Student Research

John Hood '22

The Heavy-Tailed Nature of Noise in Stochastic Gradient Descent

Supported by the Scott and Anne Perper Internship Fund

Advisor: Thomas Pietraho

distributions graph

Stochastic Gradient Descent (SGD) is a surprisingly effective variant of a standard optimization method which underlies modern machine learning techniques. We investigate its stochastic nature, reaffirm that it is heavy-tailed as opposed to normally-distributed, and that this is true irrespective of the underlying data.  We also show that the heaviness of the tails for SGD deployed on a neural network increases with the number of parameters in the model.



Madeleine Genereux ’19 and Sean McParland '19

Approximation of Discrete DSWs through PDE approximations 

Supported by the NSF through grant DMS-1615037

Advisor: Christopher Chong

dispersive shockwaves graph

Dispersive shock waves (DSWs), which connect states of different amplitude via an expanding wave train, are known to form in nonlinear dispersive media when subject to sharp changes in the initial state. Such extreme changes in a medium can occur when simulating explosions in shock tubes, when gasses or fluids are compressed in a piston chamber, or when water dams break. Sean and Madeline investigated DSWs in a system consisting of masses coupled via nonlinear springs. Using a multiple-scale analysis, they were able to derive PDEs, and from there to approximate the formation and structure of the DSWs. They also conducted numerical simulations to investigate the dynamics.

Kevin Chen '19 

Parametric Instability in Granular Chains with Time-Dependent Stiffness

Supported by the NSF through grant DMS-1615037

Advisor: Christopher Chong

A granular chain is a one dimensional network of particles that interact elastically.  Kevin studied a granular chain where the stiffness of the particles would vary in time. In particular, the stiffness would vary periodically (according to a step-function). He explored how instabilities can be introduced into the system via variable stiffness through the computation of the dispersion relationship and numerical simulations.

Waverly Harden ’19student working with drill

Dynamics of Hollow Elliptical Cylinder Arrays

Stipend and travel support by the NSF through grant DMS-1615037; additional travel support provided by the Grua/O’Conell Fund

Advisor: Christopher Chong

Strain-softening materials have gained significant interest in the physics, mathematics and material science communities recently. Most materials that we are familiar with are strain-hardening, meaning as they deform, they become harder to deform. Strain-softening materials have the opposite property; as they deform, they become easier to deform. The dynamics of strain-softening materials in an experimental setting have been rarely studied. For seven weeks, Waverly worked on the modeling and simulation of a system that exhibits strain-softening behavior. It consists of a chain of hollow elliptical cylinders (HEC). She visited the laboratory of her advisors collaborator, Professor Jinkyu Yang, at the University of Washington for the final three weeks of her project to work with the team who developed the concept of the HEC and the experimental set-up. Using the experimental data, she was able to refine her mathematical model. She also worked on an experimental apparatus that controls the impact frequency of the chain, and worked on processing of the experimental data that was acquired through digital image correlation.

Patrick Blackstone ’17

Lp-Harmonic Analysis on SO(3,1)

Supported by Kibbe Institutional Research Fellowship

Advisor: William Barker

The long term goal of this project was to determine the image of the Lp-Schwartz space (0 < p  2) under the operator-valued Fourier transform on the semi-simple Lie group SO(3,1), the proper Lorentz group. This would extend research conducted in the 1980s for the Lie group SL(2,R). Patrick’s work on SO(3,1) mirrored that of Andrew Pryhuber and Justin Dury-Agri for SU(2,1), though SO(3,1) presents a slightly more tractable group since it is only six dimensional as opposed to SU(2,1) which is eight dimensional. Of more significance, SO(3,1) has no “discrete series” of representations, a major simplification over SU(2,1). Patrick concentrated on the more algebraic foundations of the project, working out the necessary algebraic structure theory of SO(3,1). As a sophomore Patrick did not have the background to explore the more analytic aspects of the representation theory of SO(3,1), a requirement for tackling the ultimate Lp-Schwartz space problem. 

Neural networks imageParikshit Sharma '17

Neural networks and transfer learning models in image subject classification

Advisor: Thomas Pietraho

In transfer learning, machine learning models trained to perform a specific task are retrained and repurposed to perform complementary tasks.  Often, lower order features learned by a neural network in one context can be reassembled into higher-order features that are useful in another.  In this project, we set out to classify the subject area of a book based solely on the image of its cover. Using transfer learning from  a number of pretrained image classification neural nets such a Google's Inception, the classification accuracy of our new models exceeded 70 percent.  This is near the accuracy of a human asked to perform the same task.

Sophie Bèrubè '16 and Tara Palnitkar '16

Diestel-Leader Groups are Graph Automatic

Supported by the NSF through grant DMS-1105407

Advisor: Jennifer Taback

Automatic groups were introduced in the 1990's in order to classify the groups associated with three dimensional "shapes," or manifolds.  A group is a set with an operation which satisfies certain basic properties, such as the integers under addition, and in general captures the symmetry of some, perhaps complicated, object.  In an automatic group, simple computational machines called finite state automata are used to describe the group elements as well as the group multiplication.  These automata streamlined computation within the group, leading to expedited solutions of geometric and computational questions in group theory.  Unfortunately, not all three manifold groups fell into this category, and so the definition was extended to that of graph automatic groups.  The latter retained many of the computational advantages of automatic groups while enlarging the class considerably.  

Sophie and Tara worked to show that a new infinite family of groups, called Diestel-Leader groups, or higher rank lampligher groups, are graph automatic.  This required gaining a deep geometric and algebraic understanding of these groups, as well as learning about different types of finite state automata.  Along the way, they proved that some related infinitely generated groups were also graph automatic.  This work will appear as an article in the Journal of Algebra.

Jacob Hart headshotJacob Hart ’17 and Carina Spiro ’18Student in lab  

Nonlinear Energy Harvesting

Stipends supported by the Maine Space Grant Consortium, travel supported by the NSF through grant DMS-1615037

Advisor: Christopher Chong

The last several decades as seen a burst of research activity in vibration energy harvesting, which deals with the conversion of mechanical energy (such as vibration) into electrical energy. Most commercial energy harvesters are based on linear responses, which have a relatively restricted range of vibration frequencies where energy harvesting is efficient. Jake and Carina explored the integration of novel aspects of nonlinear wave equations to try to harvest energy in new ways. In particular,they studied an array of cantilevered beams that are coupled via magnetic links. For the first six weeks of the project, they worked on the modeling of the system, its numerical simulation, and the computation of periodic orbits. They visited the laboratory of their advisors collaborator, Professor Chiara Daraio, at the ETH Zurich for their final two weeks to work with the team developing the experimental set-up of the array of cantilevers. Using the experimental data, they were able to refine their mathematical model.