Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-02-01 DOI:10.4208/nmtma.oa-2023-0058

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

引用次数: 0

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.

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半线性椭圆方程的深里兹方法分析

本文提出了一种使用具有 ${rm ReLU}^2$ 激活的 ResNet 来求解半线性椭圆方程的方法。首先，我们基于椭圆方程的惩罚变分形式提出了一个综合公式。然后，我们应用了适用于多种方程的深度里兹方法。我们根据{rm ReLU}^2$ResNet的深度$\mathcal{D}、宽度$\mathcal{W}$和训练样本数$n$，得出了获得的解与真实解之间的误差上限。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.