二维Navier-Stokes方程中的激励与可辨识性
Excitation and Identifiability in the 2D Navier-Stokes equations
AI总结：
本文为二维Navier-Stokes方程建立了首个针对非线性PDE的激励理论，通过可计算的激励条件识别未知粘度，并基于能量方法证明误差估计与观测密度的关系，适用于其他耗散系统。
AI中文摘要：
参数估计中的收敛性经典地要求“持续激励”，这是对依赖于轨迹的信号的非退化条件。据我们所知，本文为非线性偏微分方程建立了第一个这样的激励理论。在从二维不可压缩流体的Navier-Stokes方程的谱观测中识别未知粘度的背景下，我们的激励条件是可计算的、可先验验证的，并且在缩放方面是尖锐的。我们的方法采用数据同化技术来处理未知的初始状态，并通过最小化流体速度的低模态观测与基于 nudging 的滤波器（该滤波器同化这些观测）的低模态投影之间的损失来选择候选粘度。我们框架的主要新颖之处在于区分一个新的泛函 $\mathsf{W}$，它表示滤波器的相关灵敏度变量在涡度上所做的功，我们的整个分析围绕该泛函展开。我们证明了 $\mathsf{W}$ 与相关高斯-牛顿迭代的可观测性格拉姆矩阵显式可比，并随后在观测损失的每个临界点建立了以下二分法：要么 $\mathsf{W}$ 超过某个阈值，在这种情况下，候选粘度满足一个显式误差估计，该估计与 $\mathsf{W}$ 和观测密度 $N$ 成反比，但与初始条件之间的误差成正比；要么 $\mathsf{W}$ 低于阈值，且观测对参数更新在定量上不敏感，这种不敏感性对用户是可检测的。值得注意的是，我们的方法是基于能量的，因此预计可适用于许多其他非线性耗散系统。
英文摘要：
Convergence in parameter estimation classically requires ``persistency of excitation," which is a non-degeneracy condition on a trajectory-dependent signal. This paper develops, to the best of our knowledge, the first such excitation theory for a nonlinear partial differential equation. In the context of identifying the unknown viscosity from spectral observations in the two-dimensional Navier--Stokes equations for incompressible fluids, our excitation condition is computable, verifiable a priori, and sharp with respect to scaling. Our approach employs a data assimilation methodology to account for an unknown initial state and select candidate viscosities by minimizing the loss between the low-mode observations of the fluid velocity and low-mode projection of a nudging-based filter that assimilates these observations. The main novelty of our framework is to distinguish a new functional, $\mathsf{W}$, representing the work done by the filter's associated sensitivity variable on the enstrophy, around which our entire analysis is centered. We show that $\mathsf{W}$ is explicitly comparable to the observability Gramian of the associated Gauss--Newton iteration, and subsequently establish the following dichotomy at every critical point of the observational loss: either $\mathsf{W}$ exceeds a certain threshold, in which case the candidate viscosity obeys an explicit error estimate that depends inversely on $\mathsf{W}$ and the observational density $N$, but directly on the error between initial conditions, or else $\mathsf{W}$ is below the threshold and the observations are quantitatively insensitive to parameter updates in a way that is detectable to the user. Notably, our approach is energy-based, and therefore expected to be adaptable to many other nonlinear dissipative systems.
