Machine-guided discovery of a real-world rogue wave model.

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 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.

Real-world rogue wave probabilities.

Rogue waves are dangerous ocean waves at least twice as high as the surrounding waves. Despite an abundance of studies conducting simulations or wave tank experiments, there is so far no reliable forecast for them. In this study, we use data mining and interpretable machine learning to analyze large amounts of observational data instead (more than 1 billion waves). This reveals how rogue wave occurrence depends on the sea state. We find that traditionally favored parameters such as surface elevation kurtosis, steepness, and Benjamin-Feir index are weak predictors for real-world rogue wave risk. In the studied regime, kurtosis is only informative within a single wave group, and is not useful for forecasting. Instead, crest-trough correlation is the dominating parameter in all studied conditions, water depths, and locations, explaining about a factor of 10 in rogue wave risk variation. For rogue crests, where bandwidth effects are unimportant, we find that skewness, steepness, and Ursell number are the strongest predictors, in line with second-order theory. Our results suggest that linear superposition in bandwidth-limited seas is the main pathway to "everyday" rogue waves, with nonlinear contributions providing a minor correction. This casts some doubt whether the common rogue wave definition as any wave exceeding a certain height threshold is meaningful in practice.