  A central area of research in nonlinear science is the study of instabilities
that drive the emergence of extreme events. Unfortunately, experimental
techniques for measuring such phenomena often provide only partial
characterization. For example, real-time studies of instabilities in nonlinear
fibre optics frequently use only spectral data, precluding detailed predictions
about the associated temporal properties. Here, we show how Machine Learning
can overcome this limitation by predicting statistics for the maximum intensity
of temporal peaks in modulation instability based only on spectral
measurements. Specifically, we train a neural network based Machine Learning
model to correlate spectral and temporal properties of optical fibre modulation
instability using data from numerical simulations, and we then use this model
to predict the temporal probability distribution based on high-dynamic range
spectral data from experiments. These results open novel perspectives in all
systems exhibiting chaos and instability where direct time-domain observations
are difficult.
