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.

翻译：耦合偏微分方程（PDEs）是许多物理过程复杂动力学建模中的关键任务。近年来，神经算子通过直接在傅里叶/小波空间中学习积分核，展现出求解PDEs的能力，因此求解耦合PDEs的困难在于处理函数之间的耦合映射。为此，我们提出了一种\textit{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与重构过程中解耦耦合积分核。与以往基于学习的求解器相比，所提模型在求解包括Gray-Scott（GS）方程和非局部平均场博弈（MFG）问题在内的耦合PDEs时，实现了显著更高的精度。根据我们的实验结果，与现有最优模型的最佳结果相比，所提模型的相对$L$2误差提升了$2\times \sim 4\times$。