Double Descent and Color Intermittent Diffusion: A Global Optimization Framework Across Disciplines Restricted; Files Only

Abstract

Global optimization problems arise across different disciplines, yet many classical optimization methods are inherently local and may become trapped in suboptimal basins of attraction. This thesis studies Double Descent and Color Intermittent Diffusion (DD-CID), a hybrid global optimization algorithm designed to explore complex objective landscapes by alternating between local descent and basin-escaping diffusion. The goal of this work is to evaluate the flexibility and effectiveness of DD-CID on challenging problems drawn from multiple disciplines.

The first test problem is the Morse potential from physical chemistry, a classical benchmark whose landscape becomes increasingly difficult as the interaction parameter grows. To improve performance, this thesis incorporates a log-barrier regularization and a continuation strategy, enabling DD-CID to locate minima that match best-recorded database values. The second test problem comes from epidemiology, where parameter estimation in SIR-type models is formulated as a data assimilation problem. To make this setting compatible with DD-CID, adjoint-based gradient and Hessian calculations are derived, allowing efficient optimization from limited observational data. Numerical results show that the method recovers model parameters with high accuracy. The third application addresses a major difficulty in physics-informed neural networks (PINNs): highly non-convex loss landscapes with multiple competing critical points. Applied to a two-mode ansatz wave equation and to viscous Burgers’ equation, DD-CID successfully identifies critical points across the landscape and produces accurate numerical solutions.

Overall, these results show that DD-CID is a robust and adaptable framework for global optimization. Beyond demonstrating the method’s cross-disciplinary applicability, this thesis shows that problem-specific enhancements (such as log-barrier methods, continuation, and adjoint formulations) can substantially expand the practical reach of DD-CID. Future work includes improving computational efficiency and extending the method to additional large-scale optimization problems.

Table of Contents

1. Introduction .................................................................. 1

2. Global Optimization ........................................................... 4

  2.1. Statement of the Optimization Problem ................................. 4

  2.2. Generic GO Algorithm .................................................. 5

  2.3. Overview of Selected Global Heuristics ............................... 6

    2.3.1. Pure Random Search ............................................ 6

    2.3.2. Basin Hopping (BH), an ILS method ............................ 7

    2.3.3. Simulated Annealing (SA) ..................................... 7

  2.4. Overview of Local Search ............................................ 8

    2.4.1. Descent Direction ............................................ 9

    2.4.2. Step Size: Line Search ..................................... 10

3. Double Descent with Color Intermittent Diffusion (DD-CID) .............. 11

  3.1. Optimality Conditions and Classification of Optimizers ............. 12

  3.2. DD-CID Algorithm ................................................... 14

    3.2.1. Double Descent .............................................. 15

    3.2.2. Color Intermittent Diffusion ................................ 16

4. Challenge in Physical Chemistry: Morse Potential ....................... 18

  4.1. Morse Potential .................................................... 18

  4.2. Observations on Candidate Solutions ................................ 20

  4.3. Problem-Specific Improvements ...................................... 22

    4.3.1. The Log-Barrier Regularization .............................. 22

    4.3.2. The Continuation Method ..................................... 24

  4.4. Numerical Results .................................................. 25

    4.4.1. ρ = 3 ....................................................... 25

    4.4.2. ρ = 6 ....................................................... 26

    4.4.3. ρ = 10 ...................................................... 27

5. Challenge in Epidemiology: Disease Modeling ............................ 28

  5.1. Introduction to SIR Models ......................................... 29

  5.2. Simple SIR ......................................................... 30

    5.2.1. The Data Assimilation Framework ............................. 30

    5.2.2. The Variational Adjoint Framework ........................... 31

    5.2.3. Computing Hessian ........................................... 34

    5.2.4. Numerical ODE Solver: Crank-Nicolson ........................ 35

    5.2.5. Numerical Results ........................................... 35

  5.3. Endemic SIR ........................................................ 37

    5.3.1. Gradient and Hessian ........................................ 37

    5.3.2. Numerical Results ........................................... 38

6. Challenge in Scientific Computing: PINN Bottleneck ..................... 41

  6.1. Introduction to PINN ............................................... 42

  6.2. Numerical Results .................................................. 44

    6.2.1. Simple Example: Wave Equation ............................... 44

    6.2.2. Viscous Burgers' Equation ................................... 46

7. Conclusion and Discussion .............................................. 49

  7.1. Conclusion ......................................................... 49

  7.2. Limitation and Future Work ......................................... 50

Bibliography .............................................................. 52

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics Mathematics
Keyword	morse potential global optimization SIR model physics-informed neural networks
Committee Chair / Thesis Advisor	Manetta, Manuela, Emory University
Committee Members	Teague, Lori, Emory University Veneziani, Alessandro, Emory University

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