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An Importance Sampling Method for Generating Optimal Interpolation Points in Training Physics-Informed Neural Networks

Author

Listed:
  • Hui Li

    (Department of Computer Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China)

  • Yichi Zhang

    (Department of Computer Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China)

  • Zhaoxiong Wu

    (Beijing Institute of Computer Technology and Applications, Beijing 100854, China)

  • Zhe Wang

    (Beijing Institute of Computer Technology and Applications, Beijing 100854, China)

  • Tong Wu

    (Beijing Institute of Computer Technology and Applications, Beijing 100854, China)

Abstract

The application of machine learning and artificial intelligence to solve scientific challenges has significantly increased in recent years. A remarkable development is the use of Physics-Informed Neural Networks (PINNs) to solve Partial Differential Equations (PDEs) numerically. However, current PINN techniques often face problems with accuracy and slow convergence. To address these problems, we propose an importance sampling method to generate optimal interpolation points during training. Experimental results demonstrate that our method achieves a 43% reduction in root mean square error compared to state-of-the-art methods when applied to the one-dimensional Korteweg–De Vries equation.

Suggested Citation

  • Hui Li & Yichi Zhang & Zhaoxiong Wu & Zhe Wang & Tong Wu, 2025. "An Importance Sampling Method for Generating Optimal Interpolation Points in Training Physics-Informed Neural Networks," Mathematics, MDPI, vol. 13(1), pages 1-20, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:1:p:150-:d:1559485
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    References listed on IDEAS

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    1. Hanchen Wang & Tianfan Fu & Yuanqi Du & Wenhao Gao & Kexin Huang & Ziming Liu & Payal Chandak & Shengchao Liu & Peter Katwyk & Andreea Deac & Anima Anandkumar & Karianne Bergen & Carla P. Gomes & Shir, 2023. "Scientific discovery in the age of artificial intelligence," Nature, Nature, vol. 620(7972), pages 47-60, August.
    2. Evans, Michael & Swartz, Timothy, 2000. "Approximating Integrals via Monte Carlo and Deterministic Methods," OUP Catalogue, Oxford University Press, number 9780198502784.
    3. Hanchen Wang & Tianfan Fu & Yuanqi Du & Wenhao Gao & Kexin Huang & Ziming Liu & Payal Chandak & Shengchao Liu & Peter Katwyk & Andreea Deac & Anima Anandkumar & Karianne Bergen & Carla P. Gomes & Shir, 2023. "Publisher Correction: Scientific discovery in the age of artificial intelligence," Nature, Nature, vol. 621(7978), pages 33-33, September.
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    Cited by:

    1. Leonid Serkin & Tatyana L. Belyaeva, 2025. "Physics-Informed Neural Networks for Higher-Order Nonlinear Schrödinger Equations: Soliton Dynamics in External Potentials," Mathematics, MDPI, vol. 13(11), pages 1-28, June.

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