🤖 AI Summary
Solving high-dimensional partial differential equations (PDEs) faces challenges including difficulty in modeling multiscale features, high computational cost, and poor adaptability to complex boundaries. To address these, this paper proposes the Multiresolution Neural Operator (M2NO). Methodologically, M2NO pioneers the deep integration of multivariate wavelet transforms with algebraic multigrid (AMG), establishing a scalable decomposition framework grounded in multiresolution analysis (MRA) that enables adaptive node selection and robust handling of intricate boundary conditions. It further incorporates high-pass and low-pass filter banks to jointly model global trends and local details. Experiments demonstrate that M2NO consistently outperforms state-of-the-art neural operators across diverse PDE benchmarks. It maintains efficiency in high-resolution and super-resolution tasks, exhibits strong generalization across unseen domains and resolutions, and achieves superior boundary robustness.
📝 Abstract
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.