  A hallmark of chaotic dynamics is the loss of information with time. Although
information loss is often expressed through a connection to Lyapunov exponents
-- valid in the limit of high information about the system state -- this
picture misses the rich spectrum of information decay across different levels
of granularity. Here we show how machine learning presents new opportunities
for the study of information loss in chaotic dynamics, with a double pendulum
serving as a model system. We use the Information Bottleneck as a training
objective for a neural network to extract information from the state of the
system that is optimally predictive of the future state after a prescribed time
horizon. We then decompose the optimally predictive information by distributing
a bottleneck to each state variable, recovering the relative importance of the
variables in determining future evolution. The framework we develop is broadly
applicable to chaotic systems and pragmatic to apply, leveraging data and
machine learning to monitor the limits of predictability and map out the loss
of information.
