Research

My research develops mathematical and computational methods for modeling complex dynamical systems and learning their hidden mechanisms from data. A recurring theme is to exploit mathematical and physical structure, such as interaction kernels and variational principles, to make data-driven inference and scientific computation stable, interpretable, and physically meaningful. My work connects scientific machine learning and inverse problems with multiscale modeling and nonlinear PDEs, with applications to collective dynamics, ion transport, urban dynamics, and physics-informed computing.

Learning Collective Dynamics & Interacting Particle Systems

How can hidden interaction mechanisms be inferred from collective behavior when observations, interaction structure, or physical laws are only partially known?

My work in collective dynamics studies both the forward and inverse relationship between microscopic interactions and emergent macroscopic behavior. Earlier work on swarmalators examined how competing attractive and repulsive interactions generate distinct collective states through modeling and analysis. This naturally motivates the inverse question: what can observed collective behavior reveal about the interactions that produced it?

Classical interaction-learning methods typically assume sufficiently rich trajectory data and a prescribed interaction architecture. My current work explores what remains identifiable in more ill-posed settings, including inference from highly limited observations, unknown interaction neighborhoods, and observational data for which even the governing physical law is not supplied beforehand.

Collective dynamics and interacting particle systems

Learning from Limited Observations

We study how interaction kernels can be recovered when only collective steady states are observed. Such data make system identification highly degenerate, and we show that identifiability is governed by the distribution of observed configurations together with the structural information encoded in the patterns themselves.

Learning Interaction Topology

We develop methods that ask not only how agents interact, but also who interacts with whom. This includes metric, nearest-neighbor, Voronoi, density-dependent interaction rules, etc., with the goal of identifying interaction structure and interaction kernels jointly.

Discovering Gravitational Structure

Using Solar-System ephemerides as a testbed, we study whether interpretable physical laws can emerge directly from trajectory observations without imposing the governing law beforehand, including source-dependent interactions and missing-source inference.

Ion Transport & PNP Equations

How can thermodynamically consistent models of ion transport be derived, analyzed, and inferred from data by exploiting their energetic and variational structure?

Ion transport and Poisson–Nernst–Planck dynamics

My work on ion transport begins with the mathematical structure of nonequilibrium transport models. In concentrated ionic systems, short-range steric interactions and friction between species introduce mechanisms absent from classical dilute Poisson–Nernst–Planck theory. Using energetic variational principles, we study how conservative interactions enter the free energy while dissipative mechanisms modify transport and mobility, and how this structure governs finite-energy solutions and long-time relaxation giving rise to cross-diffusion.

This forward understanding motivates a complementary inverse question: what constitutive physics is actually supported by concentration data? Rather than fixing a local constitutive law in advance, my current work studies how observations distinguish effective local models from genuinely nonlocal interactions and which components of the underlying constitutive physics are identifiable.

Modeling & Analysis of Cross-Diffusion PNP Systems

We develop thermodynamically consistent PNP-type models for crowded ionic systems, where steric interactions modify the free energy while interspecies drag modifies dissipation and transport. These mechanisms generate nonlinear cross-diffusion through distinct energetic and dissipative pathways. We analyze the resulting PDE systems with emphasis on global finite-energy solutions, entropy dissipation, and long-time behavior.

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Learning & Identifying Constitutive Physics

We study structure-preserving inverse problems for identifying constitutive physics from concentration data, including when nonlocal steric interactions admit effective local closures and when genuinely nonlocal structure remains identifiable. More broadly, we investigate learnable components of free energy and dissipation, such as steric interactions, and transport mechanisms, while preserving the underlying gradient-flow and Onsager structure and understanding the mathematical limits of their identifiability.

Multiscale Urban Dynamics

How do individual-level behaviors give rise to collective urban crime patterns across scales, and what hidden mechanisms can be identified from observed data?

My work on urban dynamics develops multiscale mathematical models that connect individual behavior with emergent spatio-temporal patterns at the population level. Starting from agent-based descriptions of residential burglary, we study continuum PDE models obtained through mean-field limits and develop computational methods for simulating these systems in heterogeneous and realistic urban environments.

A complementary direction reverses this perspective: what hidden urban crime mechanisms can be identified from observed dynamics? My current work develops PDE-constrained optimization and weak-form finite-element inverse methods for identifying hidden dynamics and environmental structure. A particular focus is inference from realistically observable crime intensity, rather than direct observations of the underlying model states.

Spatiotemporal urban crime dynamics in Chicago

Realistic Scientific Computing

We connect probabilistic agent-level descriptions to nonlinear continuum models and develop robust finite-element methods for spatially heterogeneous systems with natural boundary conditions and realistic urban geometries.

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Feedback & Intervention Dynamics

We study how adaptive response and delayed information affect the stability and spatio-temporal behavior. In particular, delayed response can destabilize otherwise stable states and generate oscillatory, moving, splitting, and merging hotspot patterns.

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Discovering Hidden Urban Dynamics

We develop PDE-constrained optimization and weak-form finite-element inverse methods for hidden dynamics, with focus on indirect observations, realistic crime data, and the mathematical limits of identifiability.

PINNs for Stiff Dynamics

How can the mathematical and physical structure of a dynamical system be built into physics-informed neural netowrks to improve stability, accuracy, and differentiability?

Structure-aware physics-informed learning

My work on PINNs studies how known mathematical structure can be incorporated directly into neural representations for challenging dynamical systems. Rather than treating governing equations, initial conditions, and dynamical interfaces solely as competing penalty terms, we develop representations that enforce essential structure by construction. The broader goal is to understand how the architecture of physics-informed learning should reflect the mathematical organization of the underlying dynamical problem, improving not only prediction accuracy but also conditioning, stability, and the scientific usefulness of the learned solution map.

Stable Learning for Stiff Dynamics

We study why physics-informed neural networks become difficult to train for stiff time-dependent PDEs and how exact enforcement of initial conditions can stabilize training, reduce spectral bias, and introduce an implicit time-marching structure.

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Event-Structured Learning of Critical Dynamics

For hybrid dynamical systems, we develop neural surrogates whose representations follow known physical phases and enforce exact state chaining across input-dependent event interfaces. The resulting differentiable trajectory families enable the extraction of critical transition boundaries and sensitivities, turning physics-informed surrogates into tools for studying scientifically meaningful quantities of interest.

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