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

2026-01-05

AI Summary

This work proposes a multi-fidelity graph learning framework for efficiently and accurately solving the steady incompressible Navier–Stokes equations over non-parametric two-dimensional domains. The approach leverages the Stokes solution as a low-fidelity approximation and iteratively refines it toward the full Navier–Stokes solution. Innovatively adapting the Mamba architecture to graph-structured data, the method integrates graph neural networks with Transformer mechanisms while incorporating physical constraints: implicit feature constraints enforcing mass conservation and an enhanced graph convolution based on PDE differential operators. An unsupervised node ordering strategy and a physics-informed encode-process-decode architecture further underpin the framework. This formulation achieves significantly reduced computational cost while yielding more stable and physically consistent flow field predictions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Low Level & Physics-based Vision
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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

Problem

• Navier-Stokes equations
• multi-fidelity learning
• graph-based neural networks
• non-parametrized domains
• physics-informed learning

Innovation

• multi-fidelity learning
• graph neural networks
• Mamba architecture
• physics-informed deep learning
• Navier-Stokes equations