Neural Networks and Numerical Methods for Bifurcation and Stability Analysis

Abstract

This dissertation presents the development and application of neural networks and numerical methods for the bifurcation and stability analysis of nonlinear partial differential equations (PDEs) and nonlinear lattices. The study begins with an investigation into shallow physics-informed neural networks (PINNs), optimized using the Levenberg–Marquardt algorithm, for solving forward and inverse problems involving nonlinear PDEs. Analytical expressions for network derivatives are derived, offering insights into the internal structure of PINNs and demonstrating that shallow architectures can achieve high accuracy when paired with an effective optimization method. Building on this foundation, the approach is extended to the construction of bifurcation diagrams and linear stability analysis through eigenvalue computations, with applications to canonical problems such as the Bratu and Burgers equations. The method is further applied to nonlinear lattices to capture intricate snaking bifurcation diagrams and to analyze stability. Additionally, a stochastic technique integrated with the Levenberg–Marquardt algorithm is proposed for solving high-dimensional nonlinear lattices. To complement the neural network approach, a symmetric finite difference method (SFDM) is introduced to exploit solution symmetries for efficiently solving the high-dimensional Bratu equation. By embedding symmetry properties and utilizing sparse matrix representations, SFDM delivers significant improvements in both accuracy and computational efficiency over standard methods. Collectively, these contributions demonstrate the versatility and effectiveness of neural networks and advanced numerical techniques for solving complex nonlinear problems, providing a scalable foundation for future research in bifurcation theory and stability analysis.

Date of Award	2025
Original language	American English
Supervisor	Hadi Susanto (Supervisor)