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，神经算子通过直接在傅里叶/小波空间中学习积分核，展现了解耦PDEs的能力，因此求解耦合PDEs的难点在于处理函数之间的耦合映射。为此，我们提出了一种称为\textit{耦合多小波神经算子}（CMWNO）的学习方案，通过在小波空间中的多小波分解和重构过程中对耦合积分核进行解耦。与之前基于学习的求解器相比，该模型在求解Gray-Scott（GS）方程和非局部平均场博弈（MFG）问题等耦合PDEs时，实现了显著更高的精度。根据我们的实验结果，该模型相对于最先进模型的最佳结果，在相对$L$2误差上展示出$2\times \sim 4\times$的提升。