Nicholas Boffi Brings Together Applied Math and Machine Learning

Generative models in machine learning transform mundane tasks in mind-blowing ways, from enhancing photographs to translating a lecture in real time.

Nicholas Boffi sees the potential to apply these advances to scientific computing and other areas of computational math.

“Machine learning is a really exciting area where we’re seeing huge advances, such as large language models and diffusion models, that are revolutionizing our day-to-day lives,” he said.

Boffi works in an emerging field at the intersection of applied math and machine learning (ML) and examines how the findings in each can inform one another. He joined Carnegie Mellon University’s Department of Mathematical Sciences this fall. Already he is forming a reading group that spans the two fields.

“We’re trying to better understand some of these problems at the interface between applied math and ML,” Boffi said. “There's quite a lot of interest from both the math and machine learning departments, so this is going to be exciting. I'm really looking forward to meeting others in the community and sparking collaborations.”

Boffi is a member of Carnegie Mellon’s Center for Nonlinear Analysis and an affiliated faculty member in the Machine Learning Department.

“One of the things I've been most excited about is how welcoming everyone is at Carnegie Mellon” Boffi said. “The people I’m meeting are open to collaboration, and so I'm constantly getting hit with new ideas and new problems every single day.”

Prior to Carnegie Mellon, he completed a three-year research and teaching postdoctoral position as a Courant Instructor at New York University. Some of the work he completed during that time included “Stochastic Interpolants: A Unifying Framework for Flows and Diffusions,” co-authored by Michael S. Albergo and Eric Vanden-Eijnden, and “Deep Learning Probability Flows and Entropy Production Rates in Active Matter,” co-authored by Vanden-Eijnden and published in the Proceedings of the National Academy of Sciences.

“My interests lie in developing algorithms based on machine learning that can solve computational problems that arise in scientific disciplines that we have no other methodology to solve,” Boffi said. Teaching is another one of his strengths. His teaching style for undergraduates focuses on generalizable principles.

“Rather than getting lost in technical details, I really try to communicate the connection between modern research any chance that I get,” he said.

At Carnegie Mellon, he teaches an undergraduate course titled “Introduction to Partial Differential Equations: A Computational Approach.”

“Partial differential equations are a class of equations that arise across math and science, and some people call them the ‘language of physics’ or ‘language of the laws of nature’ because they describe everything from heat transfer to electrodynamics to fluid flow,” Boffi said.

He hopes to develop a graduate course covering some topics in machine learning in the future.

A Connecticut native, Boffi has moved often in recent years to advance his work. He received a Ph.D. in applied mathematics from Harvard University in 2021.

Prior to earning his doctorate, Boffi was a research intern at Google Brain, a visiting graduate student researcher at the Massachusetts Institute of Technology, a computational science graduate fellow at Harvard University and a Fulbright Research Scholar at Tel Aviv University.

He said he looks forward to living in Pittsburgh, which has impressed him.

“Carnegie Mellon has a world-class machine learning department, a world-class math department, and a math faculty whose areas of expertise align perfectly with my interests in partial differential equations and probability theory,” he said. “And I really don't think there are many other universities with this combination of strengths.”