Rapid Bayesian Inference for Fluid Flow Modeling and Control

We give a new framework for rapid Bayesian inference for flow modeling and control, based on Bayes' rule $p(\vec{\theta} | \vec{x}) = p(\vec{x} | \vec{\theta}) p(\vec{\theta}) / p(\vec{x})$, where $p$ is a probability density function, $\vec{x}$ are multivariate data and $\vec{\theta}$ is one model drawn from a continuous model space $\Omega_{\vec{\theta}}$. We thus seek the pdf of the model $\vec{\theta}$, given the data $\vec{x}$. Traditionally, Bayesian inference requires marginalization of the integral $p(\vec{x}) =\int d\vec{\theta}\, p(\vec{x} | \vec{\theta}) p(\vec{\theta})$, which is highly computationally expensive and may not even be feasible. Instead, we propose initial order reduction of the data, such as by k-means clustering, to generate discretized data $c_i$ on a reduced-order data space $C$, followed by Bayesian inference to infer the conditional probability $P(\gamma_m | c_k)$ of the discretized model $\gamma_m$ in a reduced-order model space $\Gamma$. If needed, an inversion to infer $p(\vec{\theta} | \gamma_m)$ can be conducted. The method substantially reduces the computational complexity of Bayesian inference, enabling real-time turbulent modeling and control. We report applications to several turbulent flow and dynamical systems.