IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i1p150-d1559485.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/1/150/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/1/150/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Evangelos Katsamakas & Oleg V. Pavlov & Ryan Saklad, 2024. "Artificial intelligence and the transformation of higher education institutions," Papers 2402.08143, arXiv.org.
    2. Fabian Dvorak & Regina Stumpf & Sebastian Fehrler & Urs Fischbacher, 2024. "Generative AI Triggers Welfare-Reducing Decisions in Humans," Papers 2401.12773, arXiv.org.
    3. Koehler, Maximilian & Sauermann, Henry, 2024. "Algorithmic management in scientific research," Research Policy, Elsevier, vol. 53(4).
    4. Giacomo Damioli & Vincent Van Roy & Daniel Vertesy & Marco Vivarelli, 2024. "AI as a new emerging technological paradigm: evidence from global patenting," DISCE - Quaderni del Dipartimento di Politica Economica dipe0038, Università Cattolica del Sacro Cuore, Dipartimenti e Istituti di Scienze Economiche (DISCE).
    5. Anil R. Doshi & Oliver P. Hauser, 2023. "Generative artificial intelligence enhances creativity but reduces the diversity of novel content," Papers 2312.00506, arXiv.org, revised Mar 2024.
    6. Naudé, Wim, 2024. "What They Don't Teach You about Artificial Intelligence at Business School: Stagnation, Oil, and War," IZA Discussion Papers 17306, Institute of Labor Economics (IZA).
    7. Mohseni, Morteza, 2023. "Deep learning in bifurcations of particle trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    8. Sani I. Abba & Mohamed A. Yassin & Auwalu Saleh Mubarak & Syed Muzzamil Hussain Shah & Jamilu Usman & Atheer Y. Oudah & Sujay Raghavendra Naganna & Isam H. Aljundi, 2023. "Drinking Water Resources Suitability Assessment Based on Pollution Index of Groundwater Using Improved Explainable Artificial Intelligence," Sustainability, MDPI, vol. 15(21), pages 1-21, November.
    9. Evangelos Katsamakas & Oleg V. Pavlov & Ryan Saklad, 2024. "Artificial Intelligence and the Transformation of Higher Education Institutions: A Systems Approach," Sustainability, MDPI, vol. 16(14), pages 1-21, July.
    10. He, Hongwen & Su, Qicong & Huang, Ruchen & Niu, Zegong, 2024. "Enabling intelligent transferable energy management of series hybrid electric tracked vehicle across motion dimensions via soft actor-critic algorithm," Energy, Elsevier, vol. 294(C).
    11. Chen Wang & Xu Wu & Ziyu Xie & Tomasz Kozlowski, 2023. "Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference," Energies, MDPI, vol. 16(22), pages 1-23, November.
    12. Van Khanh Pham & Duc Minh Le, 2024. "Impact of Artificial Intelligence on Environmental Quality through Technical Change: A Free Dynamic Equilibrium Approach," Papers 2410.06501, arXiv.org.
    13. Damioli, Giacomo & Van Roy, Vincent & Vertesy, Daniel & Vivarelli, Marco, 2024. "Is Artificial Intelligence Generating a New Paradigm? Evidence from the Emerging Phase," IZA Discussion Papers 17183, Institute of Labor Economics (IZA).
    14. Stefano Bianchini & Moritz Muller & Pierre Pelletier, 2023. "Drivers and Barriers of AI Adoption and Use in Scientific Research," Papers 2312.09843, arXiv.org, revised Feb 2024.
    15. Hermann, Erik & Puntoni, Stefano, 2024. "Artificial intelligence and consumer behavior: From predictive to generative AI," Journal of Business Research, Elsevier, vol. 180(C).
    16. Nicoleta Mihaela Doran & Gabriela Badareu & Marius Dalian Doran & Maria Enescu & Anamaria Liliana Staicu & Mariana Niculescu, 2024. "Greening Automation: Policy Recommendations for Sustainable Development in AI-Driven Industries," Sustainability, MDPI, vol. 16(12), pages 1-17, June.
    17. Almeida, Derick & Naudé, Wim & Sequeira, Tiago Neves, 2024. "Artificial Intelligence and the Discovery of New Ideas: Is an Economic Growth Explosion Imminent?," IZA Discussion Papers 16766, Institute of Labor Economics (IZA).
    18. Wenhao Wan & Yongzhong Tian & Jinglian Tian & Chengxi Yuan & Yan Cao & Kangning Liu, 2024. "Research Progress in Spatiotemporal Dynamic Simulation of LUCC," Sustainability, MDPI, vol. 16(18), pages 1-18, September.
    19. Singh, Kuldeep & Chatterjee, Sheshadri & Mariani, Marcello, 2024. "Applications of generative AI and future organizational performance: The mediating role of explorative and exploitative innovation and the moderating role of ethical dilemmas and environmental dynamis," Technovation, Elsevier, vol. 133(C).
    20. repec:jss:jstsof:33:i11 is not listed on IDEAS
    21. Bo Wang & Jian Qing Shi, 2014. "Generalized Gaussian Process Regression Model for Non-Gaussian Functional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1123-1133, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:1:p:150-:d:1559485. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.