A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks
A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks
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DOI：
10.1016/j.jcp.2021.110242
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发表时间：
2021-03-04
影响因子：
4.1
通讯作者：
Ni, Naxian
中科院分区：
文献类型：
--
作者：
Dong, Suchuan;Ni, Naxian
本文提出了一种简单有效的方法来表示周期函数，并精确地施加周期边界条件来求解深度神经网络（DNN）微分方程。该方法源于涉及周期函数的函数组合的一些简单性质。它本质上是由dnn表示的任意函数与一组具有可调（训练）参数的独立周期函数组成。我们区分了两种类型的周期条件：一种是对函数及其所有导数（无限阶）施加周期性要求的条件，另一种是对函数及其导数施加周期性要求的条件，直到有限阶k（k >= 0）。前者称为c -∞周期条件，后者称为C-k周期条件。我们定义了构成c -∞周期层和C-k周期层（对于任意k >= 0）的运算。将c -∞（或C-k）周期层作为第二层的深度神经网络自动且精确地满足c -∞（或C-k）周期条件。我们对具有c -∞和C-k周期边界条件的常微分方程和偏微分方程进行了广泛的数值实验，以验证和证明所提出的方法确实精确地执行了DNN解及其导数的周期性，以达到机器精度。(c) 2021爱思唯尔公司版权所有。
We present a simple and effective method for representing periodic functions and enforcing exactly the periodic boundary conditions for solving differential equations with deep neural networks (DNN). The method stems from some simple properties about function compositions involving periodic functions. It essentially composes a DNN-represented arbitrary function with a set of independent periodic functions with adjustable (training) parameters. We distinguish two types of periodic conditions: those imposing the periodicity requirement on the function and all its derivatives (to infinite order), and those imposing periodicity on the function and its derivatives up to a finite order k(k >= 0). The former will be referred to as C-infinity periodic conditions, and the latter C-k periodic conditions. We define operations that constitute a C-infinity periodic layer and a C-k periodic layer (for any k >= 0). A deep neural network with a C-infinity (or C-k) periodic layer incorporated as the second layer automatically and exactly satisfies the C-infinity (or C-k) periodic conditions. We present extensive numerical experiments on ordinary and partial differential equations with C-infinity and C-k periodic boundary conditions to verify and demonstrate that the proposed method indeed enforces exactly, to the machine accuracy, the periodicity for the DNN solution and its derivatives. (c) 2021 Elsevier Inc. All rights reserved.