Alec Linot: Low-dimensional models for forecasting and controlling chaotic dynamical systems

Predicting and controlling chaotic systems is of broad interest for tasks such as minimizing drag in a pipe, maintaining lift during a gust encounter, and increasing turbulence for mixing or delaying flow separation. For perspective, turbulent drag accounts for 5% of manmade CO2, thus, there are major benefits to even small improvements in drag reduction. Computational fluid dynamics (CFD) allows us to simulate these flows, but these simulations can be computationally expensive and difficult to interpret. This problem is further exacerbated by the need for repeated simulations in finding control policies, computing invariant solutions, or determining stability to perturbations. 

 

Here, I discuss machine learning methods for forecasting the Kuramoto-Sivashinsky equation, Kolmogorov flow, and turbulent Couette flow. These are all systems that are known, or expected, to have long-time dynamics that are attracted to an invariant manifold. This suggests that it should be possible to “exactly” predict the dynamics of these systems in far fewer dimensions than are needed in the numerical simulations. We can parameterize these manifolds using autoencoders and forecast in this coordinate system using neural ODEs. Throughout the talk, I will highlight the importance of enforcing physical properties into the models such as energy conservation, equivariance to symmetries, and attraction to the manifold. When care is taken in constructing these models, we are able to reduce the Couette problem from O(10⁵) degrees of freedom down to ~20. I will end by showing how these models can be used with reinforcement learning to rapidly find drag reduction policies in a turbulent flow.