Research

My research interests center around utilizing mathematical modeling and computational techniques to address challenges spanning a range of disciplines, such as:

Mean-field games,
Mean-field control,
Data science,
Machine learning,

and related areas, with a focus on optimization and partial differential equations.

Modeling and Inference of Mean-Field Game Systems

Modeling large populations of rational agents in complex systems is vital. Mean-field game (MFG) systems offer a breakthrough in describing such systems, from finance competition to traffic congestion. MFG combines population approximation with optimal control for a representative player. My work applies MFG to pandemic control and addresses related inverse problems, advancing our understanding of population dynamics.

Controlling propagation of epidemics via mean-field controls and vaccine distributions.

The coronavirus disease 2019 (COVID-19) pandemic is changing and impacting lives on a global scale. As the propagation of COVID-19 has a significant spatial characteristic, it is crucial to have a spatial-type SIR model to study the spread of the infectious disease and movement of individuals. Beyond this, mean-field games (controls) provide a unique perspective to study and understand the underlying population dynamics.

In [1], we introduce a mean-field control model to control the propagation of epidemics on a spatial domain. The control variable - the spatial velocity -is first introduced for the classical disease models. We provide fast numerical algorithms, where numerical experiments demonstrate that our proposed model illustrates how to separate infected patients in a spatial domain effectively. With the invention of the COVID-19 vaccine, shipping and distributing is crucial in controlling the pandemic. We extend this work by using the concept of mean-field control to study the optimal vaccine distribution strategy in [2].

From left to right, the pictures display the density distribution ρ at time t = 0.1, 0.5, 0.9. The red line represents the boundary of Ω where we obtain partial measurements. In this MFG, the density travels from the left towards the right, crossing the boundary ∂Ω twice.

Inverse problem in mean-field games

While mean-field games provide a powerful framework for modeling large population behavior, a significant challenge arises from unknown or partially known system parameters. In [*], we addressed this challenge by recovering model parameters governing population interactions based on limited, noisy observations within a restricted aperture. This problem is highly ill-posed, yet vital for understanding population dynamics.

Our research focuses on efficiently recovering the running cost and interaction energy in MFG equations using boundary measurements of population profiles and movements. We formalize this as a constrained optimization problem with a least-squares residual functional. We've developed a fast and robust operator splitting algorithm, incorporating techniques like harmonic extensions and the primal-dual hybrid gradient method. Numerical experiments validate the effectiveness and robustness of our approach,

Efficient Computational Methods for PDEs

Nonlinear PDEs are crucial in various fields. Solving them relies on numerical methods due to their complexity. I focus on turning PDEs into optimization problems, developing specialized iterative methods, and using machine learning for solutions.

My PhD work focus on computational methods for mean-field games, where I focus on non-local MFG and a special class of non-potential MFG. [a1][a2]
We proposed a saddle point framework for solving nonlinear PDEs with implicit-in-time scheme, applying to various type of differential equations. [b1][b2]
Recently, I also start to work on in-context learning towards PDE computations, leveraging machine learning methods (large language models). [c1][c2]

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