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Error Analysis of Deep Ritz Methods for Elliptic Equations

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arxiv 2107.14478 v2 pith:B45HDIW4 submitted 2021-07-30 math.NA cs.NA

Yuling Jiao , Yanming Lai , Yisu Lo , Yang Wang , Yunfei Yang This is my paper · ORCID
classification math.NAcs.NA
keywords deepanalysisconvergenceellipticequationsfunctionsmethodnetworks
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abstract

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \cite{Weinan2017The} for second order elliptic equations with Drichilet, Neumann and Robin boundary condition, respectively. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of number of training samples.

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Cited by 2 Pith papers

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  1. An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks

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    RSmote uses SMOTE oversampling of high-residual points to train physics-informed neural networks with lower memory use and comparable accuracy to RAD.

  2. Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\alpha$-networks: A regularity and approximation perspective

    math.AP 2024-12 conditional novelty 6.0 of 10

    For half-plane Dirichlet data equal to ReLU^alpha, the harmonic solution lies in the ReLU^alpha-Barron space for 0<alpha<1, fails to lie in it for integer alpha, and admits approximations whose Barron norm grows only ...

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