COUPLED MULTIWAVELET NEURAL OPERATOR LEARN-ING FOR COUPLED PARTIAL 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 for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a 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 2ˆ" 4ˆimprovement relative L2 error compared to the best results from the state-of-the-art models.

1. INTRODUCTION

Human perception relies on detecting and processing waves. While our eyes detect waves of electromagnetic radiation, our ears detect waves of compression in the surrounding air. Going beyond waves, from complex dynamics of blood flow to sustain tissue growth and life, to navigating underwater, ground and aerial vehicles at high speeds requires discovering, learning and controlling the partial differential equations (PDEs) governing individual or webs of biological, physical and chemical phenomena (Lacasse et al., 2007; Henriquez, 1993; Laval & Leclercq, 2013; Ghanavati et al., 2017; Radmanesh et al., 2020) . Within this context, neural operators have been successfully used to learn and solve various PDEs. By representing the integral kernel termed as Green's function in the Fourier or Wavelet spaces, the fourier neural operator (Li et al., 2020b) and the multiwaveletbased neural operator (Gupta et al., 2021b;a)) exhibit significant improvements on solving PDEs compared with previous work. However, when it comes to coupled systems characterized by coupled differential equations such as mean field games (MFGs) (Lasry & Lions, 2007; Achdou & Capuzzo-Dolcetta, 2010) , analysis of coupled cyber-physical systems (Xue & Bogdan (2017), or analysis of the surface currents in the tropical Pacific Ocean Bonjean & Lagerloef (2002) , the interactions between the variables/functions need to be considered to decouple the system. Without the knowledge of underlying PDEs, the complex interactions can be hardly represented in the data-driven model. To build a data-driven model that can give a general representation of the interactions to efficiently solve coupled differential equations, we propose the coupled multiwavelets neural operator (CMWNO). Neural Operators. Neural operators (Li et al., 2020b; c; a; Gupta et al., 2021b; Bhattacharya et al., 2020; Patel et al., 2021) focus on learning the mapping between infinite-dimensional spaces of functions. The critical feature for neural operators is to model the integral operator namely the Green's function through various neural network architectures. The graph neural operators (Li et al., 2020b; c) 



use the graph kernel to model the integral operator inspired by graph neural networks; the ˚Equal Contribution 1

