论文标题
傅立叶神经网络作为函数近似器和微分方程求解器
Fourier Neural Networks as Function Approximators and Differential Equation Solvers
论文作者
论文摘要
我们提出了一个傅立叶神经网络（FNN），可以直接映射到傅立叶分解。激活和损耗函数的选择产生的结果可以密切复制傅立叶系列扩展，同时保留具有单个隐藏层的直接体系结构。该网络体系结构的简单性促进了在数据预处理或后处理阶段与任何其他高复杂性网络的集成。我们在自然周期性的平滑函数和分段连续周期性函数上验证该FNN。我们展示了该FNN在具有周期性边界条件下建模或求解部分微分方程的使用。当前方法的主要优点是训练区域之外的解决方案的有效性，受过训练的模型的解释性以及使用的简单性。
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.
