A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks

TL;DR

This paper introduces a novel method for representing periodic functions within deep neural networks, enabling exact enforcement of periodic boundary conditions for differential equations, including conditions on derivatives up to any finite order.

Contribution

The authors propose a new approach that composes DNNs with periodic functions to exactly satisfy both $C^{ ext{infinity}}$ and $C^k$ periodic boundary conditions, improving solution accuracy.

Findings

Method enforces periodicity to machine precision.

Applicable to both ordinary and partial differential equations.

Supports conditions on derivatives up to any finite order.

Abstract

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 $k$ ($k\geqslant 0$). 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 $k\geqslant 0$). 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.