Numerical Methods for Gradient Flows and Their Applications (RS-2024-00342949)

Gradient flows provide a powerful mathematical framework for describing dissipative processes in physics, materials science, biology, and data science. Our group develops efficient, accurate, and structure-preserving numerical schemes for gradient flow models, with particular emphasis on unconditional energy stability, gradient stability, solvability, and long-time robustness. 

We study both classical and emerging phase-field type equations, including the Allen–Cahn, Cahn–Hilliard, Swift–Hohenberg, phase-field crystal, and related nonlinear evolution equations. These models are widely used to describe interface motion, pattern formation, phase separation, crystal growth, and other complex phenomena involving evolving structures.

Our recent work focuses on designing numerical methods that preserve the essential mathematical structures of the underlying models. These include energy-stable schemes, maximum-principle-preserving methods, compact finite-difference methods, Fourier spectral methods, operator splitting methods, and efficient linear or explicit algorithms for nonlinear gradient flows.

Main topics

• Allen–Cahn, Cahn–Hilliard, Swift–Hohenberg, and phase-field crystal equations

• Energy-stable and gradient-stable numerical schemes

• Maximum principle, mass conservation, positivity preservation, and unique solvability

• Finite-difference methods, compact schemes, Fourier spectral methods, and operator splitting methods

• Applications to interface dynamics, pattern formation, image processing, and data classification

Mathematical Data Science

Phase-field models were originally developed to describe interfacial phenomena, but their ability to represent complex geometries and evolving boundaries also makes them useful for modern data science. Our group investigates how PDE-based models can be used for supervised learning, data classification, decision-boundary evolution, and feature extraction.

In this direction, we develop phase-field-inspired learning frameworks in which classification boundaries are treated as evolving interfaces. This approach provides an interpretable mathematical structure for separating data while maintaining connections to variational principles, energy minimization, and numerical PDE theory.

We are also interested in combining phase-field methods with topological data analysis, feature engineering, and machine learning techniques for structured and unstructured data. Recent research topics include Turing pattern classification, PDE-informed classification models, and geometric data generation through manifold evolution.

Main topics

• PDE-based supervised learning and data classification

• Phase-field approaches to decision-boundary evolution

• Turing pattern classification and feature engineering

• Topological data analysis and geometric data representation

• PDE-informed machine learning for structured and unstructured data

Gait Data Analysis with AIT STUDIO (RS-2024-00467438)

In collaboration with AIT STUDIO, we develop AI-assisted mathematical and computational frameworks for gait data analysis. Human gait contains rich information about physical condition, motor function, aging, rehabilitation, and disease-related abnormalities. Our goal is to extract meaningful quantitative features from gait data and build reliable models for abnormal gait detection and bio-healthcare applications.

This project combines mathematical modeling, time-series analysis, computer vision, and machine learning. In particular, we are interested in developing sensor-light or sensor-free approaches using video-based data, including 2D RGB camera inputs, to make gait analysis more accessible and scalable.

The long-term goal is to contribute to digital healthcare technologies that can support screening, monitoring, rehabilitation, and clinical decision-making through interpretable and robust computational tools.

Main topics

• Bio-healthcare data analysis

• Gait feature extraction and abnormal gait detection

• Time-series data analysis

• AI-assisted movement analysis

• Video-based and camera-based healthcare data processing

• Collaboration with industry partners for practical deployment

Biomedical Modeling and Mathematical Biology

Our group develops mathematical and computational models for biological and medical systems in collaboration with medical doctors, biologists, and domain experts. We are particularly interested in how mathematical models can help explain biological mechanisms, test hypotheses, and provide quantitative insight into complex biomedical phenomena.

Previous and ongoing research topics include reaction–diffusion systems, spatially heterogeneous biochemical reactions, circadian timekeeping, anatomical structure-function relationships, eye movement modeling, cytokinesis, and biomedical image-based simulation.

A central theme of this research is to connect mechanistic mathematical models with experimentally or clinically observed phenomena. We use tools from partial differential equations, stochastic simulation, numerical analysis, and scientific computing to study how spatial structure, geometry, and physical constraints influence biological function.

Main topics

• Mathematical biology and biomedical modeling

• Reaction–diffusion systems and spatial heterogeneity

• Stochastic simulations of gene regulation

• Circadian rhythm and biochemical reaction modeling

• Anatomical structure-function modeling

• Eye movement, muscle mechanics, and medical simulation

• Collaboration with medical doctors and experimental researchers