Recently, machine learning tools in particular neural networks have been widely used to solve differential equations. One main advantage of using machine learning, in this case, is that one does not need to mesh the computational domain and can instead randomly draw data points to solve the differential equations of interest. In this work, we propose a simple neural network to approximate low-frequency periodic functions or seek such solutions of differential equations. To this end, we build a Fourier Neural Network (FNN) represented as a shallow neural network (i.e with one hidden layer) based on the Fourier Decomposition. As opposed to traditional neural networks, which feature activation functions such as the sigmoid, logistic, ReLU, hyperbolic tangent and softmax functions, Fourier Neural Networks are composed using sinusoidal activation functions. We propose a strategy to initialize the weights of this FNN and showcase its performance against traditional networks for function approximations and differential equations solutions .
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 .
R
Recently, machine learning tools in particular neural networks have been widely used to solve differential equations. One main advantage of using machine learning, in this case, is that one does not need to mesh the computational domain and can instead randomly draw data points to solve the differential equations of interest. In this work, we propose a simple neural network to approximate low-frequency periodic functions or seek such solutions of differential equations. To this end, we build a Fourier Neural Network (FNN) represented as a shallow neural network (i.e with one hidden layer) based on the Fourier Decomposition. As opposed to traditional neural networks, which feature activation functions such as the sigmoid, logistic, ReLU, hyperbolic tangent and softmax functions, Fourier Neural Networks are composed using sinusoidal activation functions. We propose a strategy to initialize the weights
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
R
and showcase its performance against traditional networks for function approximations and differential equations solutions
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