Scholarlink

<p>Physics-informed neural networks (PINNs) have recently emerged as a powerful tool to solve differential equations for nonlinear mechanics. However, PINNs struggle with singular perturbation problems due to their locally abrupt behavior and singularities. The Poincaré-Lighthill-Kuo (PLK) method has been efficiently used to address these problems by applying perturbation expansions to both dependent and independent variables. This paper proposes a combination of the PLK method and PINNs, termed PLK-PINNs. The PLK-PINNs employ a parametric expression through two neural networks: one representing the mapping from parametric variables to independent variables, and the other approximating the solution of dependent variables with respect to parametric variables. Moreover, an auxiliary loss term is proposed to constrain the Jacobian determinant of the mapping within a constant sign interval to ensure the bijectivity of the mapping. The effectiveness of the proposed method is demonstrated through tests on typical singularity-shift and secular-term problems, with conventional PINNs in comparison.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Physics-informed neural networks with Poincaré-Lighthill-Kuo method for singular perturbation problems

  • 期刊论文

  • 发表日期：

    2026年2月26日

查看全文
  • Qingyong Luo,
  • Lei Zhang,
  • Guowei He

摘要

Physics-informed neural networks (PINNs) have recently emerged as a powerful tool to solve differential equations for nonlinear mechanics. However, PINNs struggle with singular perturbation problems due to their locally abrupt behavior and singularities. The Poincaré-Lighthill-Kuo (PLK) method has been efficiently used to address these problems by applying perturbation expansions to both dependent and independent variables. This paper proposes a combination of the PLK method and PINNs, termed PLK-PINNs. The PLK-PINNs employ a parametric expression through two neural networks: one representing the mapping from parametric variables to independent variables, and the other approximating the solution of dependent variables with respect to parametric variables. Moreover, an auxiliary loss term is proposed to constrain the Jacobian determinant of the mapping within a constant sign interval to ensure the bijectivity of the mapping. The effectiveness of the proposed method is demonstrated through tests on typical singularity-shift and secular-term problems, with conventional PINNs in comparison.