论文标题
深度神经网络，用于解决高维问题引起的大型线性系统
Deep neural networks for solving large linear systems arising from high-dimensional problems
论文作者
论文摘要
本文研究了深层神经网络，用于解决高度问题引起的极大的线性系统。由于维度的诅咒，将解决方案和右侧向量存储在如此大的线性系统中是昂贵的。我们的想法是使用神经网络来表征与无基质设置下解决方案大小的参数更少的解决方案。我们提出了提出方法的错误分析，表明解决方案误差是由矩阵的条件数和神经网络近似误差界定的。提出了一些来自部分微分方程，排队问题和概率布尔网络的数值示例，以证明可以非常准确地学习线性系统的解决方案。
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.
