  Solving partial differential equations (PDEs) effectively necessitates a
multi-scale approach, particularly critical in high-dimensional scenarios
characterized by increasing grid points or resolution. Traditional methods
often fail to capture the detailed features necessary for accurate modeling,
presenting a significant challenge in scientific computing. In response, we
introduce the Multiwavelet-based Algebraic Multigrid Neural Operator (M2NO), a
novel deep learning framework that synergistically combines multiwavelet
transformations and algebraic multigrid (AMG) techniques. By exploiting the
inherent similarities between these two approaches, M2NO overcomes their
individual limitations and enhances precision and flexibility across various
PDE benchmarks. Employing Multiresolution Analysis (MRA) with high-pass and
low-pass filters, the model executes hierarchical decomposition to accurately
delineate both global trends and localized details within PDE solutions,
supporting adaptive data representation at multiple scales. M2NO also automates
node selection and adeptly manages complex boundary conditions through its
multiwavelet-based operators. Extensive evaluations on a diverse array of PDE
datasets with different boundary conditions confirm M2NO's superior
performance. Furthermore, M2NO excels in handling high-resolution and
super-resolution tasks, consistently outperforming competing models and
demonstrating robust adaptability in complex computational scenarios.
