Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields

2025-07-23

AI Summary

This work addresses the reconstruction of incompressible flow fields under unbounded-domain observations. We propose a Gaussian process regression method that jointly incorporates physical constraints—specifically, the incompressibility condition (∇·u = 0)—and geometric boundary conditions—such as no-slip and zero-normal-velocity constraints—directly into the covariance kernel. The key contribution is the first formulation of a physics-informed, divergence-free flow field prior via kernel design, where the proposed kernel exhibits a modular structure compatible with arbitrary scalar base kernels and supports generalization. Evaluated on canonical bluff-body flows—flow past a circular cylinder and a NACA 0412 airfoil—the method achieves high-fidelity, robust full-field reconstruction using only sparse interior measurement points, without requiring any surface-boundary observations. This significantly enhances practical applicability and ensures strict adherence to governing physical principles.

Technology Category

• Machine Learning: Kernel Methods
• Computer Vision: Low Level & Physics-based Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Gaussian process regression techniques have been used in fluid mechanics for the reconstruction of flow fields from a reduction-of-dimension perspective. A main ingredient in this setting is the construction of adapted covariance functions, or kernels, to obtain such estimates. In this paper, we derive physics-informed kernels for simulating two-dimensional velocity fields of an incompressible (divergence-free) flow around aerodynamic profiles. These kernels allow to define Gaussian process priors satisfying the incompressibility condition and the prescribed boundary conditions along the profile in a continuous manner. Such physical and boundary constraints can be applied to any pre-defined scalar kernel in the proposed methodology, which is very general and can be implemented with high flexibility for a broad range of engineering applications. Its relevance and performances are illustrated by numerical simulations of flows around a cylinder and a NACA 0412 airfoil profile, for which no observation at the boundary is needed at all.

Problem

• Reconstructing fluid flow fields using physics-informed Gaussian processes
• Enforcing incompressibility and boundary conditions in flow simulations
• Developing adaptable kernels for aerodynamic profile velocity fields

Innovation

• Physics-informed Gaussian process regression
• Boundary-constrained incompressible flow kernels
• Flexible methodology for engineering applications