Machine learning (ML) has shown great promise for accelerating numerical simulation, with examples such as numerical weather prediction and molecular dynamics with MLpredicted potentials. At the same time, many data-driven ML approaches face wellknown challenges, including limited interpretability, reduced reliability, and substantial demands on training data and computational resources such as GPUs.
In this talk, I will present a new approach aimed at accelerating classical finite element methods while preserving their key advantages, such as interpretability, robustness, and applicability to complex geometry. The central idea is to train a machine learning model on a single reference element, which allows the training to be done cheaply with much less data and much lower GPU demands. Once trained, the model will be deployed to accelerate computation across all physical elements in the domain through techniques such as hybridization and static condensation.
I will demonstrate this hybrid FEM–ML methodology on a representative but challenging model problem: the radiative transfer equation. I will also discuss some ongoing theoretical work, including error analysis for the proposed scheme on second-order elliptic equations.

About the

Shukai Du is an assistant professor of mathematics. Before that, he served as a visiting assistant professor at the University of Wisconsin-Madison from 2020 to 2024. He obtained his PhD in applied mathematics from the University of Delaware in 2020.