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Enhancing PINNs for solving PDEs via adaptive collocation point movement and adaptive loss weighting

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Abstract

Physics-informed neural networks (PINNs) are an emerging method for solving partial differential equations (PDEs) and have been widely applied in the field of scientific computing. In this paper, we introduce a novel adaptive PINN model for solving PDEs. The model draws on the idea of traditional adaptive methods and incorporates the adaptive collocation point movement method into the PINNs model. It can use residual information from the PDE or characteristics of the solution function itself to guide the movement of collocation points, giving an appropriate distribution of collocation points for specific problems, improving the predictive accuracy of the model, and avoiding overfitting. Additionally, the model introduces an adaptive loss weighting strategy, which updates adaptive weights continuously by minimizing negative log-likelihood estimation to achieve adaptive weighting of the loss function, thereby improving the convergence rate and accuracy of the model. Finally, we conduct extensive experiments, including the one-dimensional Poisson equation, two-dimensional Poisson equation, Burgers equation, Klein–Gordon equation, Helmholtz equation, and Lid-Driven problem, to demonstrate the effectiveness and accuracy of the proposed model. The experimental results show that the model can significantly improve predictive accuracy and generalization ability. The data and code can be found at https://github.com/hsbhc/AMAW-PINN.

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Data availability

The data and code can be found at https://github.com/hsbhc/AMAW-PINN.

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Funding

This work is supported by the National Key Research and Development Program of China (No. 2021YFA1003004).

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Authors and Affiliations

  1. School of Computer Engineering and Science, Shanghai University, Baoshan, Shanghai, 200444, People’s Republic of China

    Jie Hou & Ying Li

  2. Department of Mathematics, School of Science, Shanghai University, Baoshan, Shanghai, 200444, People’s Republic of China

    Shihui Ying

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  1. Jie Hou
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All authors contributed to the study conception and design. The experiments, data collection, and analysis were performed by JH. The first draft of the manuscript was written by JH, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Ying Li.

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Hou, J., Li, Y. & Ying, S. Enhancing PINNs for solving PDEs via adaptive collocation point movement and adaptive loss weighting. Nonlinear Dyn 111, 15233–15261 (2023). https://doi.org/10.1007/s11071-023-08654-w

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