We propose a novel neural network architecture, termed Multi-Scale Separable Fourier Neural Networks (MS-SFNN), for the accurate and efficient solution of linear and nonlinear high-frequency partial differential equations (PDEs). MS-SFNN exploits a separable representation: given a $d$-dimensional input, it employs $d$ independent subnetworks -- each acting on a single coordinate -- and constructs basis functions via element-wise multiplication of their outputs. The PDE solution is approximated as a linear combination of these basis functions, with coefficients determined by least squares. Critically, all network weights and biases are randomly initialized once, from a uniform distribution with unit variance, and remain fixed thereafter. To enhance expressivity, a tunable scaling factor is introduced in each subnetwork to modulate the frequency content of the resulting basis functions. Fourier features are explicitly embedded through cosine activations, endowing the method with strong spectral approximation capabilities. To mitigate the memory bottleneck associated with dense collocation in high-frequency or three-dimensional problems, we replace automatic differentiation with analytically derived basis function derivatives and develop a memory-efficient batched QR decomposition algorithm for solving large-scale least-squares systems. Numerical experiments demonstrate that MS-SFNN achieves unprecedented accuracy across a range of challenging PDEs, significantly outperforming state-of-the-art methods such as Physics-Informed Neural Networks (PINN) and Separated-Variable Spectral Neural Networks (SV-SNN).

翻译：我们提出了一种新颖的神经网络架构，称为多尺度可分离傅里叶神经网络（MS-SFNN），用于精确高效地求解线性和非线性高频偏微分方程（PDE）。MS-SFNN利用可分离表示：对于$d$维输入，它采用$d$个独立子网络（每个子网络作用于单个坐标），并通过逐元素乘法构造基函数。PDE解近似为这些基函数的线性组合，其系数由最小二乘法确定。关键的是，所有网络权重和偏置仅从单位方差的均匀分布中随机初始化一次，此后保持不变。为增强表达能力，每个子网络中引入可调缩放因子，以调节所得基函数的频率内容。通过余弦激活显式嵌入傅里叶特征，赋予该方法强大的谱逼近能力。为缓解高频或三维问题中密集配置带来的内存瓶颈，我们用解析推导的基函数导数替代自动微分，并开发了内存高效的批量QR分解算法，用于求解大规模最小二乘系统。数值实验表明，MS-SFNN在一系列具有挑战性的PDE上达到了前所未有的精度，显著优于物理信息神经网络（PINN）和分离变量谱神经网络（SV-SNN）等最先进方法。