提出在线梯度流方法，高效优化湍流统计平均结果。
OGF: An Online Gradient Flow Method for Optimizing the Statistical Steady-State Time Averages of Unsteady Turbulent Flows
- 采用在线梯度估计与参数同步更新，避免混沌导致的梯度发散。
- 在洛伦兹-63、库拉莫托-希瓦辛斯基及可压缩湍流中实现数倍降损。
- 适合高自由度湍流仿真中的参数优化，如几何设计与控制。
湍流具有混沌与非定常特性，但其统计分布趋于稳态。工程关注量通常为时间平均统计量：$ \frac{1}{t} \int_0^t f ( u(x,τ; θ) ) dτ\overset{t \rightarrow \infty}{\rightarrow} F(x; θ)$，其中 $u(x,t; θ)$ 是带参数 $θ$ 的纳维-斯托克斯方程解。对 $F(x; θ)$ 的优化在几何设计、流动控制和模型闭合中有广泛应用，但现有方法难以扩展到高网格点规模。根本障碍是湍流的混沌性：伴随法计算的梯度随 $t \rightarrow \infty$ 指数发散。本文提出一种新型在线梯度流（OGF）方法，可扩展至高自由度系统，实现对混沌、非定常、解析湍流模拟的稳态统计量优化。该方法在前向传播中在线估计 $F(x; θ)$ 梯度，并同步更新参数 $θ$。关键特点是完全在线结构，加速优化进程，并结合有限差分估计器，避免因混沌引起的梯度发散。在三个混沌系统（洛伦兹-63、库拉莫托-希瓦辛斯基、可压缩强迫各向同性湍流的纳维-斯托克斯解）上验证，OGF 方法成功将基于 $F(x; θ)$ 的损失降低数个数量级，并准确恢复最优参数。
原文摘要 · Abstract (English)
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,τ; θ) ) dτ\overset{t \rightarrow \infty}{\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 $t \rightarrow \infty$. 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 $F(x; θ)$ 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.
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