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.

中文翻译：

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作为函数逼近器和微分方程求解器的傅立叶神经网络

我们提出了一个可以直接映射到傅立叶分解的傅立叶神经网络（FNN）。激活函数和损失函数的选择产生的结果紧密复制了傅立叶级数展开，同时保留了具有单个隐藏层的简单架构。这种网络架构的简单性有助于在数据预处理或后处理阶段与任何其他更复杂的网络集成。我们在自然周期平滑函数和分段连续周期函数上验证了这个 FNN。我们展示了使用此 FNN 建模或求解具有周期性边界条件的偏微分方程。当前方法的主要优点是训练区域外解决方案的有效性、训练模型的可解释性以及使用的简单性。