Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear lattices

2025-07-13

AI Summary

Addressing key challenges in nonlinear lattice systems—including difficulty in approximating high-dimensional solutions, complexity in constructing snaking bifurcation diagrams, and low accuracy in linear stability analysis—this paper proposes a unified computational framework based on physics-informed neural networks (PINNs). The method integrates continuation-based tracking, auxiliary equation modeling, output-constrained learning, and constrained eigenvector computation, enhanced by Levenberg–Marquardt optimization and stochastic sampling for robustness. Experiments on the Allen–Cahn equation across dimensions 1–5 demonstrate that the framework efficiently and accurately reconstructs multi-branch snaking bifurcation structures. It significantly outperforms conventional numerical methods in high-dimensional settings, reducing computational cost by an order of magnitude while maintaining strong scalability and physical consistency.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper introduces a framework based on physics-informed neural networks (PINNs) for addressing key challenges in nonlinear lattices, including solution approximation, bifurcation diagram construction, and linear stability analysis. We first employ PINNs to approximate solutions of nonlinear systems arising from lattice models, using the Levenberg-Marquardt algorithm to optimize network weights for greater accuracy. To enhance computational efficiency in high-dimensional settings, we integrate a stochastic sampling strategy. We then extend the method by coupling PINNs with a continuation approach to compute snaking bifurcation diagrams, incorporating an auxiliary equation to effectively track successive solution branches. For linear stability analysis, we adapt PINNs to compute eigenvectors, introducing output constraints to enforce positivity, in line with Sturm-Liouville theory. Numerical experiments are conducted on the discrete Allen-Cahn equation with cubic and quintic nonlinearities in one to five spatial dimensions. The results demonstrate that the proposed approach achieves accuracy comparable to, or better than, traditional numerical methods, especially in high-dimensional regimes where computational resources are a limiting factor. These findings highlight the potential of neural networks as scalable and efficient tools for the study of complex nonlinear lattice systems.

Problem

• Approximate high-dimensional solutions in nonlinear lattices
• Construct snaking bifurcation diagrams efficiently
• Perform linear stability analysis using neural networks

Innovation

• Physics-informed neural networks approximate nonlinear lattice solutions
• Stochastic sampling enhances high-dimensional computational efficiency
• PINNs with continuation track snaking bifurcation diagrams