论文标题
基于细胞平均的神经网络方法，用于高维抛物线差分方程
Cell-average based neural network method for high dimensional parabolic differential equations
论文作者
论文摘要
在本文中，我们介绍了基于细胞平均的神经网络（CANN）方法来解决高维抛物线部分微分方程。该方法基于部分微分方程的积分或弱公式。前馈网络被认为可以在相邻时间训练细胞的溶液平均值。通过高阶数值方法获得的$ t =ΔT$的初始值和近似解决方案分别作为网络的输入和输出。我们使用有监督的培训与简单的反向传播算法来训练网络参数。我们发现，神经网络已接受了针对高维问题的最佳培训，CFL条件并不严格限制CANN方法，并且训练有素的网络用于解决具有不同初始值的相同问题。对于高维抛物线方程，观察到收敛性并显示出与空间网格大小相关的误差，但与时间步长的大小无关。
In this paper, we introduce cell-average based neural network (CANN) method to solve high-dimensional parabolic partial differential equations. The method is based on the integral or weak formulation of partial differential equations. A feedforward network is considered to train the solution average of cells in neighboring time. Initial values and approximate solution at $t=Δt$ obtained by high order numerical method are taken as the inputs and outputs of network, respectively. We use supervised training combined with a simple backpropagation algorithm to train the network parameters. We find that the neural network has been trained to optimality for high-dimensional problems, the CFL condition is not strictly limited for CANN method and the trained network is used to solve the same problem with different initial values. For the high-dimensional parabolic equations, the convergence is observed and the errors are shown related to spatial mesh size but independent of time step size.
