耦合多小波神经算子学习用于耦合偏微分方程
Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations
AI总结：
提出耦合多小波神经算子（CMWNO），通过在小波空间中解耦积分核来高效求解耦合PDE，在Gray-Scott方程和平均场博弈问题上相比现有方法将相对L2误差降低2至4倍。
摘要：
耦合偏微分方程（PDE）是模拟许多物理过程复杂动力学的关键任务。近年来，神经算子已展现出通过在傅里叶/小波空间中直接学习积分核来求解PDE的能力，因此求解耦合PDE的难点在于处理函数之间的耦合映射。为此，我们提出了一种耦合多小波神经算子（CMWNO）学习方案，通过在小波空间中的多小波分解和重构过程中解耦耦合积分核。与先前的基于学习的求解器相比，所提出的模型在求解耦合PDE（包括Gray-Scott（GS）方程和非局部平均场博弈（MFG）问题）时实现了显著更高的精度。根据我们的实验结果，与最先进模型的最佳结果相比，所提出的模型在相对L2误差上表现出2倍至4倍的改进。
英文摘要：
Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a \textit{coupled multiwavelets neural operator} (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a $2\times \sim 4\times$ improvement relative $L$2 error compared to the best results from the state-of-the-art models.
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