Upcoming Event: Babuška Forum

Scientific Machine Learning for Distributional Data: From Densities to Particles and Probability Measures

Nicholas Nelsen, Assistant Professor, Aerospace Engineering & Engineering Mechanics

Abstract

Scientific data increasingly takes the form of evolving point cloud ensembles, interacting particles, or varying-size collections of sensor measurements, rather than fixed-dimensional data representations. These data arise in applications such as probabilistic weather forecasting, LiDAR imaging, fusion energy science, and multi-agent control. This talk outlines a framework for scientific machine learning on the space of probability distributions, which is a curved, nonlinear space that contains much more than Gaussian “bell curves.” It discusses two complementary formulations: neural operator methods that act on probability density functions, and transformer architectures that act directly on probability measures through particle representations. Toward analyzing the capabilities and limitations of such architectures, the talk presents recent approximation, stability, and discretization theory that underpin their computational cost and accuracy. A running motivation throughout the talk is amortized Bayesian inference, where the goal is to solve scientific inverse problems with quantified uncertainty in near real-time using a single trained model.

Biography

Nicholas Nelsen is an Assistant Professor at UT Austin, where he holds a joint appointment in the Oden Institute (Principal and GSC Faculty) and the Department of ASE/EM. Nick’s research centers on computational mathematics, machine learning, inverse problems, and uncertainty quantification. Before joining UT in August 2026, he was a Klarman Fellow at Cornell University and an NSF Mathematical Sciences Fellow at MIT. Nick earned his Ph.D. from Caltech in 2024, where his work on data-efficient operator learning was awarded multiple thesis prizes and a SIGEST recognition from SIAM.