A review of Bayesian sensor-based estimation and uncertainty quantification of aerodynamic flows

Many applications in aerodynamics depend on the use of sensors to estimate the evolving state of the flow. In particular, a wide variety of traditional and learning-based strategies for closed-loop control rely on some knowledge of the aerodynamic state in order to decide on actions. This estimation task is inherently accompanied by uncertainty due to the noisy measurements of sensors or the non-uniqueness of the underlying mapping, and knowledge of this uncertainty can be as important for decision-making as that of the state itself. The tracking of uncertainty is challenged by the often-nonlinear relationship between the sensor measurements and the flow state. For example, a collection of passing vortices leaves a footprint in wall pressure sensors that depends nonlinearly on the strengths and positions of the vortices. In this paper, we will review the recent body of work on flow estimation. We will discuss the basic tools of probability, including sampling and estimation, in the powerful setting of Bayesian inference and demonstrate these tools in static flow estimation examples. We will then proceed to unsteady examples and illustrate the application of sequential estimation, and particularly, the ensemble Kalman filter. Finally, we will discuss uncertainty quantification in neural network approximations of the mappings between sensor measurements and flow states. Recent aerodynamic applications of neural networks have shown that the flow state can be encoded into a very low-dimensional latent space, and we will discuss the implications of this encoding on uncertainty.