Abstract: Deep Learning and Artificial Intelligence have attracted enormous attention recently. The race to design and manufacture âbrain-likeâ computers is on and several companies have produced various such chips. Yet, the current state of affairs is very unsatisfactory and ad hoc. We describe a mathematical framework we have developed that provides a hierarchical architecture for learning and cognition. The architecture combines a wavelet preprocessor, a group invariant feature extractor and a hierarchical (layered) learning algorithm. There are two feedback loops, one back from the learning output to the feature extractor and one all the way back to the wavelet preprocessor. We show that the scheme can incorporate all typical metric differences but also non-metric dissimilarity measures like Bregman divergences. The learning module incorporates two universal learning algorithms in their hierarchical tree-structured form, both due to Kohonen, Learning Vector Quantization (LVQ) for supervised learning and Self-Organizing Map (SOM) for unsupervised learning. We demonstrate the superior performance of the resulting algorithms and architecture on a variety of practical problems including: speaker and sound identification, simultaneous determination of sound direction of arrival speaker and vowel ID, face recognition. We demonstrate how the underlying mathematics can be used to provide systematic models for design, analysis and evaluation of deep neural networks. We describe current work and plans on mixed signal (digital and analog) micro-electronic implementations that mimic architectural abstractions of the cortex of higher-level animals and humans, for sound and vision perception and cognition. The resulting architecture is non-von Neumann (i.e. computing and memory are not separated in the hardware) and neuromorphic. We call the resulting chip class âCortex-on-a-Chip.â
Abstract: A common problem of atomistic materials modelling is to determine properties of
crystalline defects, such as structure, energetics, mobility, from which
meso-scopic material properties or coarse-grained models can be derived (e.g.,
Kinetic Monte-Carlo, Discrete Dislocation Dynamics, Griffith-type fracture
laws). In this talk I will focus on one the most basic tasks, computing the
equilibrium configuration of a crystalline defect, but will also also comment on
free energy and transition rate computations.
A wide range of numerical strategies, including the classical supercell method
(periodic boundary conditions) or flexibe boundary conditions (discrete BEM),
but also more recent developments such as atomistic/continuum and QM/MM hybrid
schemes, can be interpreted as Galerkin discretisations with variational crimes,
for an infinite-dimensional nonlinear variational problem. This point of view is
effective in studying the structure of exact solutions, identify approximation
parameters, derive rigorous error bounds, optimise and construct novel schemes
with superior error/cost ratio.
Time permitting I will also discuss how this framework can be used to analyse
model errors in interatomic potentials and how this can feed back into the
developing of new interatomic potentials by machine learning techniques.
Abstract: Inferring the laws of interaction of particles and agents in complex dynamical systems from observational data is a fundamental challenge in a wide variety of disciplines. We start from data consisting of trajectories of interacting agents, which is in many cases abundant, and propose a non-parametric statistical learning approach to extract the governing laws of interaction. We demonstrate the effectiveness of our learning approach both by providing theoretical guarantees, and by testing the approach on a variety of prototypical systems in various disciplines, with homogeneous and heterogeneous agents systems, ranging from fundamental physical interactions between particles to systems-level interactions, with such as social influence on people's opinion, prey-predator dynamics, flocking and swarming, and cell dynamics.
Jen Rezeppa has 10+ years of experience in Silicon Valley working for and leading teams at top companies including Apple, GoPro during the IPO, and presently at Tesla. She earned a BS in Mathematics and over the course of her career, her UMCP degree has led to opportunities in teaching, reliability engineering, and most recently, demand planning and channel planning. Letâs learn about Jenâs career progression and her thoughts on leadership for students and new graduates as they explore how a math degree can lay the foundation for their professional goals.
Abstract: The talented students of Math299m - "Visualization Through Mathematica" will be presenting their final projects where they use Mathematica to model and investigate a diverse array of topics, including fractional derivatives, gravitation, seismology, machine learning, musical harmonies, financial models and more.
Abstract: Control of multi-agent systems has application to many different domains
(including traffic, biology and other) and can be addressed at many different scale. At microscopic scale sparsity is a desired property for applicability, while passing to mean-field limits pose mathematical challenges, as controls may become singular.
We first show some recent results at microscopic scale and in rigorously passing to the limit. Then we introduce a new concept of differential equation for measures which appear to be a promising framework to deal with this problems and, finally, show application to traffic.
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