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

Adrian Padilla-Segarra, Pascal Noble, Olivier Roustant, Éric Savin

TL;DR

This work addresses reconstructing two-dimensional incompressible flow fields around aerodynamic profiles from limited data while strictly enforcing physics. It introduces a boundary-constrained GP framework that uses a spectral Karhunen–Loève construction to force $Z_0(\Gamma)=0$ on a boundary and represents the velocity field as $\mathbf{u}=\textbf{curl}\,Z_0$, yielding a divergence-free prior with continuous boundary conditions via a kernel $G_0$. The approach provides a general method for arbitrary compact sets, a scalable numerical scheme, and a physics-informed GP regression (GPR) that achieves improved boundary fidelity and quantified uncertainty, demonstrated on cylinder and NACA 0412 airfoil cases with no boundary observations required. The results show reduced posterior uncertainty near the profile and enhanced predictive accuracy, highlighting the framework's potential for mesh-free data assimilation and Lagrangian particle simulations in fluid dynamics.

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 present a general method for constraining a prescribed Gaussian process on an arbitrary compact set. The kernel of the pre-defined process must be at least continuous and may include other information about the studied phenomenon. This general boundary-constraining framework can be implemented with high flexibility for a broad range of engineering applications. From this, 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. We describe an adapted numerical method for the boundary-constraining procedure parameterized by a measure on the compact set. The relevance of the methodology 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.

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

TL;DR

This work addresses reconstructing two-dimensional incompressible flow fields around aerodynamic profiles from limited data while strictly enforcing physics. It introduces a boundary-constrained GP framework that uses a spectral Karhunen–Loève construction to force on a boundary and represents the velocity field as , yielding a divergence-free prior with continuous boundary conditions via a kernel . The approach provides a general method for arbitrary compact sets, a scalable numerical scheme, and a physics-informed GP regression (GPR) that achieves improved boundary fidelity and quantified uncertainty, demonstrated on cylinder and NACA 0412 airfoil cases with no boundary observations required. The results show reduced posterior uncertainty near the profile and enhanced predictive accuracy, highlighting the framework's potential for mesh-free data assimilation and Lagrangian particle simulations in fluid dynamics.

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 present a general method for constraining a prescribed Gaussian process on an arbitrary compact set. The kernel of the pre-defined process must be at least continuous and may include other information about the studied phenomenon. This general boundary-constraining framework can be implemented with high flexibility for a broad range of engineering applications. From this, 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. We describe an adapted numerical method for the boundary-constraining procedure parameterized by a measure on the compact set. The relevance of the methodology 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.

Paper Structure

This paper contains 25 sections, 3 theorems, 72 equations, 14 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Constrained GPs.
  3. Physics-informed learning models.
  4. Reminder on Gaussian Process Regression
  5. Constraining a GP over a continuum
  6. Constraints over a compact set
  7. Constraints over parameterized sets
  8. A numerical method for approximating constrained kernels
  9. Benefits and limitations with respect to discrete constraints
  10. Physics-informed GPR for incompressible flows in aerodynamics
  11. Fluid dynamics context
  12. Divergence-free, boundary-constrained GPs
  13. Energy decay GPs
  14. GPR observation design
  15. Derivatives of a constrained kernel
  16. ...and 10 more sections

Key Result

Proposition 3.1

Consider an arbitrary scalar Gaussian process ${Z} =\{{Z}({\boldsymbol x}),{\boldsymbol x}\in{\Omega}\} \sim{\mathcal{GP}}\left( 0,{G}\right)$, with continuous covariance kernel ${G}:{\Omega}\times{\Omega}\rightarrow\mathbb{R}$, and a compact subset $\Gamma$ of its domain ${\Omega}$. Let $\lambda_n, is a centered GP that satisfies an homogeneous Dirichlet boundary condition on $\Gamma$, i.e.: Mor

Figures (14)

  • Figure 1: GPR setting for approximating the velocity field of an incompressible fluid flow around a profile in a rectangular computational domain with discrete wall boundary observations (blue dots $\bullet$), flow inlet and outlet boundary data (orange dots $\bullet$), and bulk flow data (red dots $\bullet$). No design observation points are used on the profile boundary.
  • Figure 2: Spatial GPR interpolations for velocity (arrows) and vorticity (colormap) fields around different profiles using BCGP kernels over the computational domain ${\Omega} = [0,2\pi]\times[0,2\pi]$ with $N_\text{b} = 40$ boundary design points (orange dots $\textcolor{orange}{\bullet}$). The base kernel is given by Eq. (\ref{['eq:AnisoGaussKernel']}). No GPR design information is used in the interior of the domain, only the physics-informed kernel by means of BCGPs is used.
  • Figure 3: Ground truth velocity field of an incompressible fluid flow around a cylinder profile in a wind-tunnel setting. The colormap represents the norm of the velocity field in m/s.
  • Figure 4: Initial reconstruction by physics-informed GPR of the velocity field of an incompressible fluid flow ($\text{Re} = 3\cdot 10^3$) around a cylinder profile in a wind-tunnel setting using observations only at the outer boundary $\partial{\Omega}$ of the domain ($N_\text{b} = 116$, black circumferences $\circ$). No design point is used on the profile boundary or in the interior of the computational domain ${\Omega}$. The colormap represents the norm of the velocity field in m/s.
  • Figure 5: Total standard deviation of the velocity field reconstructed by GPR with or without BCGP kernels. Design observations of the velocity are considered only at the outer boundary $\partial{\Omega}$ of the computational domain ${\Omega}$ ($N_\text{b} = 116$, black circumferences $\circ$). (a) Estimate obtained by using the BCGP kernel ${{G}_0}$; (b) estimate obtained by using the base kernel ${G}$ instead of ${{G}_0}$, under the same observation design. The colormap represents the total standard deviation in m/s.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Proposition 3.1
  • Corollary 3.2: Uniform measure on ${\mathcal{P}}$
  • Corollary 3.3: Surface measure on $\Gamma$
  • Remark 4.1