  A dynamical systems approach to turbulence envisions the flow as a trajectory
through a high-dimensional state space transiently visiting the neighbourhoods
of unstable simple invariant solutions (E. Hopf, Commun. Appl. Maths 1, 303,
1948). The hope has always been to turn this appealing picture into a
predictive framework where the statistics of the flow follows from a weighted
sum of the statistics of each simple invariant solution. Two outstanding
obstacles have prevented this goal from being achieved: (1) paucity of known
solutions and (2) the lack of a rational theory for predicting the required
weights. Here we describe a method to substantially solve these problems, and
thereby provide the first compelling evidence that the PDFs of a fully
developed turbulent flow can be reconstructed with a set of unstable periodic
orbits. Our new method for finding solutions uses automatic differentiation,
with high-quality guesses constructed by minimising a trajectory-dependent loss
function. We use this approach to find hundreds of new solutions in turbulent,
two-dimensional Kolmogorov flow. Robust statistical predictions are then
computed by learning weights after converting a turbulent trajectory into a
Markov chain for which the states are individual solutions, and the nearest
solution to a given snapshot is determined using a deep convolutional
autoencoder. To our knowledge, this is the first time the PDFs of a
spatio-temporally-chaotic system have been successfully reproduced with a set
of simple invariant states, and provides a fascinating connection between
self-sustaining dynamical processes and the more well-known statistical
properties of turbulence.
