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.

💡 Research Summary

This paper introduces M2NO (Multiwavelet-based Multigrid Neural Operator), a novel deep learning framework designed to efficiently solve high-dimensional, multi-scale partial differential equations (PDEs). The core innovation of M2NO lies in its strategic integration of the classical multigrid method with multiresolution analysis (MRA) using multiwavelets within a neural network architecture.

The authors begin by outlining the challenges faced by existing neural operators, such as Fourier Neural Operators (FNO) and DeepONets, which include poor generalization to unseen resolutions, difficulty in capturing both global trends and local features simultaneously, and inferior iterative efficiency compared to handcrafted multigrid solvers. To overcome these limitations, M2NO draws a formal analogy between the multigrid structure and wavelet-based MRA. It identifies the low-pass filter (𝐻) from the multiwavelet transformation as the natural counterpart to the multigrid restriction operator, and its transpose (𝐻^𝑇) as the prolongation operator.

The methodology involves constructing a hierarchical neural network that mimics the V-cycle of a multigrid solver. At each level, the input is decomposed using the predefined multiwavelet-based operators: low-frequency components (approximations) are restricted to a coarser grid for efficient error correction, while high-frequency components (details) are preserved on finer grids. The smoothing operations at each grid level, which are crucial for error reduction, are parameterized as learnable single-layer convolutional neural networks. This design allows M2NO to learn optimal smoothing kernels from data while being guided by the mathematically sound, multi-resolution structure.

The paper provides extensive empirical validation across a diverse set of PDE benchmarks, including 1D Poisson equations, 2D Darcy flow, Navier-Stokes equations, and large-scale real-world climate data from the ERA5 dataset. Evaluations are conducted in three key settings: high-resolution learning and inference, super-resolution (learning from low-res and predicting high-res data), and use as a preconditioner for conventional iterative solvers like Conjugate Gradient. The results consistently demonstrate that M2NO outperforms state-of-the-art neural operators (FNO, Galerkin Transformer, DeepONet) in terms of prediction accuracy, generalization capability to different resolutions, and convergence speed. Spectral analysis further confirms M2NO’s effectiveness across all frequency bands. Notably, when employed as a preconditioner, M2NO significantly accelerates the convergence of traditional solvers, showcasing its practical utility in hybrid scientific computing workflows.

In conclusion, M2NO successfully bridges the gap between data-driven learning and principled numerical analysis. By embedding the multi-resolution, error-correcting mechanism of multigrid methods into a trainable neural operator via multiwavelets, it achieves a robust, efficient, and accurate solution for dynamic multi-scale PDE problems, marking a significant advance in the field of scientific machine learning.