Differentiable flow solvers and online learning for the dynamical systems view of turbulence

Online learning is a technique in which a differentiable flow solver is included inside the training loop for a neural network. One common use case in fluid mechanics is subgrid scale parameterizations in turbulence. In that problem, online learning produces schemes that are more stable than their “offline” counterparts, but the approach can also be used in a variety of other problems, including solution discovery and state estimation. In this talk I will describe three problems rooted in a “dynamical systems” view of fluid motion in which differentiable solvers play a critical role. First, I will show how the search for unstable periodic orbits – exact solutions of the Navier-Stokes equations which contain a particular recurrent process relevant to the chaotic flow – can be framed as an optimization problem. A scalar loss function which measures the distance between the state and its location a time T later is minimized using a combination of automatic differentiation through the solver in combination with a high-dimensional optimizer. The approach converges an order of magnitude more solutions in a two-dimensional Kolmogorov flow than have been computed by earlier methods. In the second problem, I will present an online-learning algorithm for super resolution. Super resolution refers to the prediction of a high-resolution flow snapshot (usually with a neural network) given coarse-grained observations. In contrast to the usual approach, our method does not require a library of high resolution snapshots: the loss function is a modification of the variational 4DVar algorithm for state estimation and seeks to match the coarse-grained evolution of the predicted state to measurements. I will compare the performance to classical 4DVar and assess the impact of known limiting lengthscales for assimilation. I will also show how these ideas can be applied in non-Newtonian fluids where key dynamical variables cannot be measured expreimentally. Finally, I will describe a method to learn mappings between a family of dynamical systems, with a training algorithm motivated by the definition of topological equivalence. The resulting neural networks allow for continuation of arbitrary solutions of the governing equations and extend the standard dynamical systems toolbox which can follow statistically-steady states only.