  Deep reinforcement learning (RL) is a data-driven method capable of
discovering complex control strategies for high-dimensional systems, making it
promising for flow control applications. In particular, the present work is
motivated by the goal of reducing energy dissipation in turbulent flows, and
the example considered is the spatiotemporally chaotic dynamics of the
Kuramoto-Sivashinsky equation (KSE). A major challenge associated with RL is
that substantial training data must be generated by repeatedly interacting with
the target system, making it costly when the system is computationally or
experimentally expensive. We mitigate this challenge in a data-driven manner by
combining dimensionality reduction via an autoencoder with a neural ODE
framework to obtain a low-dimensional dynamical model from just a limited data
set. We substitute this data-driven reduced-order model (ROM) in place of the
true system during RL training to efficiently estimate the optimal policy,
which can then be deployed on the true system. For the KSE actuated with
localized forcing ("jets") at four locations, we demonstrate that we are able
to learn a ROM that accurately captures the actuated dynamics as well as the
underlying natural dynamics just from snapshots of the KSE experiencing random
actuations. Using this ROM and a control objective of minimizing dissipation
and power cost, we extract a control policy from it using deep RL. We show that
the ROM-based control strategy translates well to the true KSE and highlight
that the RL agent discovers and stabilizes an underlying forced equilibrium
solution of the KSE system. We show that this forced equilibrium captured in
the ROM and discovered through RL is related to an existing known equilibrium
solution of the natural KSE.
