Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations
This addresses the problem of efficiently modeling complex physical dynamics for researchers and practitioners in computational physics and machine learning, representing a strong domain-specific advancement.
The paper tackled solving coupled partial differential equations (PDEs) by proposing a coupled multiwavelets neural operator (CMWNO) that decouples integral kernels in wavelet space, achieving a 2× to 4× improvement in relative L2 error compared to state-of-the-art models on tasks like Gray-Scott equations and non-local mean field games.
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.