OGF: An Online Gradient Flow Method for Optimizing the Statistical Steady-State Time Averages of Unsteady Turbulent Flows

TL;DR

This paper introduces the OGF method, an online gradient flow approach designed to optimize steady-state statistics of turbulent flows, overcoming the divergence issues caused by chaos in traditional adjoint methods.

Contribution

The paper presents a scalable online gradient flow algorithm that effectively optimizes steady-state turbulence statistics, avoiding gradient divergence in chaotic systems.

Findings

Successfully applied to Lorenz-63, Kuramoto--Sivashinsky, and Navier--Stokes equations.

Achieved several orders of magnitude reduction in the loss function.

Accurately recovered optimal parameters in chaotic flow simulations.

Abstract

Turbulent flows are chaotic and unsteady, but their statistical distribution converges to a statistical steady state. Engineering quantities of interest typically take the form of time-average statistics such as $ \frac{1}{t} \int_0^t f ( u(x,\tau; \theta) ) d\tau \overset{t \rightarrow \infty}{\rightarrow} F(x; \theta)$, where $u(x,t; \theta)$ are solutions of the Navier--Stokes equations with parameters $\theta$. Optimizing over $F(x; \theta)$ has many engineering applications including geometric optimization, flow control, and closure modeling. However, this remains an open challenge, as existing computational approaches are incapable of scaling to physically representative numbers of grid points. The fundamental obstacle is the chaoticity of turbulent flows: gradients calculated with the adjoint method diverge exponentially as $t \rightarrow \infty$. We develop a new online gradient-flow (OGF) method that is scalable to large degree-of-freedom systems and enables optimizing for the steady-state statistics of chaotic, unsteady, turbulence-resolving simulations. The method forward-propagates an online estimate for the gradient of $F(x; \theta)$ while simultaneously performing online updates of the parameters $\theta$. A key feature is the fully online nature of the algorithm to facilitate faster optimization progress and its combination with a finite-difference estimator to avoid the divergence of gradients due to chaoticity. The proposed OGF method is demonstrated for optimizations over three chaotic ordinary and partial differential equations: the Lorenz-63 equation, the Kuramoto--Sivashinsky equation, and Navier--Stokes solutions of compressible, forced, homogeneous isotropic turbulence. In each case, the OGF method successfully reduces the loss based on $F(x; \theta)$ by several orders of magnitude and accurately recovers the optimal parameters.