Introduction to generative modeling with flows and diffusions

SEMESTRE 2

Generative models based on dynamical transport have recently led to significant advances in unsupervised learning. At mathematical level, these models are primarily designed around the construction of a map between two probability distributions that transform samples from the first into samples from the second. While these methods were first introduced in the context of image generation, they have found a wide range of applications, including in scientific computing where they offer interesting ways to reconsider complex problems once thought intractable because of the curse of dimensionality. In this class, we will discuss the mathematical underpinning of generative models based on flows and diffusions, with special focus on understanding how to structure the transport to best reach complex target distributions while maintaining computational efficiency, both at learning and sampling stages. We will also discuss applications of generative AI in scientific computing, in particular in the context of Monte Carlo sampling, with applications to the statistical mechanics and Bayesian inference, as well as probabilistic forecasting, with application to fluid dynamics and atmosphere/ocean science.

Fundamentals of measure transport theory
Flow matching with stochastic interpolants
Diffusive Transport with stochastic interpolants
Link with normalizing flows and score-based diffusion models
Link with probabilistic denoising methods
Connections to optimal transport theory and Schrödinger bridges
Algorithmic aspects
Training strategies for flow-based models
Efficient sampling techniques for diffusion models
Balancing expressivity and computational tractability
Recent algorithmic innovations and efficiency improvements
Evaluation metrics for generative models (likelihood measures, FID, Inception Score)
Challenges and limitations of current models
Applications in Scientific Computing
Monte Carlo sampling applications: Statistical mechanics simulations, Bayesian inference and uncertainty quantification
Probabilistic forecasting: Fluid dynamics predictions, Climate and atmospheric/oceanic modeling, Domain-specific adaptations and constraints, Key breakthrough papers and state-of-the-art applications, Future research directions and open problems
Prerequisites
Probability theory and statistics
Basics of ordinary and stochastic differential equations
Elements of partial differential equations
Machine learning fundamentals
Programming experience (Python recommended)

Créneaux

Lundi 13h30-15h30 et mercredi 15h15-17h15 en salle Bourbaki, à partir du 2 mars 2026.