Applications of topological data analysis in parameter estimation for partial differential equations

Baltushkin, Mikhail (2023) Applications of topological data analysis in parameter estimation for partial differential equations. Masters thesis, Northern Arizona University.

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Abstract

This thesis focuses on the use of topological data analysis (TDA) to study the parameters of an anisotropic Kuramoto-Sivashinsky type equation used to model the pyramidal patterns formed by the bombardment of germanium. The aim is to better understand the influence of the parameters and geometry of the pattern, particularly the kappa parameter, and classify their values based on the time evolution of the model. The traditional methods of parameter estimation rely on the iterative approximation of solutions, making it computationally intensive and producing large errors in the estimation process. Instead, this research proposes a machine learning algorithm that uses the relationship between the geometry of approximated solutions and parameter values to predict the parameter values. TDA's algebraic gadgets, homology and homotopy groups, are used to summarize the complicated topological structure of the approximated solution to the differential equation. In order to perform the computations of homology groups, we define the cubical complexes that are naturally induced by the approximated solution surfaces. In this study, we conducted a wide grid search over variables in classification models and parameters in representations of the patterns. The best results were obtained using the flipped surface and 0-dimensional homology groups, with the Random Forest Classifier performing the best with approximately 80\% accuracy. The model accuracy without TDA applied did not exceed 11% accuracy. We also created a confusion matrix for the best-performing random forest model, observing some misclassification, particularly within the mid-range of parameters. Interestingly, we found that combining 0- and 1-dimensional homology groups did not result in higher accuracy than using 0-homology group on its own. We tested various weighting functions when computing the persistence images and found that the linear and sigmoid functions did not significantly affect the accuracy. These findings are discussed in detail in the following sections, along with a description of the model parameters and the confusion matrix.

Item Type:	Thesis (Masters)
Publisher’s Statement:	© Copyright is held by the author. Digital access to this material is made possible by the Cline Library, Northern Arizona University. Further transmission, reproduction or presentation of protected items is prohibited except with permission of the author.
Keywords:	Partial differential equations; Topological data analysis; Kuramoto-Sivashinsky; Germanium; Kappa parameter; Homotogpy groups
Subjects:	Q Science > QA Mathematics
NAU Depositing Author Academic Status:	Student
Department/Unit:	Graduate College > Theses and Dissertations College of the Environment, Forestry, and Natural Sciences > Mathematics and Statistics
Date Deposited:	03 Aug 2023 16:23
Last Modified:	03 Aug 2023 16:23
URI:	https://openknowledge.nau.edu/id/eprint/6083

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