  We propose a physics-constrained machine learning method-based on reservoir
computing- to time-accurately predict extreme events and long-term velocity
statistics in a model of turbulent shear flow. The method leverages the
strengths of two different approaches: empirical modelling based on reservoir
computing, which it learns the chaotic dynamics from data only, and physical
modelling based on conservation laws, which extrapolates the dynamics when
training data becomes unavailable. We show that the combination of the two
approaches is able to accurately reproduce the velocity statistics and to
predict the occurrence and amplitude of extreme events in a model of
self-sustaining process in turbulence. In this flow, the extreme events are
abrupt transitions from turbulent to quasi-laminar states, which are
deterministic phenomena that cannot be traditionally predicted because of
chaos. Furthermore, the physics-constrained machine learning method is shown to
be robust with respect to noise. This work opens up new possibilities for
synergistically enhancing data-driven methods with physical knowledge for the
time-accurate prediction of chaotic flows.
