Research Symposium

BIO

Mekhi is a class of 2029 Presidential Scholar at Florida State University working on dual degrees in Applied Mathematics and Music, with a minor in interdisciplinary sciences. He has worked on two mathematical research projects; the first, completed alongside Dr. Mark Sussman (FSU) and Dr. Alireza Moradi (Blue Origin), examines how Neural Networks can solve differential equations over long time periods. The second, completed in collaboration with fellow undergraduates Edmarly Ariol and Abriana Thorpe, used linear algebraic tools to analyze sound waves, gaining insight on musical content and virtual audio production tools. He intends to continue undergraduate research in the fields of partial differential equations and mathematical modeling, with the goal of completing a Ph.D. in Applied Mathematics at Johns Hopkins University in a similar concentration. With this Ph.D. he will work as a mathematician for NASA (or a similar organization). Regardless, the versatile nature of mathematics is, to Mekhi, an exciting opportunity to work in various applied fields.
His goals in music are no less ambitious. He will complete a Bachelor of Arts in Music, using the degree’s freeform nature to work towards several goals, namely: proficiency on the saxophone, clarinet and piano, core musical theory, worldwide musical culture and composition skills. With these abilities Mekhi intends to work as composer for video games, using music to tell stories of fictional worlds and characters. He will complete this work as a freelancer, alongside small game developers, until larger projects become available through experience and industry exposure.

A Comparative Analysis: Solving Differential Equations with LSTM and Attention-Based Neural Networks

Abstract

Neural Networks (NNs) are multilayer, computerized algorithms that mimic human cognition to complete tasks. We are interested in LSTM and Attention-Based models (two of many types) designed for time series data, which we will compare to assess which is most effective at finding numerical solutions to ordinary differential equations. These solutions are relevant to solving for cryogenic fuel tank sloshing dynamics, for which analytical solutions falter over long time periods; NNs learn from early solutions to predict long-term behavior. Solving differential equations is traditionally done analytically or through numerical methods. Generally, analytical solutions do not exist for all differential equations and numerical methods can suffer from numerical instability, among other issues. It has been demonstrated that solving ODEs through NNs (including specifically LSTM and Attention-Based) is entirely possible. Traditional means of solving differential equations are, for some problems, too slow. NNs will speed this process. Our research was conducted through literature review, analyzing code and altering parameters to test solution behavior. Our findings are in progress. Upon completion, we hope to gain insight on the utilization of neural networks as tools. These findings are relevant to solve differential equations of increasing complexity in an efficient manner. Through NNs, these solutions can be found feasibly and give valuable insight into physical and natural processes.

Keywords: Neural Networks, Differential Equations, Mathematics