Edge dynamics
On each edge, the governing equations may describe advection, diffusion, reaction, shallow-water propagation, elasticity, lubrication flow, or pore-radius evolution.
Research overview
PDE and stochastic dynamics on networks
I develop reduced mathematical models for transport, waves, and evolving geometry on networks. The common structure is simple to state: PDE dynamics evolve on edges, conservation and compatibility laws couple those dynamics at vertices, and graph topology determines the global behavior.
Unifying framework
Many physical systems have complicated spatial structure but admit a useful graph-level representation. Edges carry local physics, vertices impose coupling laws, and the network evolves through geometry, stochastic events, or nonlinear feedback.
Vertex coupling
At junctions, conservation of flux and continuity of physical variables determine how waves, particles, heat, chemical species, or pressure fields pass through the network.
Network evolution
Graph structure and weights can change in time through fouling, blocking, reaction-driven deformation, tissue growth, or stochastic particle motion.
Research areas
The projects below share the same mathematical spine, but differ in the edge equations, coupling laws, and application scale.
Hyperbolic dynamics · metric graphs · junction scattering
Shallow-water waves on graphs
Branched rivers, canal systems, and narrow coastal geometries motivate reduced graph models for long water waves. The central question is which vertex conditions correctly transmit and reflect waves at junctions.
In one dimension, the linearized Saint-Venant equations decompose into characteristic modes on each edge. At a junction, height continuity and width-weighted flux conservation give an explicit scattering law,
\[
S_{ab}=\frac{2B_b}{\sum_{e} B_e}-\delta_{ab},
\]
where \(B_e\) is the channel width. This makes reflection a source-dependent property: a junction can be balanced for one set of incoming waves and reflective for another. In cyclic networks, local balance is still insufficient; transparency also depends on synchronized travel times along parallel paths.
A second direction compares the graph model with thin two-dimensional channel domains. This asks whether angles, finite width, and junction geometry leave effective corrections in the graph limit.
Saint-Venant equations
scattering
river networks
2D-to-1D limits
Reactive transport · porous media · model reduction
Reactive transport and geothermal networks
Aquifer thermal energy storage drives carbonate dissolution and precipitation inside fractured rock. This changes pore geometry, permeability, and long-time storage performance.
The model resolves pressure-driven flow, thermal transport, species transport, and reaction-driven radius evolution in slender reactive pores. At network scale, pores are treated as edges and junctions impose continuity and conservation of mass, heat, and species flux.
A current direction builds a spectral representation for the coupled thermal and reactive transport problem near chemical equilibrium. The goal is to preserve pore-scale physics while reducing the computational cost of network-scale parameter studies.
ATES
carbonate kinetics
spectral methods
fracture networks
Stochastic networks · fouling · graph operators
Membrane filtration and stochastic fouling
Membrane filters are porous networks whose performance depends on geometry, connectivity, pore-size distribution, and fouling dynamics.
In the graph formulation, pore junctions are vertices and pores are weighted edges. Pressure satisfies a conductance-weighted graph Laplacian problem, fluxes follow Hagen-Poiseuille scaling, and pore radii evolve as foulant adsorbs to the walls.
Later work adds stochastic sieving: large particles arrive at the membrane surface, perform flow-guided random walks through the network, and block pores that are too small to pass. This gives a stochastic temporal graph whose topology and weights evolve simultaneously.
Related work uses persistent homology to quantify how topological features such as connected components and loops influence filtration performance at fixed porosity.
membrane filters
adsorption
sieving
random walks
persistent homology
Biological transport · deformable channels · networked microstructure
Physiological flows: tissue scaffolds and the blood-brain barrier
Biological transport problems often couple flow, chemistry, mechanics, and evolving geometry. Network models provide a reduced framework for tracking these feedback loops in complex microstructures.
In tissue-engineering scaffolds, interconnected pores carry nutrient transport and shear stress; growth responds to nutrient availability and mechanical forcing, reshaping the scaffold geometry over time.
In blood-brain-barrier models, the focus shifts to immune-cell extravasation through narrow endothelial gaps. Chemokine transport, receptor binding, endothelial viscoelasticity, pressure, and lubrication drag combine to determine whether a cell passes through, stalls, or alters the local barrier geometry.
The longer-term direction is networked BBB transport: the glycocalyx may act as a porous filtering layer, while tight-junction geometry may form a tortuous intercellular network controlling molecular and cellular passage.
BBB
diapedesis
scaffolds
viscoelastic confinement
biological transport
Connectivity · tortuosity · random media
Tortuosity and connectivity in random media
In porous and biological media, transport is governed not only by local conductance but also by path structure. Network representations make it possible to compute tortuosity and connectivity metrics without brute-force particle simulations.
A recurring goal is to connect geometric descriptors of a medium to transport performance. For networks, asymmetric random walks and graph operators provide direct access to mean transit times, path lengths, bottlenecks, and connectivity-sensitive performance measures.
tortuosity
connectivity
random media
transport metrics
Research agenda
The next stage is to sharpen the mathematical theory and computational methods behind dynamic network models, while continuing to pursue applications in porous media, biological transport, and wave propagation.
Stochastic processes on evolving networks
Sieving can be viewed as a killed random walk on a graph whose adjacency, conductance, and topology evolve after each blocking event. This connects filtration to stochastic temporal networks and queueing-type models on edges.
Stable reduced-order methods
Spectral representations and generalized eigenvalue problems can compress edge-level PDE dynamics while preserving vertex coupling, stability, and conservation structure.
Diffusion-dominated microscale transport
At small scales, diffusion can compete with or dominate advection. Incorporating bidirectional diffusion into filtration and biological network models changes both the governing equations and the correct coupling laws.
Thermal coupling in the solid matrix
Reactive-pore models can be extended by resolving heat transfer in the surrounding matrix, including interfacial flux conditions and possible conformal-mapping reductions of complex geometries.
Large-graph and continuum limits
As the number of vertices grows, dynamic network models may admit mean-field or continuum limits. These limits could explain how local stochastic events produce macroscopic transport laws.
Student-scale computational projects
Many subproblems naturally split into projects involving numerical PDEs, graph algorithms, random walks, image-based network generation, and visualization of evolving transport networks.
For collaborators and students
Collaborations
I am interested in problems where network structure, transport, and evolving geometry interact. Good fits include porous media, filtration, biological transport, fracture networks, water-wave propagation, and graph-based reduced models derived from continuum physics.
Student projects
Projects can be theoretical, computational, or just about making cool simulations. Useful background includes differential equations, numerical methods, linear algebra, probability, graph theory, MATLAB or Python, and scientific computing.