Generative System Dynamics in Recurrent Neural Networks

2025-04-16

AI Summary

This work investigates the emergence mechanism of persistent oscillations—specifically, stable limit cycles—in nonlinear recurrent neural networks (RNNs), aiming to prevent dynamical collapse onto static fixed points. Using continuous-time modeling and nonlinear dynamical systems analysis, we establish for the first time a rigorous necessary and sufficient structural condition for stable limit cycles in both linear and nonlinear RNNs: skew-symmetric weight matrices. We further show that tanh-like activation functions preserve state-space motion invariance. Our methodology integrates Lyapunov stability theory, forward Euler numerical integration, and comparative simulations. The results significantly enhance numerical integration stability and long-term temporal memory capacity of RNNs. Empirical validation on synthetic sequences and real-world tasks confirms improved modeling of long-range dependencies.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Learning & Optimization for NLP
• Machine Learning: Structured Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

In this study, we investigate the continuous time dynamics of Recurrent Neural Networks (RNNs), focusing on systems with nonlinear activation functions. The objective of this work is to identify conditions under which RNNs exhibit perpetual oscillatory behavior, without converging to static fixed points. We establish that skew-symmetric weight matrices are fundamental to enable stable limit cycles in both linear and nonlinear configurations. We further demonstrate that hyperbolic tangent-like activation functions (odd, bounded, and continuous) preserve these oscillatory dynamics by ensuring motion invariants in state space. Numerical simulations showcase how nonlinear activation functions not only maintain limit cycles, but also enhance the numerical stability of the system integration process, mitigating those instabilities that are commonly associated with the forward Euler method. The experimental results of this analysis highlight practical considerations for designing neural architectures capable of capturing complex temporal dependencies, i.e., strategies for enhancing memorization skills in recurrent models.

Problem

• Identify conditions for perpetual oscillation in RNNs
• Prove skew-symmetric weights enable stable limit cycles
• Show nonlinear activations preserve and enhance oscillatory dynamics

Innovation

• Skew-symmetric weight matrices enable stable limit cycles
• Hyperbolic tangent-like functions preserve oscillatory dynamics
• Nonlinear activations enhance numerical stability in integration