Fourier Neural Networks as Function Approximators and Differential Equation Solvers

TL;DR

This paper introduces a Fourier neural network (FNN) that closely replicates Fourier series, offering a simple, interpretable model for function approximation and solving periodic differential equations, with advantages in validity, interpretability, and ease of use.

Contribution

The paper presents a novel FNN architecture that directly maps to Fourier decomposition, enabling effective approximation and PDE solving with a simple, interpretable network.

Findings

FNN closely replicates Fourier series expansion.

FNN effectively models periodic functions.

FNN can solve PDEs with periodic boundary conditions.

Abstract

We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a straightforward architecture with a single hidden layer. The simplicity of this network architecture facilitates the integration with any other higher-complexity networks, at a data pre- or postprocessing stage. We validate this FNN on naturally periodic smooth functions and on piecewise continuous periodic functions. We showcase the use of this FNN for modeling or solving partial differential equations with periodic boundary conditions. The main advantages of the current approach are the validity of the solution outside the training region, interpretability of the trained model, and simplicity of use.