改进傅里叶神经算子，高效建模参数化与耦合偏微分方程
Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs
- 用超网络调节算子，实现对物理参数的精准建模
- 在等离子体和格雷-斯科特系统上误差降低55%-72%
- 兼顾共享结构与变量交互，适合科学计算场景
参数化与耦合偏微分方程（PDEs）在科学与工程建模中至关重要，但现有神经算子方法难以同时处理二者。本文在傅里叶神经算子（FNO）基础上进行最小架构修改：针对参数化动态，提出基于超网络的调制机制以条件化算子；针对耦合系统，系统探索了架构选择，分析如何在保持标准FNO效率的同时平衡共享结构与跨变量交互。在基准测试中，包括一维电容耦合等离子体方程与格雷-斯科特系统，所提方法相比强基线误差降低55%-72%，验证了有原则的调制与系统性设计的有效性。
原文摘要 · Abstract (English)
Parameterized and coupled partial differential equations (PDEs) are central to modeling phenomena in science and engineering, yet neural operator methods that address both aspects remain limited. We extend Fourier neural operators (FNOs) with minimal architectural modifications along two directions. For parameterized dynamics, we propose a hypernetwork-based modulation that conditions the operator on physical parameters. For coupled systems, we conduct a systematic exploration of architectural choices, examining how operator components can be adapted to balance shared structure with cross-variable interactions while retaining the efficiency of standard FNOs. Evaluations on benchmark PDEs, including the one-dimensional capacitively coupled plasma equations and the Gray-Scott system, show that our methods achieve up to 55-72% lower errors than strong baselines, demonstrating the effectiveness of principled modulation and systematic design exploration.
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