发表机构
Guangdong Technion–Israel Institute of Technology; Shandong University（广东以色列理工学院; 山东大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究对比周期PINNs中傅里叶谱微分与空间AD的性能，发现前者在五种方程-框架设置中训练加速达2.90-18.52倍、内存降68.7%-94.1%，精度相当但需均匀空间网格。
AI 中文摘要
物理信息神经网络（PINNs）通常使用自动微分（AD）评估偏微分方程残差中出现的空间导数，当需要多个或高阶导数时，其计算和内存成本会变得很高。我们对物理空间周期PINNs中的空间AD和傅里叶谱微分进行了受控比较。在每个配对实验中，神经表示、时间微分、优化器、采样流程和训练计划均保持固定，因此两种情况仅在空间微分过程上存在差异。对于傅里叶变体，网络输出在均匀周期网格上求值并转换到傅里叶空间，空间导数通过谱乘法获得，且相同的傅里叶系数可跨导数阶数复用。我们在标准PINNs中针对Allen-Cahn和Korteweg-de Vries方程，以及在因果PINNs中针对Allen-Cahn、Korteweg-de Vries和Kuramoto-Sivashinsky方程，比较了这两种过程。在这五种方程-框架设置中，傅里叶微分的平均配对端到端训练加速比范围为2.90倍至18.52倍，峰值已分配图形处理单元（GPU）内存减少68.7%至94.1%。最终相对L₂误差保持在同一数量级，两种微分过程均未表现出一致的精度优势。对于此处考虑的一维周期基准，傅里叶谱微分在保留相当解误差的同时，提供了比空间AD低得多的训练时间和内存使用，代价是需要均匀结构化空间网格。
英文摘要
Physics-informed neural networks (PINNs) commonly evaluate the spatial derivatives appearing in partial differential equation residuals using automatic differentiation (AD), whose computational and memory costs can become substantial when multiple or high-order derivatives are required. We perform a controlled comparison of spatial AD and Fourier spectral differentiation in periodic physical-space PINNs. Within each paired experiment, the neural representation, temporal differentiation, optimizer, sampling procedure, and training schedule are held fixed, so that the two cases differ only in the spatial differentiation procedure. For the Fourier variant, network outputs are evaluated on a uniform periodic grid and transformed to Fourier space, where spatial derivatives are obtained through spectral multiplication and the same Fourier coefficients are reused across derivative orders. We compare the two procedures in standard PINNs for the Allen--Cahn and Korteweg--de Vries equations and in Causal PINNs for the Allen--Cahn, Korteweg--de Vries, and Kuramoto--Sivashinsky equations. Across these five equation--framework settings, Fourier differentiation yields mean paired end-to-end training speedups ranging from $2.90\times$ to $18.52\times$ and reduces peak allocated graphics processing unit (GPU) memory by $68.7\%$--$94.1\%$. The final relative $L_2$ errors remain of the same order, with neither differentiation procedure showing a consistent accuracy advantage. For the one-dimensional periodic benchmarks considered here, Fourier spectral differentiation therefore provides substantially lower training time and memory usage than spatial AD while retaining comparable solution error, at the cost of requiring a uniform structured spatial grid.
Comments17 pages, 4 figures, 1 table
