On Surrogate Learning for Linear Stability Assessment of Navier-Stokes Equations with Stochastic Viscosity

TL;DR

This paper explores surrogate modeling techniques like polynomial chaos, Gaussian process regression, and neural networks to efficiently assess the linear stability of Navier-Stokes solutions with stochastic viscosity, reducing computational costs.

Contribution

It introduces and compares surrogate methods for stability analysis of stochastic Navier-Stokes equations, offering a computationally efficient alternative to Monte Carlo simulations.

Findings

Surrogates closely match Monte Carlo stability assessments.

Polynomial chaos and Gaussian processes outperform neural networks in accuracy.

Surrogates significantly reduce computational time for stability analysis.

Abstract

We study linear stability of solutions to the Navier\textendash Stokes equations with stochastic viscosity. Specifically, we assume that the viscosity is given in the form of a~stochastic expansion. Stability analysis requires a solution of the steady-state Navier-Stokes equation and then leads to a generalized eigenvalue problem, from which we wish to characterize the real part of the rightmost eigenvalue. While this can be achieved by Monte Carlo simulation, due to its computational cost we study three surrogates based on generalized polynomial chaos, Gaussian process regression and a shallow neural network. The results of linear stability analysis assessment obtained by the surrogates are compared to that of Monte Carlo simulation using a set of numerical experiments.