  The Bellman equation and its continuous form, the Hamilton-Jacobi-Bellman
(HJB) equation, are ubiquitous in reinforcement learning (RL) and control
theory. However, these equations quickly become intractable for systems with
high-dimensional states and nonlinearity. This paper explores the connection
between the data-driven Koopman operator and Markov Decision Processes (MDPs),
resulting in the development of two new RL algorithms to address these
limitations. We leverage Koopman operator techniques to lift a nonlinear system
into new coordinates where the dynamics become approximately linear, and where
HJB-based methods are more tractable. In particular, the Koopman operator is
able to capture the expectation of the time evolution of the value function of
a given system via linear dynamics in the lifted coordinates. By parameterizing
the Koopman operator with the control actions, we construct a ``Koopman
tensor'' that facilitates the estimation of the optimal value function. Then, a
transformation of Bellman's framework in terms of the Koopman tensor enables us
to reformulate two max-entropy RL algorithms: soft value iteration and soft
actor-critic (SAC). This highly flexible framework can be used for
deterministic or stochastic systems as well as for discrete or continuous-time
dynamics. Finally, we show that these Koopman Assisted Reinforcement Learning
(KARL) algorithms attain state-of-the-art (SOTA) performance with respect to
traditional neural network-based SAC and linear quadratic regulator (LQR)
baselines on four controlled dynamical systems: a linear state-space system,
the Lorenz system, fluid flow past a cylinder, and a double-well potential with
non-isotropic stochastic forcing.
