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1. Clustering Dynamics on Graphs: From Spectral Clustering to Mean Shift Through Fokker–Planck Interpolation.

   https://doi.org/10.1007/978-3-030-93302-9_4  

   Craig, Katy; Garcia Trillos, Nicolas; Slepcev, Dejan (December 2021, Modeling and simulation in science engineering technology)

   Bellomo, N.; Carrillo, J.A.; Tadmor, E. (Ed.)

   In this work, we build a unifying framework to interpolate between density-driven and geometry-based algorithms for data clustering and, specifically, to connect the mean shift algorithm with spectral clustering at discrete and continuum levels. We seek this connection through the introduction of Fokker–Planck equations on data graphs. Besides introducing new forms of mean shift algorithms on graphs, we provide new theoretical insights on the behavior of the family of diffusion maps in the large sample limit as well as provide new connections between diffusion maps and mean shift dynamics on a fixed graph. Several numerical examples illustrate our theoretical findings and highlight the benefits of interpolating density-driven and geometry-based clustering algorithms. 

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2. Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

   https://doi.org/10.1017/apr.2020.47  

   Arapostathis, Ari; Pang, Guodong; Zheng, Yi (March 2021, Advances in Applied Probability)

   null (Ed.)

   Abstract We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied. 

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3. Potentials and challenges of high-field PFG NMR diffusion studies with sorbates in nanoporous media

   https://doi.org/10.1007/s10450-020-00255-y  

   Baniani, Amineh; Berens, Samuel J.; Rivera, Matthew P.; Lively, Ryan P.; Vasenkov, Sergey (August 2020, Adsorption)

   High magnetic fields (up to 17.6 T) in combination with large magnetic field gradients (up to 25 T/m) were successfully utilized in pulsed field gradient (PFG) NMR studies of gas and liquid diffusion in nanoporous materials. In this mini-review, we present selected examples of such studies demonstrating the ability of high field PFG NMR to gain unique insights and differentiate between various types of diffusion. These examples include identifying and explaining an anomalous relationship between molecular size and self-diffusivity of gases in a zeolitic imidazolate framework (ZIF), as well as revealing and explaining an influence of mixing different linkers in a ZIF on gas self-diffusion. Different types of normal and restricted self-diffusion were quantified in hybrid membranes formed by dispersing ZIF crystals in polymers. High field PFG NMR studies of such membranes allowed observing and explaining an influence of the ZIF crystal confinement in a polymer on intra-ZIF self-diffusion of gases. This technique also allowed measuring and understanding anomalous single-file diffusion (SFD) of mixed sorbates. Furthermore, the presented examples demonstrate a high potential of combining high field PFG NMR with single-crystal Infrared Microscopy (IRM) for obtaining greater physical insights into the studied diffusion processes. 

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4. Cover times with stochastic resetting

   https://doi.org/10.1063/5.0260643  

   Linn, Samantha; Lawley, Sean D (April 2025, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   Cover times quantify the speed of exhaustive search. In this work, we approximate the moments of cover times of a wide range of stochastic search processes in d-dimensional continuous space and on an arbitrary discrete network under frequent stochastic resetting. These approximations apply to a large class of resetting time distributions and search processes including diffusion, run-and-tumble particles, and Markov jump processes. We illustrate these results in several examples; in the case of diffusive search, we show that the errors of our approximations vanish exponentially fast. Finally, we derive a criterion for when endowing a discrete state search process with minimal stochastic resetting reduces the mean cover time. 

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5. Data-driven discovery of Green’s functions with human-understandable deep learning

   https://doi.org/10.1038/s41598-022-08745-5  

   Boullé, Nicolas; Earls, Christopher J.; Townsend, Alex (March 2022, Scientific Reports)

   Abstract There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, we develop a novel data-driven approach for creating a human–machine partnership to accelerate scientific discovery. By collecting physical system responses under excitations drawn from a Gaussian process, we train rational neural networks to learn Green’s functions of hidden linear partial differential equations. These functions reveal human-understandable properties and features, such as linear conservation laws and symmetries, along with shock and singularity locations, boundary effects, and dominant modes. We illustrate the technique on several examples and capture a range of physics, including advection–diffusion, viscous shocks, and Stokes flow in a lid-driven cavity. 

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