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Solving Partial Differential Equations Using Point-Based Neural Networks

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Neural Information Processing (ICONIP 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13623))

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Abstract

Recently, solving partial differential equations (PDEs) using neural networks (NNs) has been attracting increasing interests with promising potential to be applied in wide areas. In this paper, we propose a theoretical model that approximates the operator from a parametric function space to a solution function space and prove its universal approximation theorem in the operator space. For practical application, by regarding the domain of a parametric (or solution) function as a discrete point cloud, we propose a novel idea that implements the theoretical model by introducing the point-based NN as the backbone. We show that the present model can approximate the solution operator of static PDEs using the training data generated on unstructured meshes while most existing methods work for the data generated on lattice grid meshes. We conduct experiments to demonstrate the performance of our model on different types of PDEs. The numerical results verify that our model possesses a higher precision and a faster inference speed compared with the existing models for data of unstructured meshes; in addition, our model has competitive performance compared with the existing works dealing with data of lattice grid meshes.

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Acknowledgments

This work is jointly supported by the National Key R &D Program of China (No. 2018AAA0100303), the Shanghai Municipal Science and Technology Major Project (No.2018SHZDZX01) and the ZHANGJIANG LAB, the National Natural Science Foundation of China under Grant 62072111.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Ning Hua & Wenlian Lu

Authors
  1. Ning Hua
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  2. Wenlian Lu
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Corresponding author

Correspondence to Wenlian Lu .

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Editors and Affiliations

  1. Indian Institute of Technology Indore, Indore, India

    Mohammad Tanveer

  2. Indian Institute of Information Technology - Allahabad, Prayagraj, India

    Sonali Agarwal

  3. Kobe University, Kobe, Japan

    Seiichi Ozawa

  4. Indian Institute of Technology Patna, Patna, India

    Asif Ekbal

  5. University of Innsbruck, Innsbruck, Austria

    Adam Jatowt

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Hua, N., Lu, W. (2023). Solving Partial Differential Equations Using Point-Based Neural Networks. In: Tanveer, M., Agarwal, S., Ozawa, S., Ekbal, A., Jatowt, A. (eds) Neural Information Processing. ICONIP 2022. Lecture Notes in Computer Science, vol 13623. Springer, Cham. https://doi.org/10.1007/978-3-031-30105-6_1

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  • DOI: https://doi.org/10.1007/978-3-031-30105-6_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-30104-9

  • Online ISBN: 978-3-031-30105-6

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