Machine learning for solving high-dimensional partial differential equation

Zijian Mei; Zhengzheng Hao; Zhengan Chen; Chongyang Zhu; Shuo Zhao; Jingrun Chen

doi:10.1117/12.2627812

22 April 2022 Machine learning for solving high-dimensional partial differential equation

Zijian Mei, Zhengzheng Hao, Zhengan Chen, Chongyang Zhu, Shuo Zhao, Jingrun Chen

Proceedings Volume 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021); 121634A (2022) https://doi.org/10.1117/12.2627812
Event: International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 2021, Nanjing, China

Abstract

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. In the Deep-Ritz method, proposed by professor E. Weinan, how to optimize the neural network to make it more accurate has become a problem worthy of attention. In our work, we have conducted a comparative study on the network structure and different optimization methods. In terms of network structure, primarily, we introduced the RBF activation function, combined it with the ResNet network, and proposed the Combined-Ritz network (CRM). Comparing it with DRM and RRM (the network simply based on RBF function), We can see that in the case of low-dimensionality, DRM converges slowly and has many parameters, but the accuracy is higher. RRM converges fast, has fewer parameters, and has a lower accuracy. CRM combines the advantages of the two with fewer parameters and higher accuracy. In addition, in the two-dimensional situation, we proposed the CNN network architecture to solve the partial differential equation problem, and achieved good success.

Citation Download Citation

Zijian Mei, Zhengzheng Hao, Zhengan Chen, Chongyang Zhu, Shuo Zhao, and Jingrun Chen "Machine learning for solving high-dimensional partial differential equation", Proc. SPIE 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 121634A (22 April 2022); https://doi.org/10.1117/12.2627812

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