AI 中文总结
研究针对神经谱偏微分方程求解器问题，提出Perturbative-NeuSA残差公式，分解目标解为背景与扰动，结合多种元素构建求解器。通过多方程实验表明其优于基线且无需训练，闭包效果有条件，重新构建神经闭包为条件可诊断校正。
AI 中文摘要
神经谱偏微分方程求解器通常会学习整个未解析向量场，即便已有低成本近似模型能捕获大部分轨迹。本文引入Perturbative-NeuSA，一种残差公式，将目标解分解为低保真背景和高分辨率扰动，仅学习未解析动力学。该方法结合固定谱算子、背景依赖校正、目标偏微分方程中的背景缺陷及可选神经闭包。通过2D Burgers、Klein-Gordon和非均匀2D波动方程实验，确定性结构化求解器优于NeuSA基线且无需神经网络训练。在Burgers方程上收益最大，确定性校正分别将训练和外推误差降低24倍和44倍。Klein-Gordon方程实验表明闭包效果取决于背景分辨率，在波动方程中闭包在特定情况下可额外降低18%误差。多初始条件诊断显示有用闭包机制取决于初始条件谱，在结构化校正已捕获主导动力学时外推中可能消失。Perturbative-NeuSA将神经闭包重新构建为由背景保真度、残差组织和与闭包模型兼容性控制的条件可诊断校正。
英文摘要
Neural spectral PDE solvers often learn an entire unresolved vector field even when an inexpensive approximate model can already capture most of the trajectory. Here we introduce Perturbative-NeuSA, a residual formulation that decomposes the target solution into a low-fidelity background and a high-resolution perturbation, so that only the unresolved dynamics is learned. Starting from the exact perturbation equation, the method combines a fixed spectral operator, a background-dependent correction, the background defect in the target PDE, and an optional neural closure. This construction makes the roles of physical structure and neural closure separately measurable. Across 2D Burgers, Klein-Gordon, and heterogeneous 2D wave equations, the deterministic structured solver outperforms the trained NeuSA baseline while requiring no neural-network training. The largest gains occur on Burgers, where the deterministic correction reduces training and extrapolation errors by factors of 24 and 44, respectively. In addition, a Klein-Gordon sweep over seven background resolutions shows that the effect of the closure is conditional: it improves a poor background by 3.6 times, becomes neutral at intermediate resolutions, and degrades a well-resolved background. For the wave equation, however, the closure provides an additional 18% reduction when the remaining residual is interface-localized. Multi-initial-condition diagnostics further show that the useful closure regime depends on the initial-condition spectrum and can disappear in extrapolation when structured correction already captures the dominant Burgers dynamics. Perturbative-NeuSA therefore reframes neural closure as a conditional, diagnosable correction governed by background fidelity, residual organization, and compatibility with the closure model.
Comments16 pages, 5 figures, 14 tables; supplementary material included
