  We use recent advances in the machine learning area known as 'reservoir
computing' to formulate a method for model-free estimation from data of the
Lyapunov exponents of a chaotic process. The technique uses a limited time
series of measurements as input to a high-dimensional dynamical system called a
'reservoir'. After the reservoir's response to the data is recorded, linear
regression is used to learn a large set of parameters, called the 'output
weights'. The learned output weights are then used to form a modified
autonomous reservoir designed to be capable of producing arbitrarily long time
series whose ergodic properties approximate those of the input signal. When
successful, we say that the autonomous reservoir reproduces the attractor's
'climate'. Since the reservoir equations and output weights are known, we can
compute derivatives needed to determine the Lyapunov exponents of the
autonomous reservoir, which we then use as estimates of the Lyapunov exponents
for the original input generating system. We illustrate the effectiveness of
our technique with two examples, the Lorenz system, and the
Kuramoto-Sivashinsky (KS) equation. In particular, we use the Lorenz system to
show that achieving climate reproduction may require tuning of the reservoir
parameters. For the case of the KS equation, we note that as the system's
spatial size is increased, the number of Lyapunov exponents increases, thus
yielding a challenging test of our method, which we find the method
successfully passes.
