用可学习的非谐波傅里叶基构建神经网络，提升科学计算建模精度。
Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning
- 采用余弦激活的前馈结构，将频率、振幅和相位设为可训练参数。
- 在偏微分方程与最优控制问题上表现优于或媲美SIREN和普通神经网络。
- 首次实现分离形式的多维傅里叶基表示，支持周期与非周期函数建模。
我们提出傅里叶学习机（FLM），一种用于表示多维非谐波傅里叶级数的神经网络架构。FLM采用简单的前馈结构，以余弦函数作为激活函数，将频率、振幅和相位作为可训练参数，从而生成适应具体问题的谱基，适用于周期与非周期函数。与以往傅里叶启发的神经网络不同，FLM是首个能以标准多层感知机结构表示具有完整可分离基的多维傅里叶级数的架构。我们证明了傅里叶系数与振幅、相位之间的唯一对应关系，实现了全分离基形式与余弦相位移形式之间的转换。此外，在多个科学计算任务中评估了FLM性能，包括基准偏微分方程（PDEs）和一类最优控制问题（OCPs）。实验表明，FLM在精度和效率上与现有架构如SIREN和传统前馈网络相当，甚至更优。
原文摘要 · Abstract (English)
We introduce the Fourier Learning Machine (FLM), a neural network (NN) architecture designed to represent a multidimensional nonharmonic Fourier series. The FLM uses a simple feedforward structure with cosine activation functions to learn the frequencies, amplitudes, and phase shifts of the series as trainable parameters. This design allows the model to create a problem-specific spectral basis adaptable to both periodic and nonperiodic functions. Unlike previous Fourier-inspired NN models, the FLM is the first architecture able to represent a multidimensional Fourier series with a complete set of basis functions in separable form, doing so by using a standard Multilayer Perceptron-like architecture. A one-to-one correspondence between the Fourier coefficients and amplitudes and phase-shifts is demonstrated, allowing for the translation between a full, separable basis form and the cosine phase-shifted one. Additionally, we evaluate the performance of FLMs on several scientific computing problems, including benchmark Partial Differential Equations (PDEs) and a family of Optimal Control Problems (OCPs). Computational experiments show that the performance of FLMs is comparable, and often superior, to that of established architectures like SIREN and vanilla feedforward NNs.
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