论文标题
部分可观测时空混沌系统的无模型预测
A Turbulent Fluid Mechanics via Nonlinear 'Mixing' of Smooth Velocity Flows With Weighted Random Fields: Stochastically Averaged Navier-Stokes Equations and Velocity Correlations
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Let $\mathfrak{D}\subset\mathbb{R}^{3}$, with ${Vol}(\mathfrak{D})\sim L^{3}$, contain an incompressible fluid of viscosity $ν$ and velocity $\mathrm{U}_{i}(x,t)$ with $(x,t)\in\mathfrak{D}\times[0,\infty)$, satisfying the Navier-Stokes equations with some boundary conditions on $\partial\mathfrak{D}$ and evolving from initial Cauchy data. Now let $\mathscr{B}(x)$ be a Gaussian random field defined for all $x\in\mathfrak{D}$ with expectation $\mathbb{I\!E}\langle\mathscr{B}(x)\rangle=0$, and a Bargmann-Fock binary correlation $\mathbb{I\!E}\big\langle\mathscr{B}(x)\otimes \mathscr{B}({y})\big\rangle=\mathsf{C}\exp(-\|{x}-{y}\|^{2}λ^{-2})$ with $λ\le {L}$. Define a volume-averaged Reynolds number $\mathbf{Re}(\mathfrak{D},t) =(|Vol(\mathfrak{D})|^{-1}\int_{\mathfrak{D}}\|\mathrm{U}_{i}(x,t)\|dμ({x}){L}/ν$. The critical Reynolds number is $\mathbf{Re}_{c}(\mathfrak{D})$ so that turbulence fully evolves within $\mathfrak{D}$ for $t$ such that $\mathbf{Re}(\mathfrak{D},t)>\mathbf{Re}_{c}(\mathfrak{D})$. Let $ψ(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)$ be an arbitrary monotone-increasing weighting functional. The turbulent flow evolving within $\mathfrak{D}$ is described by the random field $\mathscr{U}_{i}(x,t)$ via a 'mixing' ansatz $\mathscr{U}_{i}(x,t)=\mathrm{U}_{i}(x,t)+β\mathrm{U}_{i}(x,t) \big\lbraceψ(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)\big\rbrace \mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)\big]\mathscr{B}(x)$ where $β\ge 1$ is a constant and $\mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)]$ an indicator function. The flow grows increasingly random if $\mathbf{Re}(\mathfrak{D},t)$ increases with $t$ so that this is a 'control parameter'. The turbulent flow $\mathscr{U}_{i}(x,t)$ is a solution of stochastically averaged N-S equations. Reynolds-type velocity correlations are estimated.
