Data assimilation is a set of methods for incorporating sparse observations of a complex dynamical system, either deterministic or stochastic, into incomplete models of these systems. Mathematically this is the problem of nonlinear filtering and computationally, they are based on a variety of techniques including Markov chain Monte Carlo, optimization, importance sampling. This tutorial will begin with a quick introduction to the Bayesian underpinnings of data assimilation followed by applications to chaotic dynamical systems.
In this series of talks, I will introduce basic elements of generative modeling with diffusion and flow models from first principles. This includes a short introduction to stochastic calculus, ordinary differential equations, evolution of probability measures, Fokker Planck equation, and the continuity equation. We will then apply these ideas to describe training and inference algorithms for diffusion models.
The course will cover the basics of reinforcement learning theory. We will start by implementing simple gradient-based algorithms in PyTorch and using them to solve standard control problems like CartPole and the Atari 2600 game Pong. Along the way, we will explore how to optimize both the sample complexity (the number of interactions with the environment) and the computational complexity (GPU hours) needed to learn an optimal policy.
Optimal transport studies the problem of rearranging one distribution into another while minimizing an associated cost. The past decade has witnessed tremendous progress in our understanding of the computational, methodological and statistical aspects of optimal transport (OT). Recent interest in OT has blossomed due to its close connections with diffusion models.
I will introduce the mathematical framework of OT, and then quickly transition to studying how well various objects in the OT framework (OT distances, and OT maps) can be estimated from samples of the underlying distributions.
In this series of talks, I will introduce basic elements of generative modeling with diffusion and flow models from first principles. This includes a short introduction to stochastic calculus, ordinary differential equations, evolution of probability measures, Fokker Planck equation, and the continuity equation. We will then apply these ideas to describe training and inference algorithms for diffusion models.
The course will cover the basics of reinforcement learning theory. We will start by implementing simple gradient-based algorithms in PyTorch and using them to solve standard control problems like CartPole and the Atari 2600 game Pong. Along the way, we will explore how to optimize both the sample complexity (the number of interactions with the environment) and the computational complexity (GPU hours) needed to learn an optimal policy.
Optimal transport studies the problem of rearranging one distribution into another while minimizing an associated cost. The past decade has witnessed tremendous progress in our understanding of the computational, methodological and statistical aspects of optimal transport (OT). Recent interest in OT has blossomed due to its close connections with diffusion models.
I will introduce the mathematical framework of OT, and then quickly transition to studying how well various objects in the OT framework (OT distances, and OT maps) can be estimated from samples of the underlying distributions.
In this series of talks, I will introduce basic elements of generative modeling with diffusion and flow models from first principles. This includes a short introduction to stochastic calculus, ordinary differential equations, evolution of probability measures, Fokker Planck equation, and the continuity equation. We will then apply these ideas to describe training and inference algorithms for diffusion models.
