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An improved physics-informed neural network with adaptive weighting and mixed differentiation for solving the incompressible Navier–Stokes equations

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Abstract

Physics-informed neural networks (PINNs) have been an effective tool for approximating the map** between points in the spatio-temporal domain and solutions of partial differential equations (PDEs). However, there are still some challenges in dealing with the nonlinear characteristics and complexity of the Navier–Stokes (N–S) equations. In this paper, the improved adaptive weighting PINNs based on the Gaussian likelihood estimation are applied to solve the N–S equations. The weights of the different loss items are allocated adaptively by the maximum likelihood estimation. The improved network structure has been designed with considering both the global and local information, making it easier to capture the part of PDEs solution with drastic changes. A combinational method of the numerical differentiation (ND) and the automatic differentiation (AD) is proposed to compute the differential operators, with the improved computational efficiency. The derivative operation of the convection and pressure-gradient terms was carried out using the combined method in solving the incompressible N–S equations. The results show that the effectiveness and training efficiency of this method are better than PINNs.

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Data availability

The code can be found at https://github.com/upwaj/aw-in-md-PINNs.

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Funding

This work is supported by the National Natural Science Foundation of China (Nos. 12361090, 12071406 and 12271465), the Natural Science Foundation of **njiang Province (Nos. 2023D01C164, 6142A05230203, 2022D01D32), and the Tianshan Talent Training Program (No. 2023TSYCQNTJ0015 and No. 2022TSYCTD0019).

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Authors and Affiliations

  1. College of Mathematics and System Science, **njiang University, Urumqi, 830046, People’s Republic of China

    Jie Wang, Xufeng **ao, **nlong Feng & Hui Xu

  2. School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Hui Xu

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  1. Jie Wang
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Contributions

Jie Wang: investigation, software, validation, writing—original draft. Xufeng **ao: conceptualization, methodology, formal analysis, software, writing—review and editing. **nlong Feng: formal analysis, writing—review and editing. Hui Xu: formal analysis, writing—review and editing.

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Correspondence to Xufeng **ao.

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Wang, J., **ao, X., Feng, X. et al. An improved physics-informed neural network with adaptive weighting and mixed differentiation for solving the incompressible Navier–Stokes equations. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09856-6

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