Interests and future directions

My research interests are often rooted in Dynamical systems theory and Information theory. The versatility of their applications has led me to successful projects in machine learning, quantum information, fluid dynamics, and number theory.

Dynamical systems theory

I enjoy leveraging the rich history of the field for the benefit of other scientific disciplines.

In fluid mechanics, my major contribution has been the construction of a theoretical model that explains the phenomenology of the laminar-turbulent regime in pipe-flow. We achieved this by demonstrating the existence of a homoclinic tangle in the regime. This necessitates gateways that guarantee the experimentally observed escape from turbulence.

My major interests lie in revealing connections between the dynamical systems and machine learning worlds. I am the founder of the Koopman toolkit research program: a quest for implementing novel, data-driven, Koopman Operator Theory inspired methods to design, benchmark, and optimize efficient Neural Network Differential Equation solvers. These KOT based methods generalize easily - they are relevant to a wide class of machine learning implementations and computing systems that are reliant on iterative optimization algorithms. By envisioning the training process as a discrete dynamical flow of parameters in training iteration time, we identify and construct Koopman operators/modes to drive this flow - side-stepping the standard optimization methods and gaining significant computational savings.

The goals of my academic work at Harvard are in the reverse direction - the creation of machine learning techniques to aid the study of dynamical systems. I achieve this by constructing strategies that leverage known information about the chosen dynamical system to optimize the Neural Networks studying it. The secondary focus of my work at Harvard is generalizing these methods, so that they are useful beyond the original scope of dynamical systems.

Information Theory

My interests in Information theory incorporate computational (quantum computing) and theoretical (connections to set/number theory) aspects of the field.

I contributed significantly to the discovery of the background radiation-induced quasiparticle generation phenomenon that limits superconducting qubit performance. As a core member of a multi-year, multi-institutional project, I handled the creation and study of computational simulations that provided a rough understanding of the limits and timescales within which the effect could be significant and discoverable. I also assisted in designing and studying the shielding experiment that ultimately isolated and discovered the effect.

The remainder of my time is devoted to establishing connections between statements on the complexity of natural numbers and a class of open problems in number theory. Most notably, I have shown that the former are intimately connected with the Sophie Germain prime conjecture and conjectures on the maximum allowed gap between the successive elements of kN*kN multiplication tables, where k is a positive integer. A secondary aim is to obtain useful generalizations of those connections between partially ordered sets and the complexity of their constituent elements (if the complexity is well defined and well behaved).

 

Selected Publications/Lectures:

A. S. Dogra and W. T. Redman, “Optimizing Neural Networks via Koopman Operator Theory,” Advances in Neural Information Processing Systems 33 (NeurIPS 2020).

Antti Vepsäläinen, et al., Impact of ionizing radiation on superconducting qubit coherence,“ Nature 584, p.p. 551-556, 2020.

N. B. Budanur, A. S. Dogra, and B. Hof, "Geometry of transient chaos in streamwise-localized pipe flow turbulence," Physical Review Fluids, vol. 4 (10), 2019.

A. S. Dogra and W. T. Redman, “Optimizing Neural Networks via Koopman Operator Theory,” Fields Institute for Research in Mathematical Sciences (Nov 16, 2020).

Journals I have reviewed/refereed for:

IEEE Transactions on Pattern Analysis and Machine Intelligence