Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 181-209.
Published online: 2024-02
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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} }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.