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

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

💡 Research Summary

The paper introduces a novel multi‑fidelity graph‑based learning framework for predicting stationary Navier‑Stokes (NS) solutions on arbitrary, non‑parametric 2‑D domains. The core idea is to guide the network through a cascade of increasingly accurate physical models: a low‑fidelity 1‑D Stokes solution is first mapped to a 2‑D Stokes field by a network called NN_ST, and then a second network NN_NS predicts the full NS velocity and pressure fields using the Stokes output as an additional input. By training both networks jointly, the approach reduces the amount of expensive high‑fidelity data required while allowing the model to learn the nonlinear convective term progressively.

Graph Neural Networks (GNNs) serve as the backbone for handling unstructured meshes. To capture both local interactions (through message passing) and long‑range dependencies, the authors integrate two global attention mechanisms: the classic Transformer and the newer State‑Space Model (SSM) called Mamba. Transformers provide the highest accuracy but suffer from quadratic O(N²) memory and compute costs as the number of mesh nodes grows. To overcome this limitation, the authors adapt Mamba to graph data by first imposing an unsupervised ordering of the graph nodes, effectively turning the mesh into a sequence. The ordering is derived from graph topology without any manual heuristics, enabling Mamba’s selective state‑space updates to operate on graphs. In practice, Mamba reduces computational cost by roughly 45 % compared with the Transformer while maintaining comparable error levels (within 5 % of the Transformer’s L2 error).

Physical knowledge is embedded directly into the architecture via an “encoding‑processing‑physics‑informed decoding” pipeline. Input coordinates, boundary conditions, and low‑fidelity fields are encoded, processed by the GNN‑Transformer/Mamba core, and finally decoded into velocity and pressure. Crucially, the decoding stage incorporates differential operators constructed with a Weighted Least Squares (WLSQ) scheme. For each node, a local least‑squares fit of a second‑order Taylor expansion over its k‑nearest neighbors yields matrices that approximate ∂/∂x, ∂/∂y, and the Laplacian. These operators are pre‑computed for each geometry, allowing O(N) application during training and inference, and they can be applied not only to the network outputs but also to selected latent features. The loss function therefore contains PDE residuals for momentum and continuity, as well as a mass‑conservation term, enforcing the solution to obey the governing equations. Moreover, the same operators are used to bias the graph convolution layers, ensuring that hidden representations live in the same functional space as the physical fields.

The methodology is evaluated on two synthetic datasets. VESSEL mimics complex vascular networks with an average of ~7,500 nodes per mesh, while CYLINDER represents the classic flow‑around‑a‑cylinder benchmark with ~3,500 nodes. Both datasets are generated with high‑order finite element solvers (P1‑P1 SUPG for VESSEL, P2‑P1 for CYLINDER) and provide low‑fidelity Stokes solutions as well as high‑fidelity NS solutions. Results show: (1) the multi‑fidelity cascade reduces L2 errors by more than 30 % relative to a single‑stage high‑fidelity model; (2) the Transformer achieves the best raw accuracy but requires roughly double the GPU memory; (3) the Mamba‑based model achieves similar accuracy with substantially lower memory and compute demand; (4) incorporating the WLSQ‑based physics operators dramatically lowers PDE residuals and virtually eliminates divergence from the incompressibility constraint.

Overall, the work demonstrates that embedding physics both in the loss and directly in the network architecture, combined with a hierarchical multi‑fidelity training strategy, yields models that are data‑efficient, physically consistent, and scalable to large unstructured meshes. The unsupervised node‑ordering technique for applying SSMs to graphs opens a promising avenue for real‑time CFD surrogates in biomedical and engineering applications where rapid, accurate flow predictions on complex geometries are essential.