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Asymptotic-Preserving Neural Networks for Multiscale Time-Dependent Linear Transport Equations

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Abstract

In this paper we develop a neural network for the numerical simulation of time-dependent linear transport equations with diffusive scaling and uncertainties. The goal of the network is to resolve the computational challenges of curse-of-dimensionality and multiple scales of the problem. We first show that a standard Physics-Informed Neural Network (PINN) fails to capture the multiscale nature of the problem, hence justifies the need to use Asymptotic-Preserving Neural Networks (APNNs). We show that not all classical AP formulations are directly fit for the neural network approach. We construct a micro-macro decomposition based neural network, and also build in a mass conservation mechanism into the loss function, in order to capture the dynamic and multiscale nature of the solutions. Numerical examples are used to demonstrate the effectiveness of this APNNs.

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Notes

  1. BAAI.2020. Suggested Notation for Machine Learning. https://github.com/mazhengcn/suggested-notation-for-machine-learning.

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Acknowledgements

This work is partially supported by the National Key R &D Program of China Project No. 2020YFA0712000 and Shanghai Municipal of Science and Technology Major Project No. 2021SHZDZX0102. Shi Jin is also supported by NSFC Grant No. 11871297. Zheng Ma is also supported by NSFC Grant No. 12031013 and partially supported by Institute of Modern Analysis—A Shanghai Frontier Research Center.

Funding

The work is supported by the National Key R &D Program of China Project No. 2020YFA0712000 and Shanghai Municipal of Science and Technology Major Project No. 2021SHZDZX0102. Shi Jin is also supported by NSFC grant No. 11871297. Zheng Ma is also supported by NSFC Grant No.12031013 and partially supported by Institute of Modern Analysis – A Shanghai Frontier Research Center.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China

    Shi Jin, Zheng Ma & Keke Wu

  2. Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, China

    Shi Jin & Zheng Ma

  3. Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, China

    Zheng Ma

  4. CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China

    Shi Jin, Zheng Ma & Keke Wu

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  1. Shi Jin
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Jin, S., Ma, Z. & Wu, K. Asymptotic-Preserving Neural Networks for Multiscale Time-Dependent Linear Transport Equations. J Sci Comput 94, 57 (2023). https://doi.org/10.1007/s10915-023-02100-0

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