Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913513281159168 |
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| author | Neufeld, Ariel Nguyen, Tuan Anh |
| author_facet | Neufeld, Ariel Nguyen, Tuan Anh |
| contents | Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has also been proved mathematically, i.e., it has been shown that the number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension $d\in \mathbb{N}$ and the reciprocal of the prescribed accuracy $ε\in (0,1)$. The main contribution of this paper is to rigorously prove for the first time that deep neural networks can also overcome the curse dimensionality in the approximation of a certain class of nonlinear PDEs with gradient-dependent nonlinearities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations Neufeld, Ariel Nguyen, Tuan Anh Numerical Analysis Probability Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has also been proved mathematically, i.e., it has been shown that the number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension $d\in \mathbb{N}$ and the reciprocal of the prescribed accuracy $ε\in (0,1)$. The main contribution of this paper is to rigorously prove for the first time that deep neural networks can also overcome the curse dimensionality in the approximation of a certain class of nonlinear PDEs with gradient-dependent nonlinearities. |
| title | Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2403.09200 |