Problem
Research questions and friction points this paper is trying to address.
Coupled Partial Differential Equations
Computational Efficiency
Accuracy Enhancement
Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations
🤖 AI Summary
Traditional neural operators—such as the Fourier neural operator—struggle to capture strong inter-variable couplings in systems governed by coupled partial differential equations (PDEs). To address this, we propose the Coupled Multiwavelet Neural Operator (CMWNO), the first neural operator framework integrating multiwavelet decomposition and reconstruction to explicitly model and decouple functional dependencies among variables in wavelet space. Its core innovation lies in a novel coupled mapping decoupling architecture, synergizing multiscale wavelet representations with end-to-end differentiable PDE solvers. Evaluated on two strongly coupled benchmark problems—the Gray–Scott reaction-diffusion system and nonlocal mean-field games—CMWNO achieves 2–4× lower L² error compared to state-of-the-art learning-based solvers. This demonstrates its superior accuracy and efficiency in modeling complex physical systems, establishing a new paradigm for high-fidelity, data-efficient PDE surrogates.
📝 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 extit{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 $2 imes sim 4 imes$ improvement relative $L$2 error compared to the best results from the state-of-the-art models.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Coupled Multiwavelet Neural Operator
Complex Partial Differential Equations
Enhanced Accuracy and Efficiency
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