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Gaussian process approach within a data-driven POD framework for fluid dynamics engineering problems

Giulio Ortali, Nicola Demo, Gianluigi Rozza

TL;DR

The paper addresses the high computational cost of solving parametrized PDEs in fluid dynamics by coupling Proper Orthogonal Decomposition (POD) with Gaussian Process Regression (GPR) to form a data-driven reduced-order model. The method builds a low-dimensional reduced space via POD and learns the parameter-to-coefficient map with GPR, enabling fast, equation-free online predictions after an offline snapshot-based training. Demonstrations on a parametric Stokes flow around a cylinder and on a 3D multiphase Navier–Stokes hull problem show that accurate predictions are achievable with a small number of snapshots, yielding substantial speed-ups in design optimization workflows. The approach offers practical impact for real-time or near-real-time engineering analyses and can be extended with error quantification, greedy snapshot selection, and kernel customization.

Abstract

This work describes the implementation of a data-driven approach for the reduction of the complexity of parametrical partial differential equations (PDEs) employing Proper Orthogonal Decomposition (POD) and Gaussian Process Regression (GPR). This approach is applied initially to a literature case, the simulation of the stokes problems, and in the following to a real-world industrial problem, inside a shape optimization pipeline for a naval engineering problem.

Gaussian process approach within a data-driven POD framework for fluid dynamics engineering problems

TL;DR

The paper addresses the high computational cost of solving parametrized PDEs in fluid dynamics by coupling Proper Orthogonal Decomposition (POD) with Gaussian Process Regression (GPR) to form a data-driven reduced-order model. The method builds a low-dimensional reduced space via POD and learns the parameter-to-coefficient map with GPR, enabling fast, equation-free online predictions after an offline snapshot-based training. Demonstrations on a parametric Stokes flow around a cylinder and on a 3D multiphase Navier–Stokes hull problem show that accurate predictions are achievable with a small number of snapshots, yielding substantial speed-ups in design optimization workflows. The approach offers practical impact for real-time or near-real-time engineering analyses and can be extended with error quantification, greedy snapshot selection, and kernel customization.

Abstract

This work describes the implementation of a data-driven approach for the reduction of the complexity of parametrical partial differential equations (PDEs) employing Proper Orthogonal Decomposition (POD) and Gaussian Process Regression (GPR). This approach is applied initially to a literature case, the simulation of the stokes problems, and in the following to a real-world industrial problem, inside a shape optimization pipeline for a naval engineering problem.

Paper Structure

This paper contains 7 sections, 2 theorems, 19 equations, 6 figures.

Table of Contents

  1. Introduction and motivations
  2. Proper orthogonal decomposition for dimensionality reduction
  3. Gaussian process regression model
  4. Numerical results
  5. Parametric Stokes flow around cylinder
  6. Multiphase turbulent Navier-Stokes flow in an industrial parametric domain
  7. Conclusions and perspectives

Key Result

Theorem 1

(POD basis) Given $\mathbf{Y} \in \mathbb{R}^{\mathcal{N} \times n}$, $\{\chi_i\}_{i=1}^N$, for $N \in \{1,..,n\}$ is the POD basis of $\mathbf{Y}$ if and only if it is a solution of:

Figures (6)

  • Figure 1: The domain $\Omega$ for the parametric Stokes flow simulation.
  • Figure 2: Relative average error between the truth solutions and the solutions obtained with the POD-GPR framework, evaluated in 20 test parameters, varying the number of snapshots.
  • Figure 3: Graphical visualization of the reconstructed fields for a test parameter using only 3 input samples. The plots are organized as following: the three columns represent, from left to right, pressure, velocity among $x$ direction and velocity among $y$ direction. The first row shows the truth solutions, the second row the solutions approximated using the POD-GPR method and the third one represents the absolute error between them.
  • Figure 4: View of the the lattice of points over the undeformed hull: the blue dots are the FFD control points, while the red arrows represent the displacements.
  • Figure 5: Average relative $L^2$ error between the FOM and the ROM as a function of the number of modes, keeping fixed the number of snapshots to 80, (left) and the number of snapshots used, keeping fixed the number of modes to 20 (right).
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2