Approximation Koopman Semigroup: The main purpose of this project is to study approximation methods for solving nonlinear systems using Bernhard Koopman’s Global Linearization Method. This approach enables the application of linear semigroup methods to a nonlinear system by focusing on the dynamics of the observables of the states rather than directly studying the dynamics of the states. In particular, we will investigate the extent to which eigenvalues and eigenfunctions of the linear Koopman operator, which generates the observations of the underlying nonlinear system, can serve as useful tools for approximating and/or studying the qualitative properties of the underlying nonlinear flow.
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The main contributions of this research project are the development of new techniques for solving nonlinear dynamical systems.
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Article: Splitting Operator Method of Lie-Koopman Operator: Analysis, learning. In preparation.
The main contributions of this research project are the development of new techniques for Topological deep
learning. The main products of this project will be source code, publications, reports, and curriculum
material.
1. Article: TopoX: A Suite of Python Packages for Machine Learning on Topological Domains. Accepted
to Journal of Machine Learning Research (JMLR).
2. Source code: Developer of three transformative Python packages on Topological Deep Learning, focusing on fast robust deep learning computations for graph generalizations such as hypergraphs, simplicial
complexes, and cellular complexes. Check out the packages on GitHub: https://github.com/pyt-team.
3. Article: What is a Topological Neural Network?. In preparation.
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Higher Order Neural-Networks, Topological Deep Learning (TDL): This project deals with the analysis and the study of complex systems that can be modeled as graphs, simplicial or cell complexes. A central methodological issue underlying my research is finding unifying principles that govern many areas in modern data science and integrate the multiple levels of organization which are commonly found in a range of systems. My research combines a mixture of tools from topological data analysis, geometric data processing, machine learning, signal processing, statistics and topology.
Broadly speaking, the research questions that I am interested are divided into two parts:
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Unifying principles on deep learning models and deep learning on higher order complexes is concerned with building deep learning models on generalized spaces such as cell complexes. This question also seeks building a precise mathematical and algorithmic theory that combines between deep learning on complexes and topological data analysis. Applications of deep learning protocols executed on such domains are massive, and they range from non-linear signal processing supported on topological spaces, computational biology and medicine, social science and art.
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Unifying principles on the general notion of data: unifying principle on the notation of data yields more cohesive algorithmic abstraction which ultimately help us to write better, easier to use, more inclusive machine learning packages.
Publications:
1- TopoX: A Suite of Python Packages for Machine Learning on Topological Domains. Accepted,
Journal of Machine Learning Research (JMLR).
2- Introduction to Topological Neural Network. In preparation.
3- Splitting Operator Method of Koopman Operator: Analysis, learning. In preparation.
4- Cohomology of GKM-sheaves. Preprint, https://arxiv.org/abs/1806.01761