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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10957_2025_2793_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ell _\text {max}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mtext>max</mtext> </msub> </math></EquationSource> </InlineEquation>, which encapsulates the degree of training. Our theoretical results are validated by experiments on a 50-dimensional and a 500-dimensional NPE problem.</p>

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

  • 期刊论文

  • 发表日期：

    2025年8月13日

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  • Dawen Wu,
  • Abdel Lisser

摘要

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.