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This paper investigates how representation learning can enable optimalcontrol in unknown and complex dynamics, such as chaotic and non-linearsystems, without relying on prior domain knowledge of the dynamics. The coreidea is to establish an equivariant geometry that is diffeomorphic to themanifold defined by a dynamical system and to perform optimal control withinthis corresponding geometry, which is a non-trivial task. To address thischallenge, Koopman Embed to Equivariant Control (KEEC) is proposed for modellearning and control. Inspired by Lie theory, KEEC begins by learning anon-linear dynamical system defined on a manifold and embedding trajectoriesinto a Lie group. Subsequently, KEEC formulates an equivariant value functionequation in reinforcement learning on the equivariant geometry, ensuring aninvariant effect as the value function on the original manifold. By derivinganalytical-form optimal actions on the equivariant value function, KEECtheoretically achieves quadratic convergence for the optimal equivariant valuefunction by leveraging the differential information on the equivariantgeometry. The effectiveness of KEEC is demonstrated in challenging dynamicalsystems, including chaotic ones like Lorenz-63. Notably, our results show thatisometric functions, which maintain the compactness and completeness ofgeometry while preserving metric and differential information, consistentlyoutperform loss functions lacking these characteristics.

KEEC: Embed to Control on An Equivariant Geometry