Abstract: In this talk, we take a dynamical systems approach toward two algorithmic questions that arise from complex systems in scientific and machine learning applications. 
In the first part, we discuss the computation of linear response: the derivative of statistics or long-time averages of a dynamical system with respect to its input parameters. In many ergodic chaotic systems, such as certain turbulent fluid flows, detailed climate models, etc., linear response exists but has been notoriously difficult to compute. Apart from the curse of dimensionality, this difficulty can be attributed to a defining aspect of chaos: infinitesimal perturbations along a given orbit grow in norm exponentially. In this talk, we present a new alternative for linear response computation called the space-split sensitivity (S3) algorithm. One key component of S3 is a fast computation of conditional scores – log gradients of probability measures conditioned on the unstable manifold. 
In the second half, we discuss a problem where taking the dynamical systems approach is insightful for generalization in machine learning: the performance of a learning algorithm on unseen data. We consider local descent training algorithms that do not converge to a fixed point but whose long-time averages converge. We redefine generalization and training errors, which traditionally use loss values at fixed parameters, in terms of loss statistics. We then extend classical generalization analyses to such non-converging regimes. Further, we show how training dynamics can provide clues for generalization. 

Abstract: In this talk, we introduce a new and unified convergence theory for general two-level iterative methods. In the case of the Galerkin coarse solver, we establish a succinct identity for the energy norm of the error propagation operator of two-level methods. More generally, we present some convergence estimates for two-level methods with approximate coarse solvers, including both linear and nonlinear types.

Abstract: Random walks have been traditionally used to explore a graph and have revealed a valuable tool to investigate and characterize the structural properties of complex networks such as modules, central nodes, and paths or to determine the typical times to reach a target. Although various types of random walks have been proposed, most, if not all, rely on the assumption of linearity and independence of the individual walkers. In recent years, we have introduced a new class of nonlinear stochastic processes describing a system of interacting random walkers moving over networks with finite node capacities [1-4]. We have shown that the asymptotic distribution of diffusing agents is a nonlinear function of the nodes’ degree and saturates to a constant value for sufficiently large connectivities at variance with standard diffusion in the absence of excluded-volume effects. Building on this observation, we have developed an inverse problem to reconstruct the a priori unknown connectivity distribution [4]. The method gathers all the necessary information by repeating a limited number of independent measurements of the asymptotic density at a single node, which can be chosen randomly. The technique is successfully tested against synthetic and real data and is also shown to estimate the total number of nodes with great accuracy.
On the other hand, we have found that, for each level of the nonlinear bias, there is an optimal crowding that maximizes the entropy rate in a given network topology [3]. The analysis suggests that a significant fraction of real-world networks are organized in such a way as to favor exploration under congested conditions. Based on such a fact, we present a targeted model of optimal social distancing on metapopulation networks, named the ESIR model, which can effectively reduce the disease spreading and, at the same time, minimize the impact on human mobility and related costs [2]. The finite carrying capacity of the network’s metanodes is modeled as a slack compartment E for the classic SIR model. It quantifies the density of vacant spaces to accommodate the diffusing individuals. Formulating the problem as a multi-objective optimization problem shows that when the walkers avoid crowded nodes, the system can rapidly approach Pareto optimality, thus reducing the spreading considerably while minimizing the impact on human mobility, as also validated in empirical transport networks. These results envisage ad hoc mobility protocols that can enhance policymaking for pandemic control.
[1] J.-F. de Kemmeter, T. Carletti, M. Asllani, Self-segregation in heterogeneous metapopulation landscapes Journal of Theoretical Biology 554, 111271 (2022)
[2] B. A. Siebert, J. P. Gleeson, M. Asllani, Nonlinear random walks optimize the trade-off between cost and prevention in epidemics lockdown measures: the ESIR model Chaos, Solitons & Fractals 161, 112322 (2022)
[3] T. Carletti, M. Asllani, D. Fanelli, V. Latora Nonlinear walkers and efficient exploration of a crowded network Physical Review Research, 2, 033012 (2020)
[4] M. Asllani, T. Carletti, F. Di Patti, D. Fanelli, F. Piazza “Hopping in the crowd to unveil network topology” Physical Review Letters, 120, 158301 (2018)

Abstract:

Plasma is one of the four fundamental states of matter. It exists widely in nature and can be artificially generated by heating or subjecting a neutral gas to a strong electromagnetic field. When the temperature is high or the density is low, the effect of collisions becomes minor compared to the effect of the electromagnetic forces, and such plasmas are modeled by the relativistic Vlasov-Maxwell equation. We present results on the stability of equilibria (time-independent solutions) for the relativistic Vlasov-Maxwell equation. In particular, linear stability criteria for certain classes of equilibria are discussed. We also give a result on the nonlinear stability of an initial-boundary value problem for the Vlasov-Poisson equation, a reduced form of the Vlasov-Maxwell equation applicable when the magnetic field is negligible.  

Abstract:

Democrats and Republicans in the United States like members of their own party more than members of the opposing party. To what extent is animosity between partisan groups motivated by dislike for partisan out-groups per se, policy disagreement, or other social group conflicts? In many circumstances, these motivations are observationally equivalent. We draw on a series of novel experiments to shed light on the nature of social animosity between partisans. First, we assess stereotypes about Democrats and Republicans. Next, we draw on vignette evaluation experiments to estimate effects of shared partisanship when additional information is or is not present, and benchmark these effects against shared policy preference effects. Results reveal limits in our ability to distinguish between possible explanations for social conflict. Nonetheless, they suggest that common measures of animosity between political parties may capture programmatic conflict more so than social identity-based partisan hostility.

Abstract:
The Nielsen-Thurston classification theorem establishes a deep connection between braids and surface dynamics. We consider braids on low-genus surfaces, with braid generators defined using maximally symmetric embedded graphs. Using a recently developed algorithm, we find unique pseudo-Anosov (pA) surface braids which maximize a suitably normalized version of the braid dilation (or equivalently the topological entropy). In particular, we will report on an interesting family of these maximally mixing braids which includes the well known golden and silver ratio Artin braids. Furthermore, we will introduce a physical system of active nematic microtubules, which realizes these braids as an emergent property of its dynamics. 

Abstract:
Our group has focused attention on the analysis of genetic and genomic data in healthy agers and centenarians from multiple studies that are still ongoing.  All these studies are in the process of augmenting genome-wide genotype data and multi-omics data that often include gene expression, serum metabolomics and proteomics, DNA methylation, and microbiome data. The promise of such massive data generation effort is to be able to identify molecular signatures of the genetic fingerprints of extreme longevity that can be used as therapeutic targets.  I will describe challenges and approaches to multi-omics integration of data from these studies that are typically small, include related individuals with repeated measures over time, and often very noisy data at the extreme of health span.

Abstract:

In this talk, I will exam various diffusion-reaction systems, employing a general energetic variational effects. The framework reveals the underlying variational structures of the systems, especially those involving different mechanisms and also boundary/surface physics. Moreover, we will systematically exams various assumptions, postulations and criterion when the temperature effects are involved. We will also discuss various modeling and analysis issues related to this topic.

Abstract:
Heat pipes are essential in every modern computer because they are reliable and passive devices per their reliance on capillarity, have an effective thermal conductivity 10-to-100 times that of a solid copper rod of same diameter and possess a sufficiently-high maximum heat load. We develop a semi-analytical procedure to capture the effect of slowly varying (streamwise) meniscus curvature on the flow resistance of the adiabatic section of an axial-groove heat pipe (AGHP). The relevant small parameter is the pitch of the grooves divided by the length of the adiabatic section. Prescribed are the geometry of the AGHP, its orientation with respect to the gravity vector and relevant thermophysical properties (and, by implication, the capillary pressure driving the flow). Our requisite consideration of the evaporator and condenser sections of the AGHP invoke the standard assumption that the radius of the meniscus in them is a constant equal to that of the lands between menisci. The deviation of the meniscus geometry from a circular arc (relative to an origin at the radial center of the AGHP) in the adiabatic section is captured using a boundary perturbation, where the small parameter is the protrusion angle between the arc defining a meniscus and that corresponding to the radius of an adjacent land. A local analysis ensures the singularities at the triple contact lines are resolved. Our procedure enables more accurate prediction of the components of the capillarity-limited maximum heat load in an AGHP and the corresponding thermal resistance of its adiabatic section.

Bio: Marc Hodes earned his BS, MS, and PhD degrees in Mechanical Engineering from the University of Pittsburgh, the University of Minnesota and the Massachusetts Institute of Technology, respectively. He spent 10 years at Bell Labs Research (Murray Hill, NJ) and has spent extended periods in residence at the National Institute of Standards and Technologies (NIST) and the University of Limerick. He joined the Department of Mechanical Engineering at Tufts University in 2008 where he is a Professor and the Director of Graduate Studies. His Groups’ research there has been funded by government agencies, e.g., NSF, DARPA and DoE, and industry, e.g., Huawei and Google. Research interests are in Transport Phenomena and, over the course of his career, four thematic areas have been addressed: 1) the thermal management of electronics, 2) mass transfer in supercritical fluids, 3) analysis of thermoelectric modules, and 4) momentum, heat, mass and charge transport in the presence of apparent slip. Pertinent to the latter topic, Marc runs the Red Lotus Project with Professors Darren Crowdy and Demetrios Papageorgiou in the Department of Mathematics at Imperial College London, where he is a regular, long-term Academic Visitor. Professor Hodes is the sole- or co-author of over 50 papers in archival journals on transport phenomena. Together with Dr. Georgios Karamanis, Marc spun Transport Phenomena Technologies, LLC, out of Tufts University in 2017, which is currently focused on a Phase II NSF SBIR project on conforming vapor chambers for the thermal management of electronics and various consulting projects.

Abstract: Materials that change shape, often dramatically, are ubiquitous in nature and beyond. Many times, this shape is determined by the system’s energetics. As a simple example, a soap bubble is driven by surface tension to form a spherical shell, minimizing its area for a given volume. Equilibrium shapes of more complicated soft materials can also be modeled by minimizing a given energy functional with respect to the shape of the domain and auxiliary fields describing the structure. Such shape optimization problems, while very general and describing many problems of interest, remain very challenging to solve numerically and there is a lack of suitable simulation tools that are both readily accessible and general purpose. 

 

In this talk, we address this gap with Morpho, an open-source programmable environment for shape optimization and shape-shifting. I will showcase how Morpho can be used to solve a host of soft matter problems that can be cast in this general framework. In particular, I will highlight our recent study of the mechanics of a swelling hydrogel sphere confined in a matrix of soil particles, finding non-trivial internal strain distributions as well as diversions away from Hertzian contact mechanics. Finally, I will talk about how to extend Morpho to resolve non-equilibrium shape dynamics of active matter. 

 

*This material is based upon work supported by the National Science Foundation under grant OAC-2003820

Abstract: Built on the theory of optimal transport, Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions. In this talk, a general data-driven framework for conditional density estimation and simulation will be presented, and a number of examples will be shown to demonstrate its applicability. This talk is based on joint works with Esteban G. Tabak (NYU) and Giulio Trigila (CUNY).