  This paper investigates how representation learning can enable optimal
control in unknown and complex dynamics, such as chaotic and non-linear
systems, without relying on prior domain knowledge of the dynamics. The core
idea is to establish an equivariant geometry that is diffeomorphic to the
manifold defined by a dynamical system and to perform optimal control within
this corresponding geometry, which is a non-trivial task. To address this
challenge, Koopman Embed to Equivariant Control (KEEC) is proposed for model
learning and control. Inspired by Lie theory, KEEC begins by learning a
non-linear dynamical system defined on a manifold and embedding trajectories
into a Lie group. Subsequently, KEEC formulates an equivariant value function
equation in reinforcement learning on the equivariant geometry, ensuring an
invariant effect as the value function on the original manifold. By deriving
analytical-form optimal actions on the equivariant value function, KEEC
theoretically achieves quadratic convergence for the optimal equivariant value
function by leveraging the differential information on the equivariant
geometry. The effectiveness of KEEC is demonstrated in challenging dynamical
systems, including chaotic ones like Lorenz-63. Notably, our results show that
isometric functions, which maintain the compactness and completeness of
geometry while preserving metric and differential information, consistently
outperform loss functions lacking these characteristics.
