# Multi-fidelity graph-based neural networks architectures to learn Navier-Stokes solutions on non-parametrized 2D domains

arXiv: https://arxiv.org/abs/2601.02157 · HTML: https://mlsanity.com/paper/2601.02157/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Time series forecasting with SSMs, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 35 of its 57 references that are in the map.
- First posted on arXiv 5 Jan 2026.

## Placed in

- State-space sequence models (S4, Mamba) / [Time series forecasting with SSMs](https://mlsanity.com/ssm/time-series-forecasting-with-ssms/)

## Builds on (most cited first, 30 of 35)

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- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
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- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
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- [Exphormer: Sparse Transformers for Graphs](https://mlsanity.com/paper/2303.06147/) (2023)
- [Understanding Attention and Generalization in Graph Neural Networks](https://mlsanity.com/paper/1905.02850/) (2019)
- [Graph-Mamba: Towards Long-Range Graph Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2402.00789/) (2024)
- [GraphNorm: A Principled Approach to Accelerating Graph Neural Network Training](https://mlsanity.com/paper/2009.03294/) (2020)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Graph Mamba: Towards Learning on Graphs with State Space Models](https://mlsanity.com/paper/2402.08678/) (2024)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Eagle: Large-Scale Learning of Turbulent Fluid Dynamics with Mesh Transformers](https://mlsanity.com/paper/2302.10803/) (2023)
- [Efficient Learning of Mesh-Based Physical Simulation with BSMS-GNN](https://mlsanity.com/paper/2210.02573/) (2022)
- [Transformers are Graph Neural Networks](https://mlsanity.com/paper/2506.22084/) (2025)
- [Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networks](https://mlsanity.com/paper/2303.07310/) (2023)
- [A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulations](https://mlsanity.com/paper/2212.14545/) (2022)
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- [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)

## Abstract (arXiv)

We propose a graph-based, multi-fidelity learning framework for the prediction of stationary Navier--Stokes solutions in non-parametrized two-dimensional geometries. The method is designed to guide the learning process through successive approximations, starting from reduced-order and full Stokes models, and progressively approaching the Navier--Stokes solution. To effectively capture both local and long-range dependencies in the velocity and pressure fields, we combine graph neural networks with Transformer and Mamba architectures. While Transformers achieve the highest accuracy, we show that Mamba can be successfully adapted to graph-structured data through an unsupervised node-ordering strategy. The Mamba approach significantly reduces computational cost while maintaining performance. Physical knowledge is embedded directly into the architecture through an encoding-processing-physics informed decoding pipeline. Derivatives are computed through algebraic operators constructed via the Weighted Least Squares method. The flexibility of these operators allows us not only to make the output obey the governing equations, but also to constrain selected hidden features to satisfy mass conservation. We introduce additional physical biases through an enriched graph convolution with the same differential operators describing the PDEs. Overall, we successfully guide the learning process by physical knowledge and fluid dynamics insights, leading to more regular and accurate predictions