# Multiscale Neural Operators for Solving Time-Independent PDEs

arXiv: https://arxiv.org/abs/2311.05964 · HTML: https://mlsanity.com/paper/2311.05964/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 26 references that are in the map.
- First posted on arXiv 10 Nov 2023.

## Builds on (most cited first, 22 of 22)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Perceiver IO: A General Architecture for Structured Inputs & Outputs](https://mlsanity.com/paper/2107.14795/) (2021)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [DRew: Dynamically Rewired Message Passing with Delay](https://mlsanity.com/paper/2305.08018/) (2023)
- [Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks](https://mlsanity.com/paper/2206.14092/) (2022)
- [Towards Fast Simulation of Environmental Fluid Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2205.02637/) (2022)
- [GNN-based physics solver for time-independent PDEs](https://mlsanity.com/paper/2303.15681/) (2023)
- [Enhancing Multi-Objective Optimization through Machine Learning-Supported Multiphysics Simulation](https://mlsanity.com/paper/2309.13179/) (2023)

## Built on by (most cited first, 5 of 5)

- [From Structured to Unstructured:A Comparative Analysis of Computer Vision and Graph Models in solving Mesh-based PDEs](https://mlsanity.com/paper/2406.00081/) (2024)
- [ConDiff: A Challenging Dataset for Neural Solvers of Partial Differential Equations](https://mlsanity.com/paper/2406.04709/) (2024)
- [PIBNet: a Physics-Inspired Boundary Network for Multiple Scattering Simulations](https://mlsanity.com/paper/2512.02049/) (2025)
- [Read, Write, Relax: Why Neural PDE Surrogates Need Both Global and Local Processing](https://mlsanity.com/paper/2608.21677/) (2026)
- [ScaGNN: a Graph Neural Network for Multiple Scattering Simulations](https://mlsanity.com/paper/2609.37509/) (2026)

## Abstract (arXiv)

Time-independent Partial Differential Equations (PDEs) on large meshes pose significant challenges for data-driven neural PDE solvers. We introduce a novel graph rewiring technique to tackle some of these challenges, such as aggregating information across scales and on irregular meshes. Our proposed approach bridges distant nodes, enhancing the global interaction capabilities of GNNs. Our experiments on three datasets reveal that GNN-based methods set new performance standards for time-independent PDEs on irregular meshes. Finally, we show that our graph rewiring strategy boosts the performance of baseline methods, achieving state-of-the-art results in one of the tasks.