M2NO: An Efficient Multi-Resolution Operator Framework for Dynamic Multi-Scale PDE Solvers

Solving high-dimensional partial differential equations (PDEs) efficiently requires handling multi-scale features across varying resolutions. To address this challenge, we present the Multiwavelet-based Multigrid Neural Operator (M2NO), a deep learning framework that integrates a multigrid structure with predefined multiwavelet spaces. M2NO leverages multi-resolution analysis to selectively transfer low-frequency error components to coarser grids while preserving high-frequency details at finer levels. This design enhances both accuracy and computational efficiency without introducing additional complexity. Moreover, M2NO serves as an effective preconditioner for iterative solvers, further accelerating convergence in large-scale PDE simulations. Through extensive evaluations on diverse PDE benchmarks, including high-resolution, super-resolution tasks, and preconditioning settings, M2NO consistently outperforms existing models. Its ability to efficiently capture fine-scale variations and large-scale structures makes it a robust and versatile solution for complex PDE simulations. Our code and datasets are available on https://github.com/lizhihao2022/M2NO.