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.

通过多小波变换和代数多网格技术，本文引入了一种名为 M2NO 的新型深度学习框架，以有效解决高维场景下部分微分方程的建模问题。M2NO 通过执行多分辨率分析和分层分解，能够准确绘制 PDE 解的全局趋势和局部细节，并通过其多小波操作实现复杂边界条件的自动选择和管理。对各种 PDE 数据集的广泛评估证实了 M2NO 在性能上的卓越表现。此外，M2NO 在高分辨率和超分辨率任务中表现出色，始终胜过竞争模型，并在复杂计算情景中展现出强大的适应能力。