In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 activations.Firstly,we present a comprehensive formulation based on the penalized variational form of the ellipt...In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 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 D,width W of the ReLU2 ResNet,and the num-ber of training samples n.Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.展开更多
基金supported by the National Key Research and Development Program of China(Grant No.2020YFA0714200)the National Nature Science Foundation of China(Grant Nos.12125103,12071362,12371424,12371441)supported by the Fundamental Research Funds for the Central Universities.The numerical calculations have been done at the Supercomputing Center of Wuhan University.
文摘In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 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 D,width W of the ReLU2 ResNet,and the num-ber of training samples n.Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.
