''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,
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''A unified framework for solving non-autonomous ODEs using operator splitting''. In progress.
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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) are organized
around the Purdue model hierarchy of sensors/actuators, Pro-
grammable Logic Controllers (PLCs), and supervisory layers.
Despite this, existing data-driven intrusion detectors treat ICS
telemetry as either a multivariate sequence or a pairwise graph,
collapsing the hierarchy and missing higher-order, cross-level
dependencies that govern control propagation and sensing
dynamics. Motivated by this critical modeling gap, we in-
troduce TMS-ICS, a hierarchy-aware, process-level industrial
intrusion detection framework that preserves this structure
in both modeling and inference. See,
''Topology Meets Security: ICS Intrusion Detection via Topological Deep Learning''. Submitted.
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.
''Generalizing Graph Embedding Algorithms to Topological Spaces: Behind the Scenes of TopoEmbedX''. In progress.
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Traffic Flow Prediction via Topological Neural Networks
In this work, we present a novel framework referred to as TNNs
for traffic flow prediction by modeling the transportation network as a combinatorial complex.
This representation enables the integration of both pairwise and higher-order relationships among road segments. The proposed framework combines temporal convolution to capture traffic dynamics over time with hierarchical message passing across different topological levels to incorporate multi-scale spatial dependencies
''TNNs: Topological Neural Networks for Traffic Flow Prediction''. submitted.
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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''
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