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 $C^{\infty }$ periodic conditions, and the latter $C^k$ periodic conditions. We define operations that constitute a $C^{\infty }$ periodic layer and a $C^k$ periodic layer (for any $k⩾0$). A deep neural network with a $C^{\infty }$ (or $C^k$) periodic layer incorporated as the second layer automatically and exactly satisfies the $C^{\infty }$ (or $C^k$) periodic conditions. We present extensive numerical experiments on ordinary and partial differential equations with $C^{\infty }$ 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.

我们提出了一种简单有效的方法，用于表示周期函数并严格执行周期边界条件，以利用深层神经网络（DNN）求解微分方程。该方法源于有关涉及周期函数的函数组成的一些简单属性。它实质上由DNN表示的任意函数和一组具有可调（训练）参数的独立周期函数组成。我们区分两种类型的周期条件：对函数及其所有导数（无穷阶）施加周期性要求的函数，以及对函数及其导数直至有限阶k的周期性赋值的周期条件（$ķ⩾0$）。前者称为$C^{\infty }$ 周期性条件，而后者 $C^{ķ}$周期性条件。我们定义了构成$C^{\infty }$ 周期层和 $C^{ķ}$ 周期性层（对于任何 $ķ⩾0$）。一个深度神经网络$C^{\infty }$ （或者 $C^{ķ}$）作为第二层的周期性层会自动准确地满足 $C^{\infty }$ （或者 $C^{ķ}$）周期性条件。我们目前对常微分方程和偏微分方程进行了广泛的数值实验，$C^{\infty }$ 和 $C^{ķ}$ 周期性边界条件，以验证和证明所提出的方法确实对机器精度确实实现了DNN解及其导数的周期性。