Comparing Manifold-Constrained Gaussian Processes and Physics-Inspired Neural Networks for Parameter Inference in Epidemiological Models Restricted; Files Only

Abstract

Accurate parameter estimation in compartmental epidemiological models remains a fundamental challenge due to observational noise, partial observability, and the nonlinear nature of disease dynamics. This thesis evaluates two approaches for parameter inference: the manifold-constrained Gaussian process (MAGI) framework and physics-informed neural networks (PINNs). Using simulated data from SIR and SIRW models, we compare the two methods across a range of noise settings.

Empirically, MAGI consistently achieves lower trajectory error and superior robustness under noisy observations, owing to its principled incorporation of Gaussian noise and uncertainty quantification through a Bayesian framework. However, MAGI’s performance degrades for parameters with low sensitivity (e.g., $\xi$ in the SIRW model), and its high computational cost limits scalability. Conversely, PINNs exhibit stability in parameter estimation across increasing noise levels due to implicit ODE regularization, but tend to overfit noisy data in trajectory reconstruction, particularly at high noise levels.

Additionally, we explore a hybrid method using a PINN-informed mean function in MAGI. Despite theoretical appeal, this approach underperforms due to amplified derivative noise introduced by the neural network predictions, which corrupts the posterior via erroneous constraint enforcement.

Overall, our findings indicate that MAGI is better suited for accurate trajectory inference in low- to moderate-noise settings, while PINNs may be more practical when computational resources are constrained or when stable parameter estimation is prioritized. Both methods struggle with identifiability in parameters weakly informed by the data, underscoring the need for improved model design or data collection strategies in such cases.

Table of Contents

1. Introduction 

2. Background 

  2.1. Overview of Compartment Models 

  2.2. Epidemic SIR Model 

  2.3. Endemic SIR Model 

  2.4. SIRW Model: Waning Immunity and Boosting 

3. Background: Gaussian Processes 

  3.1. Introduction 

  3.2. Derivation of Standard GP Regression with Noisy Data 

  3.3. MAGI 

  3.4. MAGI Model Setup 

  3.5. Mean Function and Kernel Choice 

  3.6. Hyperparameter Optimization 

  3.7. Hamiltonian Monte Carlo (HMC) 

4. PINN Background 

  4.1. Neural Networks 

  4.2. Theoretical Background 

  4.3. Physics-Informed Neural Networks (PINNs) 

  4.4. Challenges in PINNs 

5. System Identification of Endemic SIR Model 

  5.1. First-Order Local Sensitivity Analysis 

  5.2. MAGI and PINN Experiments 

  5.3. Discussion 

6. System Identification of the SIRW Model 

  6.1. Sensitivity Analysis 

  6.2. Experiments 

  6.3. Results 

7. Combining the GP and PINN Models 

  7.1. Results 

8. Conclusion 

  8.1. Conclusion 

  8.2. Future Directions 

Appendix 

  A.1. Neural Network Training Details 

  A.2. HMC Sampling Details 

Bibliography

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Statistics Applied Mathematics
Mot-clé	parameter inference sir models machine learning physics-inspired neural networks gaussian processes
Committee Chair / Thesis Advisor	Yuanzhe Xi, Emory University Alessandro Veneziani, Emory University
Committee Members	David Hirshberg, Emory University

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