A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks
This addresses a specific challenge in computational physics and engineering for solving differential equations with periodic boundary conditions, but it is incremental as it builds on existing DNN frameworks with a novel layer design.
The authors tackled the problem of representing periodic functions and enforcing exactly periodic boundary conditions in solving differential equations with deep neural networks, by introducing a method that composes a DNN with periodic layers to automatically satisfy conditions to machine accuracy, as verified through extensive numerical experiments.
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.