# MAgNet: Mesh Agnostic Neural PDE Solver

arXiv: https://arxiv.org/abs/2210.05495 · HTML: https://mlsanity.com/paper/2210.05495/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 56 references that are in the map.
- First posted on arXiv 11 Oct 2022.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [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)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge](https://mlsanity.com/paper/1711.07970/) (2017)
- [Visual Interaction Networks](https://mlsanity.com/paper/1706.01433/) (2017)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework](https://mlsanity.com/paper/2005.01463/) (2020)

## Built on by (most cited first, 28 of 28)

- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2411.15178/) (2024)
- [Knowledge-guided Machine Learning: Current Trends and Future Prospects](https://mlsanity.com/paper/2403.15989/) (2024)
- [Improving day-ahead Solar Irradiance Time Series Forecasting by Leveraging Spatio-Temporal Context](https://mlsanity.com/paper/2306.01112/) (2023)
- [Multi-Scale Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2302.03580/) (2023)
- [Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.03919/) (2024)
- [Local Off-Grid Weather Forecasting with Multi-Modal Earth Observation Data](https://mlsanity.com/paper/2410.12938/) (2024)
- [Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids](https://mlsanity.com/paper/2402.08187/) (2024)
- [Space and Time Continuous Physics Simulation From Partial Observations](https://mlsanity.com/paper/2401.09198/) (2024)
- [Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models](https://mlsanity.com/paper/2505.10930/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Breaking the Discretization Barrier of Continuous Physics Simulation Learning](https://mlsanity.com/paper/2509.17955/) (2025)
- [EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equations](https://mlsanity.com/paper/2304.03689/) (2023)
- [Learning Space-Time Continuous Neural PDEs from Partially Observed States](https://mlsanity.com/paper/2307.04110/) (2023)
- [Discovering Physical Directions in Weight Space: Composing Neural PDE Experts](https://mlsanity.com/paper/2605.14546/) (2026)
- [Learning Efficient Surrogate Dynamic Models with Graph Spline Networks](https://mlsanity.com/paper/2310.16397/) (2023)
- [Topology-Agnostic Graph U-Nets for Scalar Field Prediction on Unstructured Meshes](https://mlsanity.com/paper/2410.06406/) (2024)
- [Accelerating PDE Surrogates via RL-Guided Mesh Optimization](https://mlsanity.com/paper/2603.02066/) (2026)
- [Physics-enhanced Neural Operator for Simulating Turbulent Transport](https://mlsanity.com/paper/2406.04367/) (2024)
- [COMPOL: A Unified Neural Operator Framework for Scalable Multi-Physics Simulations](https://mlsanity.com/paper/2501.17296/) (2025)
- [Reconstructing Turbulent Flows Using Physics-Aware Spatio-Temporal Dynamics and Test-Time Refinement](https://mlsanity.com/paper/2304.12130/) (2023)
- [A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equations](https://mlsanity.com/paper/2408.14461/) (2024)
- [How to Re-enable PDE Loss for Physical Systems Modeling Under Partial Observation](https://mlsanity.com/paper/2412.09116/) (2024)
- [Modeling Continuous Spatial-temporal Dynamics of Turbulent Flow with Test-time Refinement](https://mlsanity.com/paper/2412.19927/) (2024)
- [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026)

## Abstract (arXiv)

The computational complexity of classical numerical methods for solving Partial Differential Equations (PDE) scales significantly as the resolution increases. As an important example, climate predictions require fine spatio-temporal resolutions to resolve all turbulent scales in the fluid simulations. This makes the task of accurately resolving these scales computationally out of reach even with modern supercomputers. As a result, current numerical modelers solve PDEs on grids that are too coarse (3km to 200km on each side), which hinders the accuracy and usefulness of the predictions. In this paper, we leverage the recent advances in Implicit Neural Representations (INR) to design a novel architecture that predicts the spatially continuous solution of a PDE given a spatial position query. By augmenting coordinate-based architectures with Graph Neural Networks (GNN), we enable zero-shot generalization to new non-uniform meshes and long-term predictions up to 250 frames ahead that are physically consistent. Our Mesh Agnostic Neural PDE Solver (MAgNet) is able to make accurate predictions across a variety of PDE simulation datasets and compares favorably with existing baselines. Moreover, MAgNet generalizes well to different meshes and resolutions up to four times those trained on.