AMCS Seminar Series

A distinguishing feature of vehicular traffic flow is that it may exhibit significant wave patterns. We first demonstrate that those frustrating (when stuck in traffic) traffic features possess an intriguing structural beauty (when seen from the outside), rendering phantom traffic jams to be mathematical analogs of detonation waves. We then show how a few well-controlled automated vehicles can mitigate traffic instabilities and waves, first in theory and simulation, then in real-world traffic experiments. These culminate in the CIRCLES (Congestion Impacts Reduction via CAV-in-the-loop Lagrangian Energy Smoothing) project: the largest field test of deployed control vehicles on a fully instrumented highway, carried out by a consortium of mathematicians, engineers, industry partners, and government agencies.

The study of fluid-structure interaction spans many disciplines, from fluid dynamics and biological locomotion to mechanical engineering, geophysical phenomena, and applied mathematics. In this talk, I will demonstrate a few examples in which initially symmetric systems bifurcate into asymmetric and unstable states due to close coupling between structures and fluids. One example shows how a symmetric wing, when flapped in a fluid, can break symmetry to generate sustained unidirectional flight. Another reveals how a model continent in a laboratory experiment, interacting with a convective fluid mantle, can enter into oscillatory or trapped states.

Further, I will discuss a recent experiment inspired by the suspected super-rotation (differential rotation) of Earth’s inner core, showing how a symmetric structure can begin to spontaneously spin within a symmetric cavity filled with turbulent thermal convection. These phenomena were first discovered experimentally, and their results have been rigorously investigated through follow-up experiments and numerical simulations.

The study of reaction-diffusion equations on metric graphs has been drawing attention recently. Here, we focus on pattern dynamics on compact metric graphs. There are eight different types of compact metric graphs that are formed by two or three finite intervals. We consider systems of reaction-diffusion equations on these compact metric graphs with natural Kirchhoff boundary conditions.

Suppose further that the system exhibits Turing or wave instability. Then, we may observe pattern formation even on compact metric graphs. However, the bifurcation structures depend on the domain. It turns out that a solution on the simple circle or interval can be naturally related to that on the double-leaf metric graph. We are going to compare the variations of other solutions and examine how the bifurcation structures differ by considering the double-leaf metric graph.

In this talk, we present randomized algorithms to construct approximations for a set of solutions of parameter-dependent partial differential equations (PDEs), where the parameter set is high dimensional. It is well-known that the Proper Orthogonal Decomposition (POD)/Principal Component Analysis (PCA) and the greedy algorithm perform (quasi-)optimally in approximating such a set. However, both the POD and the greedy algorithm rely on a training set of finite cardinality that is chosen such that every point in the admissible parameter set is close to a point in the training set. Therefore, both algorithms suffer from the curse of dimensionality. We suggest breaking the curse of dimensionality by exploiting the concentration of measure phenomenon, which is also sometimes called the “blessing of dimensionality”.

In detail, we will present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than only for the parameters in the training set. Moreover, we will present a randomized POD and a corresponding error analysis that shows that for exponentially decaying eigenvalues of the randomized POD which uses the exact correlation operator (integral in the expectation) the approximation error between any solution corresponding to a parameter in the admissible parameter set and the approximation with the POD that uses a Monte-Carlo approximation converges exponentially as well.

Key challenges in studying conformational changes in biomolecules and atomic clusters via molecular dynamics simulations are high dimensionality, complexity of the system, and a broad range of timescales. The time step is determined by the smallest timescale, about 1fs, while the waiting times to observe transitions of interest may be of the order of microseconds or milliseconds. These challenges are often addressed by introducing collective variables (CVs), i.e., functions of atomic coordinates whose dynamics capture the processes happening on slow timescales. 

We investigate the problem of learning CVs with two goals: (1) to minimize the error in transition rates in the reduced model, and (2) to preserve the symmetries present in the potential energy function of the system. We propose a computational algorithm to learn collective variables based on the theory of effective dynamics (Legoll and Lelievre, 2010; Duong et al., 2018). The algorithm involves a symmetry-preserving feature map, learning the residence manifold via diffusion maps, and learning CVs via autoencoders. We present three case studies: normal butane C4H10, Lennard-Jones-7 (LJ7) in 2D, and Lennard-Jones-8 (LJ8) in 3D.

Joint work with Shashank Sule, Jiaxin (Margot) Yuan, Arnav Mehta, and Yeuk Yin Lam

In conventional simulation of PDEs, the finite element exterior calculus provides rich variational and algebraic structures which can be used to ensure discrete models preserve geometric properties of their continuous counterparts. Through this process, physical properties related to conservation, energy stability, thermodynamic compatibility can be guaranteed. In scientific machine learning, the black-box nature of contemporary transformer architectures runs contrary to this – with no exploitable structure, it is challenging to design architectures that are numerically stable and produce physical predictions. In this talk, we introduce a data-driven finite element exterior calculus which allows us to build data-driven finite element models which preserve physical structure; we achieve this by encapsulating self-attention transformers within mixed finite element spaces. In contrast to the majority of machine learning models, we can exploit this structure to prove well-posedness, numerical stability, and other properties. We demonstrate examples using these for several applications: developing real time digital twins of fusion power systems, distributed robotic systems, and to achieve state-of-the-art on conventional computing tasks.

As this era’s biggest game-changer, autonomous vehicles (AV) are expected to exhibit new driving and travel behaviors, thanks to their sensing, communication, and computational capabilities. However, the majority of studies assume AVs are essentially human drivers but react faster, “see” farther, and “know” the road environment better. We believe AVs’ most disruptive characteristic lies in its intelligent goal-seeking and adapting behavior. Building on this understanding, we propose a dynamic game based control leveraging the notion of mean-field games (MFG). I will first introduce how MFG can be applied to the decision-making process of a large number of AVs. Then I will talk about how the MFG-based control is generalized to road networks, in which the optimal controls of both velocity and route choice need to be solved for AVs, by resorting to nonlinear complementarity problems. Last but not the least, the latest advance in learning MFGs will be discussed, which further motivates why a game-theoretic framework is needed when individual data becomes available.

In many applications, both spatial transport and stochasticity in chemical reaction processes play critical roles in system dynamics. Mesoscopic particle-based stochastic reaction-diffusion (PBSRD) models have been successfully used to study a variety of such reaction processes, particularly for problems arising in cell biology (such as T cell signaling). In this talk I’ll present our recent work investigating how PBSRD models represent an intermediate physical scale, bridging more microscopic Langevin Dynamics models to more macroscopic reaction-diffusion PDE models. I will first illustrate how the rigorous large population, mean-field limit of PBSRD models gives rise to non-local reaction-diffusion partial integro-differential equations, from which standard reaction-diffusion PDE models arise as a short-range limit for reactive interactions. 

As commonly used, PBSRD models typically assume overdamped transport, ignoring inertial forces. I will discuss how to construct more microscopic particle-based reactive Langevin Dynamics (PBRLD) models that include inertial forces, formulating models that are consistent with detailed balance of reaction fluxes at equilibrium. We show via asymptotic analysis that with appropriate scaling assumptions for the dependence of reaction kernels on friction/mass, PBRLD models converge to common PBSRD models in the overdamped limit. Finally, we identify and prove the large population mean-field limit of the new PBRLD models, obtaining systems of nonlocal kinetic reactive-transport equations.

Shifted divergences provide a principled way of making information theoretic divergences (e.g. KL) geometrically aware via optimal transport smoothing. In this talk, I will argue that shifted divergences provide a powerful approach towards unifying central problems in optimization, sampling, privacy, functional inequalities, and beyond. For concreteness, I will describe these connections by mentioning several recent highlights, focusing on the first.

Based on joint works with Kunal Talwar, with Sinho Chewi, and with Jinho Bok and Weijie Su.

The remarkable diversity of macroscopic material behaviors—such as plasticity, phase transformations, viscoelasticity, and diffusion—fundamentally arises from underlying atomistic or particle-level dynamics. However, establishing a direct and predictive link between these scales, especially for non-equilibrium phenomena, remains a formidable challenge from both theoretical and computational standpoints. This gap in understanding currently hinders predictive simulations and material discovery, resulting in significant economic losses and innovation bottlenecks across various industries. In this talk, we present recent advances in the predictive modeling of non-equilibrium mechanics by integrating tools from continuum mechanics, statistical physics, applied mathematics, and machine learning. We demonstrate how this interdisciplinary approach opens new pathways for material modeling under non-equilibrium conditions.

Over the past decade, significant progress has been made by the scientific community in developing low-rank methods for solving time-dependent problems. In particular, the reduced storage of low-rank solutions allows us to address the curse of dimensionality often associated with solving high-dimensional problems, especially in kinetic simulations. Naturally, scientists also desire structure preservation and conservation incorporated into the low-rank framework.

In this talk, I will overview my recent work developing low-rank integrators that are also implicit, high-order accurate in time, and structure-preserving. A particular emphasis will be placed on convection-diffusion equations, with the Vlasov-Fokker-Planck equation acting as our motivating example.

Information to follow