Related papers: Subspace method based on neural networks for eigen…
We present a subspace method based on neural networks for solving the partial differential equation in weak form with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…
We present a subspace method based on neural networks (SNN) for solving the partial differential equation with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…
Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence…
This study proposes a class of augmented subspace schemes for the weak Galerkin (WG) finite element method used to solve eigenvalue problems. The augmented subspace is built with the conforming linear finite element space defined on the…
We adapt a symmetric interior penalty discontinuous Galerkin method using a patch reconstructed approximation space to solve elliptic eigenvalue problems, including both second and fourth order problems in 2D and 3D. It is a direct…
In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential…
A type of parallel augmented subspace scheme for eigenvalue problems is proposed by using coarse space in the multigrid method. With the help of coarse space in multigrid method, solving the eigenvalue problem in the finest space is…
In this paper, we propose a method for computing eigenvalues of elliptic problems using Deep Learning techniques. A key feature of our approach is that it is independent of the space dimension and can compute arbitrary eigenvalues without…
We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as…
Eigenvalue problems are critical to several fields of science and engineering. We present a novel unsupervised neural network for discovering eigenfunctions and eigenvalues for differential eigenvalue problems with solutions that…
In this paper, we present a nonnested augmented subspace algorithm and its multilevel correction method for solving eigenvalue problems with curved interfaces. The augmented subspace algorithm and the corresponding multilevel correction…
It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of…
In this article, we study eigenvalue problems associated to self-adjoint operators and their approximation obtained by subspace projection, as used in the reduced basis method for instance. We provide error bounds between the exact…
Learning the principal eigenfunctions of an integral operator defined by a kernel and a data distribution is at the core of many machine learning problems. Traditional nonparametric solutions based on the Nystr{\"o}m formula suffer from…
How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the…
We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses the setting…
This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique. A high order lower bound can be obtained at a relatively low cost via the proposed method. The error…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…