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

TL;DR

This paper introduces a multi-fidelity graph neural network framework that combines advanced architectures and physical constraints to accurately predict stationary Navier-Stokes solutions in complex 2D geometries, reducing computational costs.

Contribution

The paper presents a novel multi-fidelity learning approach integrating graph neural networks, Transformers, and physics-informed operators for efficient Navier-Stokes solution prediction on non-parametrized domains.

Findings

Transformers achieve highest accuracy in predictions.

Mamba reduces computational cost significantly.

Physics-informed operators improve solution regularity.

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