标题： OGF：一种用于优化非定常湍流统计稳态时间平均值的在线梯度流方法 显示英文标题

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

摘要： 湍流是混沌且非定常的，但其统计分布会收敛到统计稳态。 感兴趣的工程量通常以时间平均统计量的形式出现，如$ \frac{1}{t} \int_0^t f ( u(x,\tau; \theta) ) d\tau \overset{t \rightarrow \infty}{\rightarrow} F(x; \theta)$，其中$u(x,t; \theta)$是参数为$\theta$的 Navier--Stokes 方程的解。 对$F(x; \theta)$进行优化在许多工程应用中都有涉及，包括几何优化、流动控制和闭合建模。 然而，这仍然是一个开放性挑战，因为现有的计算方法无法扩展到物理上有代表性的网格点数量。 根本障碍是湍流的混沌性：使用伴随法计算的梯度会随着$t \rightarrow \infty$指数级发散。 我们开发了一种新的在线梯度流（OGF）方法，该方法可以扩展到高自由度系统，并能够对混沌、非定常、解析湍流的稳态统计量进行优化。 该方法在同时执行参数$\theta$的在线更新的同时，向前传播对$F(x; \theta)$梯度的在线估计。 一个关键特性是算法的完全在线性质，以促进更快的优化进展，并将其与有限差分估计器结合，以避免由于混沌性导致的梯度发散。 所提出的 OGF 方法已在三个混沌常微分和偏微分方程的优化中进行了演示：Lorenz-63 方程、Kuramoto--Sivashinsky 方程以及可压缩、强迫、均匀各向同性湍流的 Navier--Stokes 解。 在每种情况下，OGF方法根据$F(x; \theta)$成功将损失减少了几个数量级，并准确恢复了最优参数。 显示英文摘要

摘要： 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.