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 691-714, August.
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