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This paper studies deep neural networks for solving extremely large linear systems arising from highdimensional problems. Because of the curse of dimensionality, it is expensive to store both the solution and right-hand side vector in such extremely large linear systems. Our idea is to employ a neural network to characterize the solution with much fewer parameters than the size of the solution under a matrix-free setting. We present an error analysis of the proposed method, indicating that the solution error is bounded by the condition number of the matrix and the neural network approximation error. Several numerical examples from partial differential equations, queueing problems, and probabilistic Boolean networks are presented to demonstrate that the solutions of linear systems can be learned quite accurately.

In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a deep adaptive basis Galerkin (DABG) method, which employs the spectral-Galerkin method for the time variable of oscillatory solutions and the deep neural network method for high-dimensional space variables. The proposed method can lead to a linear system of differential equations having unknown DNNs that can be trained via the loss function. We establish a posterior estimates of the solution error, which is bounded by the minimal loss function and the term $O(N^{-m})$, where $N$ is the number of basis functions and $m$ characterizes the regularity of the e'quation. We also show that if the true solution is a Barron-type function, the error bound converges to zero as $M=O(N^p)$ approaches to infinity, where $M$ is the width of the used networks, and $p$ is a positive constant. Numerical examples, including high-dimensional linear evolution equations and the nonlinear Allen-Cahn equation, are presented to demonstrate the performance of the proposed DABG method is better than that of existing DNNs.

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic It\^o diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows. First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Plank equation associated with the estimated drift and diffusion coefficients with a neural-network-based least-squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients, and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term, and generalization error results of nonparametric regression and of PDE regression solution obtained with neural-network models. The effectiveness of this method is reflected by numerical simulations of a two-dimensional Student's t distribution and a 20-dimensional Langevin dynamics.