Zhihao Li

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, Ting Wang, Guojian Zou et al.

Traffic state estimation (TSE) fundamentally involves solving high-dimensional spatiotemporal partial differential equations (PDEs) governing traffic flow dynamics from limited, noisy measurements. While Physics-Informed Neural Networks (PINNs) enforce PDE constraints point-wise, this paper adopts a physics-informed deep operator network (PI-DeepONet) framework that reformulates TSE as an operator learning problem. Our approach trains a parameterized neural operator that maps sparse input data to the full spatiotemporal traffic state field, governed by the traffic flow conservation law. Crucially, unlike PINNs that enforce PDE constraints point-wise, PI-DeepONet integrates traffic flow conservation model and the fundamental diagram directly into the operator learning process, ensuring physical consistency while capturing congestion propagation, spatial correlations, and temporal evolution. Experiments on the NGSIM dataset demonstrate superior performance over state-of-the-art baselines. Further analysis reveals insights into optimal function generation strategies and branch network complexity. Additionally, the impact of input function generation methods and the number of functions on model performance is explored, highlighting the robustness and efficacy of proposed framework.