We present a method to construct low-dimensional predictive models from data sets representing essentially nonlinear (or non-linearizable) dynamical systems. Our approach, which we call dynamics-based machine learning, seeks to learn sparse, nonlinear models directly from attracting phase space structures (or spectral submanifolds) inferred from data, which are often the same even for systems governed by different physics. We illustrate the capabilities of our method in finding low-dimensional dynamical normal forms on high-dimensional numerical data sets and experimental measurements. Specifically, we deal with mechanical oscillations in mechanical systems, vortex shedding in fluid dynamics and sloshing in a water tank, where we find that our models, trained on unforced data, also generalize to predict forced dynamics.

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