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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.15849 |
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| _version_ | 1866917712421191680 |
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| author | Jiao, Yuling Li, Di Lu, Xiliang Yang, Jerry Zhijian Yuan, Cheng |
| author_facet | Jiao, Yuling Li, Di Lu, Xiliang Yang, Jerry Zhijian Yuan, Cheng |
| contents | With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15849 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs Jiao, Yuling Li, Di Lu, Xiliang Yang, Jerry Zhijian Yuan, Cheng Machine Learning Computational Physics 68T07, 65N99 With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers. |
| title | GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs |
| topic | Machine Learning Computational Physics 68T07, 65N99 |
| url | https://arxiv.org/abs/2303.15849 |