I am an Assistant Professor in the Department of Computational Mathematics, Science, and Engineering at Michigan State University, with a joint appointment in the Department of Statistics and Probability.

My research focuses on computational mathematics, particularly in scientific machine learning (SciML) and multi-scale modeling related to fluid physics, materials science, and biophysics. As part of the broader field of AI for Science, I develop numerical algorithms for constructing accurate and structure-preserving ML-based models of multi-scale systems directly from the first-principle-based descriptions. My research goal is to establish accurate modeling of multi-scale dynamical systems relevant to non-Newtonian hydrodynamics, non-equilibrium kinetic processes, and meso-scale stochastic reduced dynamics, which are central to various science and engineering applications.

Research Overview

Accurate modeling of multi-scale systems has been a long-standing problem in both computational mathematics and broad scientific applications. A fundamental challenge arises from their multi-scale nature and high-dimensionality. There is generally no simple set of modes that can be used to project and predict the dynamics in a self-contained manner. Existing approaches often rely on sophisticated micro-macro coupling and empirical constitutive closures. Despite their broad applications, these empirical models generally show limitations in retaining the molecular-level interactions. Currently, there is still a lack of reliable models to quantify complex multi-scale processes by faithfully modeling the micro-interactions in a transferable and integrated manner.

Recent progress in the machine learning (ML) approach, with its unprecedented capability to approximate high-dimensional functions, has opened up many new possibilities in computational science. Meanwhile, ML is often perceived as a “black-box” approach lacking fundamental principles. This has been an essential obstacle to making further progress in physical modeling and scientific computing. To construct truly reliable ML-models for multi-scale problems, fundamental challenges remain: (1) How to effectively transfer micro-scale physical laws across scales while retaining physical interpretability; (2) How to strictly preserve the physical constraints and mathematical structures of the ML-based partial and ordinary differential equations (PDEs and ODEs) that ensures the well-posedness and numerical stability?

My research aims to address these challenges by developing numerical algorithms for constructing accurate ML-based models of multi-scale systems directly from first-principle-based descriptions. As part of the AI for Science initiative, a key objective is to retain the micro-model fidelity while strictly preserving canonical structures and symmetry constraints. Examples include hydrodynamics of multi-scale fluids, kinetic transport, and mesoscale stochastic reduced dynamics. The long-term goal is to enable predictive modeling of multi-scale systems that extends beyond phenomenological understanding, facilitating integrated control across multiple scales.

Research Interests

• Scientific machine-learning
• Multi-scale modeling
• Model reduction and stochastic simulation

Representative publications

• P. Ge, Z. Zhang, and H. Lei. Data-driven learning of the generalized Langevin equation with state-dependent memory. Phys. Rev. Lett. 133:077301, 2024. [link].

• L. Lyu and H. Lei. Construction of coarse-grained molecular dynamics with many-body non-Markovian memory. Phys. Rev. Lett. 131:177301, 2023. [link].

• L. Fang , P. Ge, L. Zhang, W. E, and H. Lei. DeePN²: A Deep Learning-Based non-Newtonian Hydrodynamic Model. Journal of Machine Learning 1: 114–140, 2022 [link].

• H. Lei, L. Wu, and W. E. Machine Learning Based non-Newtonian Fluid Model with Molecular Fidelity. Phys. Rev. E 102: 043309, 2020 [link].

• H. Lei, N. A. Baker, and X. Li. Data-Driven Parameterization of the Generalized Langevin Equation. Proc. Natl. Acad. Sci. 113 (50):14183–14188, 2016 [link].

Opening

My group is recruiting Ph.D. students, Postdocs, and Visiting students (more info). If you are interested, please contact me by email.