While substantial progress has been made in the past 50 years on the modelling of turbulent flow systems - and in the past 5 years, on their closed-loop control - a missing dimension of most modelling and control strategies developed to date is an embedding probabilistic inferential framework. Furthermore, it can be proven that the only consistent and rigorous foundation for probabilistic inference is given by Bayes’ theorem (also derived by Laplace):
where P is a probability, H is a hypothesis or model, D are the data and | indicates conditioning. This equation expresses the probability P(H|D) of an hypothesis or model subject to data, given the probability P(D|H) of data subject to the hypothesis , and the prior probabilities of the hypothesis P(H) and data P(D). The probability P(H|D) is therefore expressed over the hypothesis space or model space ΩH rather than the state space of the data ΩD. Note that Bayes’ theorem invokes an extended definition of probability as a plausibility or degree of belief. This definition of probability is mathematically rigorous (Cox 1961), and is more natural - and closer to its original meaning - than its narrower interpretation as a measurable frequency.
For modelling, a Bayesian framework enables the investigator to move beyond traditional deterministic or stochastic models – as well as popular “data-driven” or “model-free” approaches – to synthesise all information (models, state space, past and present data) into a seamless inference of the present and future states of the system. For control, a Bayesian strategy permits the continuous updating of control signals based on all available information (sensory and model), while simultaneously taking account of sensor and control variability and other uncertainties. Although a Bayesian probabilistic framework has not proved necessary for the linear or weakly nonlinear dynamical system models or control strategies developed to date, it is becoming increasingly important – in some cases, critically so – for the modelling and control of highly complex nonlinear systems, especially those which involve turbulent flow. For these reasons, Bayesian inference is now being recognised as the foundation of inference throughout many fields, including machine learning, medical research, genetics, astronomy, political forecasting and legal practice.
These themes were explored in the 1st International Workshop on Bayesian Inference for Modelling and Control in Fluid Mechanics (“BayesianControl 2015”), which was held in Poitiers, France, on 14 April 2015. This was the first meeting of its kind. The workshop examined (i) the purpose and foundations of Bayesian inference; (ii) the implementation of Bayesian inference for modelling and control of turbulent flow systems; and (iii) recent experimental developments, especially in cluster-based reduced-order models and closed-loop flow control. Participants in the workshop were able to discuss and formulate strategies for the implementation of Bayesian inference, especially for closed-loop flow control, based on the theoretical and practical insights.
The detailed program for the Workshop is posted here.
During the workshop, it was clear that there was strong interest in this topic. Interested participants are therefore urged to join the SIAM-UQ 16 conference to be held in Lousanne, Switzerland, in 5-8 April 2016, which will have several sessions on Bayesian inference in nonlinear and flow systems. All interested participants should register immediately !
The Organising Committee
Robert K. Niven, UNSW Canberra, Australia.
Bernd Noack, Institut Pprime, Poitiers, France.
Laurent Cordier, Institut Pprime, Poitiers, France.
Email: maxent (AT) iinet.net.au
(alternative: r.niven (AT) adfa.edu.au)
This workshop was proudly supported by:
- Institut Pprime, Poitiers, France
- The University of New South Wales, Australia
- MaxEnt Solutions Pty Ltd, Australia (ABN 82 602 277 243).