Can AI simulate complex angles?

Imagine designing a world where every direction matters, like ocean currents or stellar movements. Researchers use mathematical models called angular variables to describe these directions in large datasets. Traditional methods, such as the von Mises-Fisher distribution, work well in simple cases but struggle when many variables are involved.

A new solution comes from generative artificial intelligence. Using neural networks like GANs, normalizing flows, and flow matching, computers learn to create realistic new examples of complex angular data. These deep models capture hidden relationships that simple methods miss.

Tests on real datasets, such as metocean data, show that AI models outperform classic approaches based on finite mixtures of vMF distributions. They can generate smoother, more realistic simulations, helping scientists and engineers predict extreme events more accurately.

The research paves the way for more flexible AI tools in data science, with potential applications in climate, finance, and health. Understanding how these models work can inspire new ways to tackle multidimensional problems across many fields.

Deep generative models for angular data rely on three main families. GANs consist of a generator network learning to produce realistic angular samples, while a discriminator learns to tell real from fake. Normalizing flows map a simple distribution to a complex angular distribution via invertible transformations, preserving total probability. Flow matching directly aligns sampling paths between a known distribution and the target using transport‑based loss.

The von Mises‑Fisher (vMF) distribution models concentration on a unit sphere; its finite mixtures increase flexibility but often need many components to capture intricate patterns. AI models overcome this by learning latent representations that compress high‑dimensional information into manageable spaces.

Evaluation metrics include mean angular distance (MAD), Kolmogorov‑Smirnov divergence applied to marginals, and estimated log‑likelihood via importance sampling. On metocean datasets, GANs achieved a 12% lower MAD than vMF mixtures, while normalizing flows improved log‑likelihood by 9%.

These findings show that deep generative methods can capture the complexity of multivariate angular data, providing more accurate simulations for climate, naval engineering, and risk modelling applications.