A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks
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Publication:6367123
DOI10.1016/J.JCP.2022.110939arXiv2105.03480WikidataQ114163373 ScholiaQ114163373MaRDI QIDQ6367123
Publication date: 7 May 2021
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. Unlike popular methods like PINN cite{raissi2019physics} and Deep Ritz cite{weinan2018deep} where the Dirichlet boundary condition is enforced solely through penalty functions and thus changes the true solution, the proposed method is able to address the boundary conditions without penalty functions and it gives the correct true solution even when penalty functions are added, thanks to the semigroup operator. For eigenvalue problems, a primal-dual method is proposed, efficiently resolving the constraint with a simple scalar dual variable and resulting in a faster algorithm compared with the BSDE solver cite{han2020solving} in certain problems such as the eigenvalue problem associated with the linear Schr"odinger operator. Numerical results are provided to demonstrate the performance of the proposed methods.
Stochastic analysis (60Hxx) Artificial intelligence (68Txx) Partial differential equations of mathematical physics and other areas of application (35Qxx)
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