Zhilu Lai

Zhihao Li, Zhilu Lai, Xiaobo Zhang et al.

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.

Zhihao Li, Yu Feng, Zhilu Lai et al.

Learning PDE dynamics for fluids increasingly relies on neural operators and Transformer-based models, yet these approaches often lack interpretability and struggle with localized, high-frequency structures while incurring quadratic cost in spatial samples. We propose representing fields with a Gaussian basis, where learned atoms carry explicit geometry (centers, anisotropic scales, weights) and form a compact, mesh-agnostic, directly visualizable state. Building on this representation, we introduce a Gaussian Particle Operator that acts in modal space: learned Gaussian modal windows perform a Petrov-Galerkin measurement, and PG Gaussian Attention enables global cross-scale coupling. This basis-to-basis design is resolution-agnostic and achieves near-linear complexity in N for a fixed modal budget, supporting irregular geometries and seamless 2D-to-3D extension. On standard PDE benchmarks and real datasets, our method attains state-of-the-art competitive accuracy while providing intrinsic interpretability.