Luka Todorovic

Luka Todorovic

Biography

Luka Todorovic is a rising senior at the University of Michigan studying mathematics
and statistics. During his time in Ann Arbor, he has harbored a love of building and working
with mathematical and statistical models meant to help us better understand the mechanisms
of the world around us through techniques of probability and dynamical systems. Luka plans
to pursue a PhD in either mathematics or statistics and then pursue a career in academia as a
professor or researcher. This summer, Luka is working with Dr. Sai Ravela on a few problems
related to the discovery and sizing of optimal neural network architecture to model chaotic and
physical dynamical systems, utilizing approaches of simpler tensor graph compilation as well
as information theoretic approaches to optimizing neural networks. Beyond the classroom,
Luka enjoys spending time in the kitchen, the gym, playing guitar, and chasing storms when
one rolls through.

Optimal Neural Structure for modeling Physical Differential Equations and
Dynamical Systems
Luka Todorovic1 and Sai Ravela2
1Department of Mathematics, University of Michigan
2Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology

Machine Learning has become a driving field of research in modeling physical systems and
processes. Notably, different machine learning models such as Physics-Informed Neural Networks
(PINNs) and Neural ODEs (NODEs) have been recognized as useful– yet limited– alternatives
to standard emulating Neural Networks for chaotic and unstable dynamical processes. Because
of this, recent focus has shifted towards mathematically showcasing optimal neural network
structures, as well as developing other neural architectures to model these systems. Our research
analyzes a new class of neural architecture called PolyNets, which treats dynamical systems as
a direct mapping to a finite-dimensional polynomial space through a tensor graph. Our analysis
compares the speed and accuracy of PolyNets to soliton solutions of the Korteweg-de Vries
equation against both traditional numerical methods on a CPU and GPU, as well as common neural
networks such as Emulator NNs, PINNs, and NODEs. Our findings demonstrate not only the speed
and accuracy of PolyNets compared to other methods, but also highlight their capabilities to model
out-of-distribution solutions due to their non-reliance on training for known systems. Future work
will strive towards modeling more complex systems and equations, such as the 2D Kuramoto-
Sivashinsky equation as well as shallow water equations.