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.

翻译：本文研究利用深度神经网络求解高维问题产生的超大规模线性系统。由于维数灾难，在存储此类超大规模线性系统的解向量和右端项时会产生高昂成本。我们的核心思想是：在无矩阵框架下，采用参数数量远小于解向量规模的神经网络来表征解。我们给出了所提方法的误差分析，表明解误差受限于矩阵条件数与神经网络逼近误差。通过来自偏微分方程、排队问题和概率布尔网络的多个数值算例，验证了该方法能够相当精确地学习线性系统的解。