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Finding proper collective variables for complex systems and processes is one of the most challenging tasks in simulations, which limits the interpretation of experimental and simulated data and the application of enhanced sampling techniques. Here, we propose a machine learning approach able to distill few, physically relevant variables by associating instantaneous configurations of the system to their corresponding inherent structures as defined in liquids theory. We apply this approach to the challenging case of structural transitions in nanoclusters, managing to characterize and explore the structural complexity of an experimentally relevant system constituted by 147 gold atoms. Our inherent-structure variables are shown to be effective at computing complex free-energy landscapes, transition rates, and at describing non-equilibrium melting and freezing processes. The effectiveness of this machine learning strategy guided by the generally-applicable concept of inherent structures shows promise to devise collective variables for a vast range of systems, including liquids, glasses, and proteins.