Data driven governing equations approximation using deep neural networks
Data driven governing equations approximation using deep neural networks
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DOI：
10.1016/j.jcp.2019.06.042
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发表时间：
2019-10-15
影响因子：
4.1
通讯作者：
Xiu, Dongbin
中科院分区：
文献类型：
--
作者：
Qin, Tong;Wu, Kailiang;Xiu, Dongbin
我们提出了一个使用观测数据和深度神经网络（DNN）近似未知控制方程的数值框架。特别是，我们建议使用残差网络（ResNet）作为方程近似的基本构建块。我们证明了ResNet块可以被认为是一个单步方法，在时间积分中是精确的。然后，我们提出了两个多步方法，递归ResNet（RT-ResNet）方法和递归ReNet（RS-ResNet）方法。RT-ResNet是统一时间步长的多步法，而RS-ResNet是使用可变时间步长的自适应多步法。这里提出的所有三种方法都是基于底层动力系统的积分形式。因此，它们不需要用于方程恢复的时间导数数据，并且可以科普相对粗糙分布的轨迹数据。几个数值例子来证明该方法的性能。(C)2019爱思唯尔公司All rights reserved.
We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods. (C) 2019 Elsevier Inc. All rights reserved.