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

arXiv: https://arxiv.org/abs/2507.05149 · HTML: https://mlsanity.com/paper/2507.05149/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 86 references that are in the map.
- First posted on arXiv 7 Jul 2025.

## Builds on (most cited first, 15 of 15)

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- [Scientific multi-agent reinforcement learning for wall-models of turbulent flows](https://mlsanity.com/paper/2106.11144/) (2021)
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- [Control of chaotic systems by Deep Reinforcement Learning](https://mlsanity.com/paper/1906.07672/) (2019)
- [Stochastic Gradient Descent in Continuous Time](https://mlsanity.com/paper/1611.05545/) (2016)
- [Embedded training of neural-network sub-grid-scale turbulence models](https://mlsanity.com/paper/2105.01030/) (2021)
- [PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence](https://mlsanity.com/paper/2105.08633/) (2021)
- [Deep Learning Closure Models for Large-Eddy Simulation of Flows around Bluff Bodies](https://mlsanity.com/paper/2208.03498/) (2022)
- [Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures](https://mlsanity.com/paper/2104.15138/) (2021)
- [Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NILSS)](https://mlsanity.com/paper/1611.00880/) (2016)
- [Two-Timescale Stochastic Gradient Descent in Continuous Time with Applications to Joint Online Parameter Estimation and Optimal Sensor Placement](https://mlsanity.com/paper/2007.15998/) (2020)
- [Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations](https://mlsanity.com/paper/2202.06637/) (2022)
- [A Forward Propagation Algorithm for Online Optimization of Nonlinear Stochastic Differential Equations](https://mlsanity.com/paper/2207.04496/) (2022)
- [Multi-agent reinforcement learning for wall modeling in LES of flow over periodic hills](https://mlsanity.com/paper/2211.16427/) (2022)
- [Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach](https://mlsanity.com/paper/2204.10066/) (2022)

## Abstract (arXiv)

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  frac1t int0t f ( u(x,τ; θ) ) dτ oversett rightarrow ∞rightarrow F(x; θ), where u(x,t; θ) are solutions of the Navier--Stokes equations with parameters θ. Optimizing over F(x; θ) 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 ∞. 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; θ) while simultaneously performing online updates of the parameters θ. 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; θ) by several orders of magnitude and accurately recovers the optimal parameters.