Calendar
9/2/2022
Speaker: Christopher Jones (UNC-CH, GMU)
Title: Dynamics is our best shot!
Abstract:
Two of the aims in using mathematics in real world applications are: (1) understanding the mechanisms responsible for different effects and phenomena, and (2) predicting the future state of the system under study. Dynamical systems provides a perspective and a lens for addressing these two questions. The system under study is formulated as an evolving set of state variables and the set of trajectories with different initializations are viewed geometrically.
I will use this lens to look at a pressing problem in climate science: how a climate subsystem might abruptly " tip " from its current state into a completely different state. This is a problem that requires dynamical systems to understand, and I will show how we can decode different ways in which the tipping might happen.
Dynamical systems models tend to be simplified; extraneous forces are ignored to produce models which attempt to capture the key mechanisms. The inclusion of data from observations is a way to connect these models with reality and I will discuss the area of data assimilation that achieves a balance between data and physical models in a systematic way.
9/9/2022
Speaker: Evelyn Sander (GMU)
Title: Stable floating configurations for 3D printed objects
Abstract:
This talk concentrates on the study of stability of floating objects through mathematical modeling and experimentation. The models are based on standard ideas of center of gravity, center of buoyancy, and Archimedes' Principle. There will be a discussion of a variety of floating shapes with two-dimensional cross sections for which it is possible to analytically and/or computationally a potential energy landscape in order to identify stable and unstable floating orientations. I then will compare the analysis and computations to experiments on floating objects designed and created through 3D printing. The talk includes a demonstration of code we have developed for testing the floating configurations for new shapes. I will give a brief overview of the methods involved in 3D printing the objects.
This research is joint work with Dr. Dan Anderson at GMU and undergraduate students Brandon G. Barreto-Rosa, Joshua D. Calvano, and Lujain Nsair, all of whom who were part of an undergraduate research program run by the MEGL at GMU.
9/16/2022
Speaker: Molei Tao (GT)
Title: When dynamics meet machine learning
Abstract:
The interaction of machine learning and dynamics can lead to both new methodology for dynamics, and deepened understanding and/or efficacious algorithms for machine learning. This talk will give examples in both directions.
Specifically, I will first discuss data-driven learning and prediction of mechanical dynamics, for which I will demonstrate one strong benefit of having physics hard-wired into deep learning models;
more precisely, how to make symplectic predictions, and how that
provably improves the accuracy of long-time predictions.
Then I will discuss how dynamics can be used to better understand the
implicit biases of large learning rates in the training of machine
learning models. They could lead to quantitative escapes from local
minima via chaos, which is an alternative mechanism to commonly known
noisy escapes due to stochastic gradients. I will also report how large
learning rates bias toward flatter minimizers, which arguably generalize
better.
9/23/2022
Speaker: Shane Kepley (VU)
Title: Efficient parameterization of invariant manifolds using deep neural networks
Abstract: Spectral methods are the gold standard for parameterizing manifolds of solutions for ODEs because of their high precision and amenability to computer assisted proofs. However, these methods suffer from several drawbacks. In particular, the parameterizations are costly to compute and time-stepping is far more complicated than other methods. In this talk we demonstrate how computing these parameterizations and accurately time-stepping can be reduced to a related manifold learning problem. The latter problem is solved by training a deep neural network to interpolate charts for a low dimensional manifold embedded in a high dimensional Euclidean space. This training is highly parallelizable and need only be performed once. Once the neural network is trained, it is capable of parameterizing invariant manifolds for the ODE and time-stepping with remarkable efficiency and precision.
9/30/2022
Speaker: Andrey Shilnikov (GSU)
Title: Overview of GPU-based tools for studying multiscale and complex dynamics
Abstract: I will review several parallel GPU-based approaches to better understand multistable dynamics of simple neural networks and global bifurcation unfolding of systems with deterministic chaos.
10/7/2022
Speaker: Marian Gidea (YU)
Title: The three-body problem and low energy space missions
Abstract:
The three-body problem, on the dynamics of three masses under mutual gravity, serves as a model for the motion of a spacecraft relative to the Earth-Moon or Sun-Earth system. We describe the equations of motion for the three-body problem and the geometric objects that organize the dynamics: equilibriums points, periodic and quasi-periodic orbits, and their stable and unstable manifolds. As it turns out, trajectories that follow these manifolds require zero energy cost.
We describe several methods to design low energy spacecraft trajectories from Earth to Moon, as well as maneuvers to change the inclination of the orbit of a satellite relative to the ecliptic. This is based on joint works with E. Belbruno, F. Topputo, A. Delshams, and P. Roldan.
10/14/2022
Speaker: Krishna Pusuluri (GSU)
Title: Parallel computations to study complex dynamics in neuroscience and other chaotic nonlinear systems
Abstract:
We will begin with a brief overview of several parallel and hybrid computing approaches including CUDA, OpenAcc, OpenMP, and OpenMPI, followed by a demonstration of how we can leverage these technologies to study complex dynamics arising from diverse nonlinear systems. First, we discuss multistable rhythms in oscillatory 4-cell central pattern generators (CPGs) of inhibitory coupled neurons. We show how network topology and intrinsic properties of the cells affect dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. We then discuss symbolic methods and parametric sweeps to analyze isolated neuron dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits and animal CPGs. We also demonstrate how such symbolic methods can help identify the universal principles governing both simple and complex dynamics, and chaotic structure in various Lorenz-like systems, their key self-similar organizing structures in 2D parameter space, as well as detailed computational reconstructions of 3D bifurcation surfaces.
10/21/2022
Speaker: Ivo Pasmans (University of Reading, UK)
Title: Computational challenges in operational data assimilation: problems and solutions
Abstract:
Operational weather and ocean forecasting proceeds as a sequence of time intervals. During each interval numerical models produce a forecast, observations are collected and a comparison between the two is made. This comparison is used, in a process called data assimilation (DA), to construct observation-informed initial conditions for the forecast in the next time interval. Many DA algorithms are in use, but they all share the need to solve a high-dimensional (>1010) system of linear equations. Constructing and solving this system in the limited amount of time available between the reception of the observations and the start of the next time interval is highly non-trivial for three reasons. 1) As the numerical models are computationally demanding, it is generally impossible to construct the full linear system. 2) Its high dimensionality makes it impossible to store the system as a matrix in memory. Consequently, it is not possible to directly invert it. 3) The operational time-constraints strongly limit the number of iterations that can be used by iterative linear solvers. By adapting DA algorithms to use parallelization, it is possible to leverage the computational power of superclusters to construct a high-rank approximation to the linear system and solve it using less then ~20 iterations. In this talk, I will first introduce the two most popular families of DA algorithms: Kalman filters and variational DA. After this, I will discuss some of the adaptations that have been developed to enable parallelization. Among these are ensemble Kalman filters, domain localization, the EVIL (Ensemble Variational Integrated Localized) and saddle point algorithms.
10/28/2022
Speaker: Maciej Capinski (AGH University of Science and Technology)
Title: Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem
Abstract:
In this talk we will discuss a shooting method designed for solving two point boundary value problems in a setting where a system has integrals of motion. We will show how it can be applied to obtain certain families of orbits in the circular restricted three body problem. These include transverse ejection/collisions from one primary body to the other, families of periodic orbits, orbits passing through collision, and orbits connecting fixed points to ejections or collisions.
This is joint work with Shane Kepley and Jason Mireles James.
11/04/2022
Speaker: Gemma Huguet (UPC)
Title: Oscillatory Dynamics in Mathematical Models of Neural Networks
Abstract:
Oscillations are ubiquitous in the brain, but their role is not completely understood. In this talk we will focus on the study of oscillations in
neuronal networks. I will introduce some neuronal models and I will show how
tools from dynamical systems theory, such as the parameterization method for
invariant manifolds or the separatrix map, can be used to provide a thorough
analysis of the oscillatory dynamics. I will show how the conclusions obtained
may contribute to unveiling the role of oscillations in certain cognitive tasks.
11/11/2022
Speaker: Christian Sampson (JCSDA)
Title: Predicting The Weather, 4d-Var, Hybrid Tangent Linear Models, and JEDI
Abstract:
Weather modeling in conjunction with Data Assimilation (DA) has proven to provide effective weather forecasts that can both help you plan your day to save your life. We often refer to the combination of weather models and DA as Numerical Weather Prediction (NWP). One of the most widely employed DA methods in NWP is a variational method called 4d-Var. In this method, a cost function involving the model background error and a series of observations over time is minimized to find the best initial condition from which to run your model so that model forecast is consistent with observations. 4d-Var has been shown to provide the most reliable weather forecasts to date, but is not without its pitfalls. In particular, 4d-Var depends heavily on a tangent linear model (TLM) and an adjoint to the tangent linear model. While conceptually simple, coding these two elements is extremely time intensive and difficult. A small change in the larger weather model can induce months of work on its TLM and adjoint delaying the benefits of improvements on the model side. In this talk I will introduce the 4d-var method in general and present work on a Hybrid Tangent Linear Model (HTLM) developed in [Payne 2021] which is aimed at improving TLMs as well as allowing the use of incomplete TLMs when model physics changes. I will also touch on the Joint Effort for Data Integration (JEDI) project which now includes an HTLM and how you can use JEDI for DA.
References:
Payne, T. J. (2021). A Hybrid Differential-Ensemble Linear Forecast Model for 4D-Var, Monthly Weather Review, 149(1), 3-19. Retrieved Oct 27, 2022,
11/18/2022
Speaker: Hannah Choi (GT)
Title: Structure and computation of data-driven brain networks
Abstract:
The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. One of many unique features of the mammalian brain network is its spatial embedding and hierarchical organization. I will discuss effects of these structural characteristics on network dynamics as well as their computational implications with a focus on the flexibility between modular and global computations and predictive coding.
Recording:
12/02/2022
Speaker: Blake Barker (BYU)
Title: Solving ODE eigenvalue problems with rigorous computation
Abstract:
ODE eigenvalue problems often arise in the study of stability of traveling waves, in showing the second variation of a functional is positive definite, and in many other applications. For many eigenvalue problems, it is not possible to obtain an explicit eigen pair. Thus, one uses numerical methods to approximate the solution. By rigorously bounding all errors in the computation, including computer rounding errors via use of an interval arithmetic package, one may obtain a computer assisted proof that the true solution lies in a small neighborhood of an approximation. This allows one to prove stability of traveling waves, for example. In this talk, we discuss recent work regarding computer assisted proof of stability of waves, and discuss other areas of application, such as in identifying most probable paths of escape in stochastic systems.
12/04-12/05
Inperson workshop-seminar at GT