RSmote uses SMOTE oversampling of high-residual points to train physics-informed neural networks with lower memory use and comparable accuracy to RAD.
Error Analysis of Deep Ritz Methods for Elliptic Equations
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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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An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks
RSmote uses SMOTE oversampling of high-residual points to train physics-informed neural networks with lower memory use and comparable accuracy to RAD.