  This paper proposes an integration of surrogate modeling and topology to
significantly reduce the amount of data required to describe the underlying
global dynamics of robot controllers, including closed-box ones. A Gaussian
Process (GP), trained with randomized short trajectories over the state-space,
acts as a surrogate model for the underlying dynamical system. Then, a
combinatorial representation is built and used to describe the dynamics in the
form of a directed acyclic graph, known as {\it Morse graph}. The Morse graph
is able to describe the system's attractors and their corresponding regions of
attraction (\roa). Furthermore, a pointwise confidence level of the global
dynamics estimation over the entire state space is provided. In contrast to
alternatives, the framework does not require estimation of Lyapunov functions,
alleviating the need for high prediction accuracy of the GP. The framework is
suitable for data-driven controllers that do not expose an analytical model as
long as Lipschitz-continuity is satisfied. The method is compared against
established analytical and recent machine learning alternatives for estimating
\roa s, outperforming them in data efficiency without sacrificing accuracy.
Link to code: https://go.rutgers.edu/49hy35en
