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
We are grateful for grant support from NSF, DOE, Ford, and MSU Foundation.
