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

TL;DR

This work develops a physics-informed, multi-fidelity framework for predicting stationary Navier–Stokes solutions in non-parametrized 2D domains using graph neural networks. By coupling 1D Stokes, 2D Stokes, and 2D Navier–Stokes solvers within a two-network pipeline and embedding PDE knowledge via an encoding–processing–decoding scheme, it achieves accurate predictions while enforcing physical constraints through Weighted Least Squares derivatives and a Grad-Lapl Graph Convolution. The study demonstrates that physics-informed channels and multi-fidelity training improve robustness and accuracy, with GraphTransformer providing the best performance and GraphMamba offering a more cost-efficient alternative. Results on VESSEL and CYLINDER datasets show meaningful gains over baselines and highlight the trade-offs between accuracy and computational cost, suggesting practical pathways for real-time, physics-consistent CFD surrogates. The framework also opens avenues for extending to unsteady problems and higher dimensions, with code and data to be released.

Abstract

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

Figures (12)

• Figure 1: Global multi-fidelity pipeline: two networks are trained together, with the output of the Stokes net that is passed as input for the final Navier-Stokes net.
• Figure 2: Examples of geometries from (a) VESSEL and (b) CYLINDER datasets.
• Figure 3: Clustering module with the proposed node ordering based on regions $r$ and levels $l$. There is a first global region order (here, $r_0$, $...$, $r_3$); then we define two orders from the $l_0$ and the $l_1$ scores. Specifically, for the first level, nodes are traversed region-wise, where within each region $r_i$, nodes are visited from the highest to the lowest $l_0$ score. The same logic is applied to the second level $l_1$.
• Figure 4: Final GraphTransformer and GraphMamba architectures with the respective steps on the right. Below the Transformer model $(*)$ with its inner Transformer Block, and the Mamba model $(**)$.
• Figure 5: The Grad-Lapl Graph Convolution layer. The required derivatives (first order and second order) are computed through the WLSQ operators $G_x, G_y \ \text{and} \ K$. The new features are concatenated and are finally processed by a GAT Convolution layer.