Generation of heavy-tailed distributions with application to rainfall generation

This project focuses on the study and development of generative models capable of reproducing heavy-tailed distributions, particularly those derived from radar-observed rainfall fields. Current generative models—VAEs, GANs, and diffusion models—excel on complex data such as images or videos, but their theoretical guarantees generally rely on assumptions incompatible with heavy distributions, which characterize extreme rainfall events. However, the realistic modeling and generation of these extremes are essential, especially in the context of climate change where their frequency is expected to increase.

Several recent studies have attempted to adapt generative models to these atypical distributions: new variants of VAEs have been proposed to better explore extremes, and "heavy-tailed" versions of diffusion models or GANs show encouraging empirical results. However, these approaches still lack theoretical guarantees, and their ability to generate spatiotemporal rainfall fields remains largely undefined. The objective of this thesis is therefore twofold: (1) to establish theoretical guarantees for the ability of generative models to approximate heavy-tailed distributions, by adapting or developing new analysis techniques; (2) to propose architectures and learning schemes specifically designed for these distributions.

In this context, several avenues will be explored: defining new a priori in latent spaces and improving VAE encoders/decoders; drawing inspiration from latent diffusion models to combine autoencoders and diffusion processes on a suitable latent space; or identifying the key properties of architectures that allow for the correct capture of queue behavior.

The developed methods will be applied to the generation of high-resolution spatio-temporal rainfall fields. Two radar datasets will serve as references: one from a relatively homogeneous region of Germany, and the other from the Swiss Alps, characterized by significant non-stationarity due to the terrain. The thesis thus aims to design new generative models that are both theoretically sound and practically usable for simulating extreme rainfall events, with important applications in urban planning, insurance, and climate risk management.