Towards Opening the Black Box of Ocean Wave Learning: From Feature-Mixing Networks to Functional Relationships

Abstract

Ocean surface waves in mixed sea states, where locally generated wind-sea coexists with remotely generated swell, remain difficult to parameterize in Earth System Models. A basic physical question is which minimal set of variables governs the wave height, and what relationships connect them. Deep learning can predict wave height accurately, but its black-box nature obscures these relationships. We address this with a stationary, diagnostic workflow that combines a deep feature-mixing neural network, an explainability analysis (Integrated Gradients), and symbolic regression; it is not a forecasting tool and does not produce future wave states. Using ERA5 reanalysis data and dimensionless variables motivated by wave-similarity theory, the neural network first establishes a predictive benchmark for the dimensionless significant wave height (a mean absolute percentage error, MAPE Ω = 21.04%, in the South Atlantic). The explainability analysis identifies the wave age β and the directional spectral partitions as the dominant drivers, a ranking that remains stable across basins even as predictive accuracy degrades. The symbolic-regression step, implemented with SymbWaves, then searches for compact equations linking these variables, distilling them into concise relationships: a linear wind-sea relation ( y ≈ 0.2 β ), basin-stable to within 3.5%, and a swell power-law relation ( y ∝ β 1.8 ) with common exponent and amplitudes agreeing within ∼15%. A directionally modulated swell relation further resolves the dependence on swell direction and identifies a hemispheric sign inversion that reflects the poleward origin of swell in each basin. The recovered relationships are compact and physically interpretable.