Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations
Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations
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DOI：
10.48550/arxiv.2303.02304
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发表时间：
2023-03
期刊：
影响因子：
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通讯作者：
Xiongye Xiao;De-An Cao;Ruochen Yang;Gaurav Gupta;Gengshuo Liu;Chenzhong Yin;R. Balan;P. Bogdan
中科院分区：
文献类型：
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作者：
Xiongye Xiao;De-An Cao;Ruochen Yang;Gaurav Gupta;Gengshuo Liu;Chenzhong Yin;R. Balan;P. Bogdan
耦合偏微分方程（PDEs）是对许多物理过程的复杂动力学进行建模的关键任务。最近，神经算子已显示出通过直接在傅里叶/小波空间中学习积分核来求解偏微分方程的能力，因此求解耦合偏微分方程的困难在于处理函数之间的耦合映射。为此，我们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，提出了一种\textit{耦合多小波神经算子}（CMWNO）学习方案。在求解包括格雷 - 斯科特（GS）方程和非局部平均场博弈（MFG）问题在内的耦合偏微分方程时，所提出的模型与先前基于学习的求解器相比，取得了显著更高的精度。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$L$2误差方面有$2\times\sim4\times$的改进。
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.