A Geometry-Aware Data-Driven Framework for Predictive Modeling and PDE Operators Restricted; Files Only

Abstract

High-dimensional data poses challenges for modern data science, including high computational costs and diminished interpretability under the “curse of dimensionality.” However, such data often lie on a lower-dimensional manifold embedded in an ambient space, which suggests an opportunity to exploit the geometry of these manifolds for more efficient and accurate methods. Motivated by manifold learning, this thesis tackles two central problems: (1) improving predictive modeling for datasets with complicated manifold structure, and (2) developing an effective solver for partial differential equations (PDEs) defined on unknown manifolds.

We propose a data-driven learning framework that incorporates a Diffusion Maps (DM) kernel into Kernel Ridge Regression (KRR) to enhance predictive performance. By capturing local geometry through a diffusion process on the observed point cloud, the DM kernel better reflects the true intrinsic distances compared to traditional Euclidean-based kernels. Building on this manifold-aware approach, we then tackle the challenge of solving elliptic PDEs on manifolds with unknown geometry. To accommodate varying coefficients in PDEs, we propose a hybrid architecture that combines kernel learning with a lightweight neural network, proposed as MF-Net. This integrated design enhances solver robustness and enables faster inference. Numerical results reveal the effectiveness of both proposed method. This work highlights how geometry-aware kernel methods and modern data-driven techniques can be combined to advance predictive modeling and PDE solving in high-dimensional, manifold-structured settings, paving the way for more efficient and robust solutions in real-world applications.

Table of Contents

1 Introduction 1

2 Background 6

2.1 Kernel Methods and Kernel Ridge Regression 6

2.1.1 Mercer Kernel 7

2.1.2 Mercer’s Theorem 9

2.1.3 Ridge Regression 10

2.1.4 Kernel Ridge Regression 11

2.2 Manifold Learning and Diffusion Maps 13

2.2.1 Manifold 13

2.2.2 Manifold Learning 14

2.2.3 Approaches of Manifold Learning 16

2.2.4 Diffusion Maps Algorithm 19

2.3 Neural Network-Based Approaches for Solving PDEs 24

2.4 Solving PDEs on Manifolds 27

3 Data-Driven Learning via Diffusion Maps 30

3.1 The limitations of Euclidean-based Kernels 30

3.2 Kernel Ridge Regression with DM Kernels 32

3.3 Numerical Experiment: Swiss Roll 33

3.3.1 Experiment Setup 34

3.3.2 Data Generation 34

3.3.3 2D Swiss Roll 35

3.3.4 Experiment Results 38

4 A Hybrid Data-Driven Framework for PDE Operators 41

4.1 Problem Formulation 41

4.2 A Hybrid Architecture 42

4.2.1 Architecture Overview 42

4.2.2 Kernel Learning 43

4.2.3 MF-Net Architecture 45

4.3 Numerical Experiment: PDEs on Torus 47

4.3.1 Experimental Design 47

4.3.2 Results 48

5 Future Works 50

6 Conclusion 52

Bibliography 54

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Computer Science Applied Mathematics Mathematics
Keyword	Partial Differential Equation Manifold Diffusion Map Machine Learning Operator Learning Kernel Methods
Committee Chair / Thesis Advisor	Xi, Yuanzhe, Emory University
Committee Members	Sanchez-Becerra, Alejandro, Emory University Nagy, James, Emory University

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