Scientific Machine Learning in High Dimensions
Many problems at the frontier of scientific machine learning reduce to solving high-dimensional partial differential equations: Hamilton–Jacobi– Bellman equations in optimal control, the continuity equation and Fokker–Planck equation in generative modeling, and coupled systems in mean-field games and Bayesian inverse problems, to give some examples. While the reach of classical numerical methods is limited by the curse of dimensionality, recent advances at the interface of numerical analysis and machine learning are beginning to change that picture by combining classical mathematical structure with neural-network approximation and Monte-Carlo-style sampling.
In this talk I will discuss the mathematical and computational foundations of these methods with a focus on high-dimensional stochastic optimal control. I will draw on selected results from our group's work on neural HJB solvers, conditional optimal transport for Bayesian inference, and mean-field-games, and place them in the broader landscape of methods that have emerged in recent years.
I will outline the main challenge of closing the gap between theoretical guarantees and
practical performance. I will emphasize the role of sampling in determining whether neural methods discover the structure of high-dimensional value functions, the design of benchmarks, and the choice of discretization and adjoint formulations when differentiating through neural SDEs. I will close with open questions and opportunities for the scientific machine learning community.
This initiative is part of the “Ph.D. Lectures” activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.
Contatto:
paolo.zunino@polimi.it
Lars Ruthotto is the Winship Distinguished Research Associate Professor in the Departments of Mathematics and Computer Science at Emory University, where he is a member of Emory's Scientific Computing Group and directs the Emory REU/RET Site for Computational Mathematics for Data Science. He earned his Diploma (2010) and PhD (2012) in Mathematics from the University of Münster, Germany, and was a postdoc at the University of British Columbia before joining Emory. His research develops computational methods at the interface of scientific computing, optimization, machine learning, and inverse problems, with recent emphasis on neural-network methods for high-dimensional partial differential equations and stochastic optimal control. He is a recipient of the NSF CAREER Award, serves as a Section Editor for the SIAM Journal on Scientific Computing, and is supported by the US NSF and the Office of Naval Research.