  Big data and large-scale machine learning have had a profound impact on
science and engineering, particularly in fields focused on forecasting and
prediction. Yet, it is still not clear how we can use the superior pattern
matching abilities of machine learning models for scientific discovery. This is
because the goals of machine learning and science are generally not aligned. In
addition to being accurate, scientific theories must also be causally
consistent with the underlying physical process and allow for human analysis,
reasoning, and manipulation to advance the field. In this paper, we present a
case study on discovering a new symbolic model for oceanic rogue waves from
data using causal analysis, deep learning, parsimony-guided model selection,
and symbolic regression. We train an artificial neural network on causal
features from an extensive dataset of observations from wave buoys, while
selecting for predictive performance and causal invariance. We apply symbolic
regression to distill this black-box model into a mathematical equation that
retains the neural network's predictive capabilities, while allowing for
interpretation in the context of existing wave theory. The resulting model
reproduces known behavior, generates well-calibrated probabilities, and
achieves better predictive scores on unseen data than current theory. This
showcases how machine learning can facilitate inductive scientific discovery,
and paves the way for more accurate rogue wave forecasting.
