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<p>Physics-informed neural network&#xa0;(PINN) exhibits remarkable capability in a wide range of fields by incorporating physical laws into the deep learning process. However, a key limitation is its difficulty in resolving tasks with high-frequency regions accurately. Although this issue can be mitigated by designing a sampling distribution of residual points, this approach may incur substantial computation costs. To overcome this inefficiency and enhance generalization, we propose a novel progressive learning-based multi-stage PINN strategy, which adjusts the adaptive sampling of residual points based on phased refinement. The strategy primarily consists of three phases: a pre-training phase on sparse data that preliminarily identifies the general characteristics and lays the foundation for the subsequent learning; an intensive training phase that improves the ability to learn local features by adding diverse and representative data; and an adaptive refinement phase that further focuses on sampling in the high-error regions to boost the reliability of the model. This framework enriches the valuable features of the training data and enhances the applicability of the samples. Experimental results demonstrate that our method achieves better accuracy in capturing high-frequency features for low- and high-dimensional nonlinear equations, such as the (1+1)-dimensional modified Korteweg-de Vries equation and the (2+1)-dimensional Kadomtsev-Petviashvili equation. The multi-stage PINN accomplishes dependable prediction in both training and extrapolated domains, and establishes a more effective balance between computational cost and performance.</p>

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Progressive learning-based multi-stage Physics-Informed Neural Network

  • 期刊论文

  • 发表日期：

    2026年1月29日

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  • Haiyi Liu,
  • Qun Liu,
  • Guoyin Wang,
  • Xiaoxia Zhang

摘要

Physics-informed neural network (PINN) exhibits remarkable capability in a wide range of fields by incorporating physical laws into the deep learning process. However, a key limitation is its difficulty in resolving tasks with high-frequency regions accurately. Although this issue can be mitigated by designing a sampling distribution of residual points, this approach may incur substantial computation costs. To overcome this inefficiency and enhance generalization, we propose a novel progressive learning-based multi-stage PINN strategy, which adjusts the adaptive sampling of residual points based on phased refinement. The strategy primarily consists of three phases: a pre-training phase on sparse data that preliminarily identifies the general characteristics and lays the foundation for the subsequent learning; an intensive training phase that improves the ability to learn local features by adding diverse and representative data; and an adaptive refinement phase that further focuses on sampling in the high-error regions to boost the reliability of the model. This framework enriches the valuable features of the training data and enhances the applicability of the samples. Experimental results demonstrate that our method achieves better accuracy in capturing high-frequency features for low- and high-dimensional nonlinear equations, such as the (1+1)-dimensional modified Korteweg-de Vries equation and the (2+1)-dimensional Kadomtsev-Petviashvili equation. The multi-stage PINN accomplishes dependable prediction in both training and extrapolated domains, and establishes a more effective balance between computational cost and performance.