Many applications in aerodynamics, particularly in closed-loop control, depend on sensors to estimate the evolving state of the flow. This estimation task is inherently accompanied by uncertainty due to the noisy measurements of sensors or the non-uniqueness of the underlying mapping. Knowledge of this uncertainty can be as important for decision-making as that of the state itself. Uncertainty tracking is challenged by the often-nonlinear relationship between the measurements and the flow state. For example, a collection of passing vortices leaves a footprint in wall pressure that depends nonlinearly on the vortices' strengths and positions. In this paper, we outline recent approaches to flow estimation and illuminate them with worked examples and selected case studies. We review relevant probability tools, including sampling and estimation, in the powerful setting of Bayesian inference and demonstrate these in static flow estimation examples. We then review unsteady examples and illustrate the application of sequential estimation, and particularly, the ensemble Kalman filter. Finally, we discuss uncertainty quantification in neural network approximations of the mappings between sensor measurements and flow states. Recent aerodynamic applications have shown that the flow state can be encoded into a very low-dimensional latent space. We discuss the uncertainty implications of this encoding.

翻译：空气动力学中的许多应用，特别是闭环控制，依赖于传感器来估计流动的演化状态。由于传感器的噪声测量或底层映射的非唯一性，该估计任务本质上伴随着不确定性。对于决策而言，了解这种不确定性可能与了解状态本身同等重要。不确定性追踪的挑战在于测量值与流动状态之间通常存在非线性关系。例如，一组通过的涡旋会在壁面压力上留下印记，该印记非线性地依赖于涡旋的强度和位置。本文概述了流动估计的最新方法，并通过计算示例和精选案例研究加以阐明。我们在贝叶斯推断的强大框架下回顾了相关的概率工具，包括采样与估计，并在静态流动估计示例中展示了这些工具的应用。随后，我们回顾了非定常示例，并阐述了序列估计的应用，特别是集合卡尔曼滤波器。最后，我们讨论了传感器测量与流动状态之间映射关系的神经网络近似中的不确定性量化。近期的空气动力学应用表明，流动状态可被编码到极低维的潜在空间中。我们探讨了这种编码对不确定性的影响。