''The purpose of mathematics is not to find the truth, but to find the best possible description of the truth.''-Bernard Koopman
My research focuses on the intersection of dynamical systems, operator theory, and machine learning. I explore data-driven methods for approximating nonlinear dynamics, particularly through Koopman operator theory and operator splitting techniques. This approach enables the use of linear analysis tools to study complex, nonlinear systems. I am also involved in the development of topological deep learning methods, which integrate concepts from algebraic topology with modern neural networks to enhance learning on structured and geometric data. These methods have practical applications in cyber-physical systems, industrial control, and social recommendation systems, where understanding and predicting dynamic behavior is critical.
Projects:
Splitting Operator: The primary objective of this project is to study approximation methods for solving nonlinear dynamical systems using Bernhard Koopman's global linearization approach. This framework enables the application of linear semigroup theory to nonlinear systems by analyzing the evolution of observable functions of the state, rather than the state trajectories themselves.
In this work, we employ operator splitting methods to approximate the Koopman operator semigroup and reconstruct the flow of nonlinear dynamics. Theoretical developments are complemented by numerical experiments that illustrate the accuracy and efficiency of the proposed methods. See,
Higher Order Neural Networks: 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.
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. See,
Topological Deep Learning Framework for Anomaly Detection: Industrial Control Systems (ICS) security requires monitor-
ing complex interactions between sensors, actuators, and Programmable Logic Controllers (PLCs) across multiple hierarchical levels. However, existing anomaly detection methods, including CNNs, LSTMs, GNNs, and physics-aware approaches, often fail to capture critical cross-level and higher-order interactions between ICS components. These limitations result in diminished detection accuracy, increased false positive rates, and an inability to localize anomalies. In this work, we address these critical challenges by introducing TopoDetect, an ICS security framework specifically designed to detect and localize anomalies. Notably, our framework employs Combinatorial Complexes (CCs) to model higher-order control hierarchies among ICS components. See,
Topological Representation Learning (TRL):
The purpose of this project is to investigate methods for
topological representation learning in TopoEmbedX (TEX) and explore how it can be applied to
represent elements of a topological domain within a Euclidean space. In progress,
''Generalizing Graph Embedding Algorithms to Topological Spaces: Behind the Scenes of TopoEmbedX''
GKM-sheaves: In this project, I studied the foundations of sheaf theory, including sheaf cohomology and its applications in modern geometry and topology. I also explored equivariant cohomology, which generalizes classical cohomological techniques to spaces with group actions, providing powerful tools for understanding symmetries in mathematical structures. This work deepened my understanding of both the algebraic and topological frameworks underpinning advanced geometric analysis. See, ''Cohomology of GKM-sheaves''