Each week, we select a recently published Open Access article to feature. This week’s article comes from Statistical Analysis and Data Mining showcases a Fourier neural network that can be use for modeling and solving partial differential equations with periodic boundary conditions.
The article’s abstract is given below, with the full article available to read here.
, , Fourier neural networks as function approximators and differential equation solvers, Stat Anal Data Min: The ASA Data Sci Journal. 1– 15. https://doi.org/10.1002/sam.11531
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.