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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

An Incomplete Constraint Method with Two Feedforward Neural Networks for Solving Linear Partial Differential Equations

Journal of Applied Nonlinear Dynamics 14(4) (2025) 981--1008 | DOI:10.5890/JAND.2025.12.016

Yanxia Lyu$^{1, 2}$, Xianchi Li$^{1}$

$^1$ School of Computer and Communication Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066000, China

$^2$ Hebei Key Laboratory of Marine Perception Network and Data Processing, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China

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Abstract

This paper proposes an incomplete constraint (IC) method, together with extreme learning machines (ELM) and multilayer perceptrons (MLP), to solve linear partial differential equations (PDEs). Unlike the physics-informed neural networks (PINN) approach, the IC method does not enforce strict adherence to initial and boundary conditions within the neural network architecture, which simplifies formulation and enhances accuracy. Specifically, by employing trial functions generated via ELM/MLP and spatiotemporal sampling-based configuration points, PDEs are discretized into weak optimization problems described as nonlinear algebraic equations. And then, optimal parameters are determined through nonlinear optimization algorithms and iterative. Finally, for a comprehensive error distribution analysis, five numerical examples are provided to verify the validity and efficiency of the proposed method. Numerical results demonstrate it has lower error levels and/or computational costs than PINN and radial basis collocation methods.

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