Coupled Multiwavelet Operator Learning for Coupled Differential Equations

Xiongye Xiao, Defu Cao, Ruochen Yang, Gaurav Gupta, Gengshuo Liu, Chenzhong Yin, Radu Balan, Paul Bogdan

Keywords: Neural operators, coupled differential equations, multiwavelet transform, partial differential equations

TL;DR: We propose a novel coupled multiwavelet operator learning scheme for efficiently solving coupled differential equations.

Abstract: 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 of 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 $2X-4X$ improvement relative $L$2 error compared to the best results from the state-of-the-art models.

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