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Volume 17, Issue 1
Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

Mo Chen, Yuling Jiao, Xiliang Lu, Pengcheng Song, Fengru Wang & Jerry Zhijian Yang

DOI: 10.4208/nmtma.OA-2023-0058

Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 181-209.

Published online: 2024-02

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  • Abstract

In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${\rm ReLU}^2$ activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations. We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth $\mathcal{D},$ width $\mathcal{W}$ of the ${\rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

  • Keywords

Semilinear elliptic equations, Deep Ritz method, ReLU$^2$ ResNet, convergence rate.

  • AMS Subject Headings

35J61, 68T07, 65N12, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-17-181, author = {Chen , MoJiao , YulingLu , XiliangSong , PengchengWang , Fengru and Yang , Jerry Zhijian}, title = {Analysis of Deep Ritz Methods for Semilinear Elliptic Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {1}, pages = {181--209}, abstract = {

In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${\rm ReLU}^2$ activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations. We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth $\mathcal{D},$ width $\mathcal{W}$ of the ${\rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2023-0058 }, url = {http://global-sci.org/intro/article_detail/nmtma/22915.html} }
TY - JOUR T1 - Analysis of Deep Ritz Methods for Semilinear Elliptic Equations AU - Chen , Mo AU - Jiao , Yuling AU - Lu , Xiliang AU - Song , Pengcheng AU - Wang , Fengru AU - Yang , Jerry Zhijian JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 181 EP - 209 PY - 2024 DA - 2024/02 SN - 17 DO - http://doi.org/10.4208/nmtma.OA-2023-0058 UR - https://global-sci.org/intro/article_detail/nmtma/22915.html KW - Semilinear elliptic equations, Deep Ritz method, ReLU$^2$ ResNet, convergence rate. AB -

In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${\rm ReLU}^2$ activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations. We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth $\mathcal{D},$ width $\mathcal{W}$ of the ${\rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

Chen , MoJiao , YulingLu , XiliangSong , PengchengWang , Fengru and Yang , Jerry Zhijian. (2024). Analysis of Deep Ritz Methods for Semilinear Elliptic Equations. Numerical Mathematics: Theory, Methods and Applications. 17 (1). 181-209. doi:10.4208/nmtma.OA-2023-0058
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