A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks

• URL: http://arxiv.org/abs/2105.03480v1
• Date: Fri, 7 May 2021 19:49:06 GMT
• Title: A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks
• Authors: Haoya Li, Lexing Ying
• Abstract summary: We propose a semigroup computation method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. For eigenvalue problems, a primal-dual method is proposed, resolving the constraint with a scalar dual variable.
• Score: 1.52292571922932
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. The main advantages are that no mixed second-order derivative computation is needed during the stochastic gradient descent training and that the boundary conditions are taken into account automatically by the semigroup operator. For eigenvalue problems, a primal-dual method is proposed, resolving the constraint with a scalar dual variable. Numerical results are provided to demonstrate the performance of the proposed methods.

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