Meshfree-based physics-informed neural networks for the unsteady Oseen equations

Keyi Peng(彭珂依)^{1,†}, Jing Yue(岳靖)^{2,‡}, Wen Zhang(张文)^{2,§}, and Jian Li(李剑)^{1,¶}

1 School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China; 2 School of Electrical and Control Engineering, Shaanxi University of Science and Technology, Xi'an 710021, China

Abstract We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations. Firstly, based on the ideas of meshfree and small sample learning, we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh. Specifically, we optimize the neural network by minimizing the loss function to satisfy the differential operators, initial condition and boundary condition. Then, we prove the convergence of the loss function and the convergence of the neural network. In addition, the feasibility and effectiveness of the method are verified by the results of numerical experiments, and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.

Fund: Project supported in part by the National Natural Science Foundation of China (Grant No. 11771259), Shaanxi Provincial Joint Laboratory of Artificial Intelligence (Grant No. 2022JCSYS05), Innovative Team Project of Shaanxi Provincial Department of Education (Grant No. 21JP013), and Shaanxi Provincial Social Science Fund Annual Project (Grant No. 2022D332).

Keyi Peng(彭珂依), Jing Yue(岳靖), Wen Zhang(张文), and Jian Li(李剑) Meshfree-based physics-informed neural networks for the unsteady Oseen equations 2023 Chin. Phys. B 32 040207

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