A novel number-theoretic sampling method for neural network solutions of partial differential equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918079709052928 |
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| author | Yang, Yu He, Pingan Peng, Xiaoling He, Qiaolin |
| author_facet | Yang, Yu He, Pingan Peng, Xiaoling He, Qiaolin |
| contents | Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling points, which is a robust approach specifically designed to handle partial differential equations with rough solutions or in high dimensions. The sample points are generated by the generating vector to achieve the smallest discrepancy. The architecture integrates Physics-Informed Neural Networks (PINNs) with rigorous mathematical guarantees demonstrating lower error bounds compared to conventional uniform random sampling. Numerical validation includes low-regularity Poisson equations, two-dimensional inverse Helmholtz problems, and high-dimensional linear/nonlinear PDEs, systematically demonstrating the algorithm's superior performance and generalization capabilities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A novel number-theoretic sampling method for neural network solutions of partial differential equations Yang, Yu He, Pingan Peng, Xiaoling He, Qiaolin Numerical Analysis Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling points, which is a robust approach specifically designed to handle partial differential equations with rough solutions or in high dimensions. The sample points are generated by the generating vector to achieve the smallest discrepancy. The architecture integrates Physics-Informed Neural Networks (PINNs) with rigorous mathematical guarantees demonstrating lower error bounds compared to conventional uniform random sampling. Numerical validation includes low-regularity Poisson equations, two-dimensional inverse Helmholtz problems, and high-dimensional linear/nonlinear PDEs, systematically demonstrating the algorithm's superior performance and generalization capabilities. |
| title | A novel number-theoretic sampling method for neural network solutions of partial differential equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2411.17039 |