Fourier Neural Networks as Function Approximators and Differential Equation Solvers
This provides a method for modeling or solving partial differential equations with periodic boundary conditions, offering advantages like solution validity outside training regions and interpretability, but it is incremental as it builds on existing neural network and Fourier analysis concepts.
The paper tackles the problem of approximating functions and solving differential equations by introducing Fourier neural networks (FNNs) that map directly to Fourier decomposition, achieving results that closely replicate Fourier series expansions with a simple single-hidden-layer architecture.
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.