Applied Mathematics Colloquium with Eric Vanden Eijnden, NYU Courant

Tuesday, March 21, 2023
2:45 PM - 3:45 PM
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Eric Vanden Eijnden, from NYU's Courant Institute, will present a talk at the Applied Mathematics Colloquium.

Title: Generative Models for Sampling and Forecasting

Abstract: Recent advances in unsupervised learning using generative models have found applications in a wide range of problems, including image generation, Monte Carlo sampling, and probabilistic forecasting. These generative 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. Progress in this domain have been made via the introduction of algorithms or inductive biases that render the learning this map, and the Jacobian of the associated change of variables, more tractable. The challenge is to choose what structure to impose on the transport to best reach a complex target distribution containing the data of interest from a simple one used as base, while maintaining computational efficiency. In this talk, I will formalize this problem and discuss how to construct such transport maps by introducing a continuous-time normalizing flow whose velocity is the minimizer of a simple quadratic loss expressed in terms of expectations that are readily amenable to empirical estimation. The flow can be used to generate samples from either the base or target, and to estimate their likelihood. In addition, this flow can be optimized to minimize the path length in Wasserstein-2 metric, thereby paving the way for building transport maps that are optimal in the sense of Monge-Ampere. I will also also contextualize this approach in its relation to score-based diffusion models that have gained a lot of popularity lately. Finally, I will discuss how such generative models can be used in the context of Monte-Carlo sampling, with applications to the calculation of free energies and Bayes factors, as well as probabilistic forecasting, with application to atmosphere/ocean science.  Based on joint works with Michael Albergo, Marylou Gabrie, and Grant Rotskoff

Bio: Eric Vanden-Eijnden is a professor of mathematics at the Courant Institute of Mathematical Sciences, New York University. He earned his doctorate in 1997 from the Université libre de Bruxelles under the supervision of Radu Bălescu. In 2009 he was awarded the Germund Dahlquist Prize of the Society for Industrial and Applied Mathematics "for his work in developing mathematical tools and numerical methods for the analysis of dynamical systems that are both stochastic and multiscale", and in 2011 he won SIAM's J.D. Crawford Prize for outstanding research in nonlinear science.

His research focuses on the mathematical and computational aspects of statistical mechanics, with applications to complex dynamical systems arising in molecular dynamics, materials science, atmosphere-ocean science, fluids dynamics, and neural networks. More recently he has become interested in the mathematical foundations of machine learning (ML) and started to explore the exciting new prospects ML offer for scientific computing. His work combines tools from probability theory, mathematical physics, numerical analysis, and optimization to uncover governing principles in complex systems and design efficient algorithms for their simulation.



This talk will be offered in a hybrid format. Please send an email to [email protected] for the Zoom link.

Event Contact Information:
APAM Department
[email protected]
LOCATION:
  • Morningside
TYPE:
  • Seminar
CATEGORY:
  • Engineering
EVENTS OPEN TO:
  • Faculty
  • Graduate Students
  • Postdocs
  • Staff
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