A Differential Manifold Perspective and Universality Analysis of Continuous Attractors in Artificial Neural Networks (2509.10514v1)

Abstract: Continuous attractors are critical for information processing in both biological and artificial neural systems, with implications for spatial navigation, memory, and deep learning optimization. However, existing research lacks a unified framework to analyze their properties across diverse dynamical systems, limiting cross-architectural generalizability. This study establishes a novel framework from the perspective of differential manifolds to investigate continuous attractors in artificial neural networks. It verifies compatibility with prior conclusions, elucidates links between continuous attractor phenomena and eigenvalues of the local Jacobian matrix, and demonstrates the universality of singular value stratification in common classification models and datasets. These findings suggest continuous attractors may be ubiquitous in general neural networks, highlighting the need for a general theory, with the proposed framework offering a promising foundation given the close mathematical connection between eigenvalues and singular values.

• The paper introduces a differential manifold framework to analyze continuous attractors by linking eigenvalue decomposition with the stability of neural networks.
• It employs singular value decomposition of local Jacobian matrices to uncover eigenvalue stratification that supports model generalization and attractor persistence.
• Empirical validations on classification models reinforce the manifold hypothesis by demonstrating that high-dimensional data reside on low-dimensional attractor manifolds.

A Differential Manifold Perspective and Universality Analysis of Continuous Attractors in Artificial Neural Networks

Introduction

The paper "A Differential Manifold Perspective and Universality Analysis of Continuous Attractors in Artificial Neural Networks" (arXiv ID: (2509.10514)) proposes an innovative framework for analyzing continuous attractors using differential manifolds. Continuous attractors play a pivotal role in various neural systems, being crucial for spatial navigation, memory, and optimization in deep learning. However, previous studies have lacked a comprehensive framework for understanding their properties across diverse dynamical systems, leading to limited generalizability.

Mathematical Framework

The authors develop a mathematical framework utilizing differential manifold theory to investigate continuous attractors in artificial neural networks. They establish connections between attractor phenomena and the eigenvalues of the local Jacobian matrix, demonstrating the universality of singular value stratification across common classification models and datasets. This framework integrates eigenvalue decomposition and singular value decomposition (SVD) of Jacobian matrices, allowing for detailed characterizations of equilibrium stability and attractor mechanisms.

The framework's strength lies in its application to both neural networks and complex systems, analyzing phase coexistence and state transitions by varying Jacobian structure parameters. This approach reinforces the manifold hypothesis, suggesting that high-dimensional data often reside on lower-dimensional manifolds.

Figure 1: The iteration trajectories of a discrete dynamical system $x(t+1)=\sin(Wx(t)+b)-Ax(t)$ with eigenvalue stratification characteristics at time steps $t=50(a), 100(b), 200(c),$ and $20000(d)$.

Theoretical Implications

The proposed framework provides a unified perspective on previous findings, offering a systematic approach to attractor analysis. It reveals that eigenvalue magnitudes and signs govern equilibrium stability, while eigenvector orientations delineate local dynamics, which collectively dictate attractor emergence. This framework also aids in understanding phase transitions and bifurcation pathways in piecewise-smooth systems, thereby bridging neural and non-neural dynamical systems research.

Furthermore, the work validates the manifold hypothesis by showing how attractors encode low-dimensional representations within high-dimensional spaces. This serves as a crucial step in aligning empirical studies with theoretical models, offering both a rigorous analytical foundation and practical insights.

Empirical Validation

The authors conducted experiments to validate their proposed theories, focusing on neural networks trained for classification tasks. The stratification of singular values observed through SVD hinted at the presence of approximate continuous attractors, supporting the manifold hypothesis. The experiments reveal that singular value stratification is prevalent in neural networks' mappings, providing empirical evidence that models retain generalization capability through structural singular value stratification.

Figure 2: Box plot of class-specific singular value decomposition (SVD) of classification results from a pre-trained ResNet-18 model (10 samples). Each dot corresponds to a singular value of a particular sample, and the $\times$ corresponds to the mean value of each group of singular values.

Implications and Future Directions

This framework opens new avenues for developing a comprehensive theory of attractors in general neural networks, offering insights into optimization landscapes, training stability, and generalization mechanisms. Such advancements could redefine AI paradigms, transitioning from simulating intelligent behavior to understanding the underlying principles of intelligence.

Future research might apply this methodology to deep neural networks where continuous attractors appear during training, potentially improving our grasp on neural network interpretability and performance. The extension of this mathematical framework to singular value stratification, beyond eigenvalue analysis, remains a promising area for further exploration.

Conclusion

The establishment of a differential manifold-based framework for continuous attractors in artificial neural networks represents a significant step towards a unified theory of attractor dynamics. The empirical and theoretical integration offered by this paper provides a robust foundation for future research, potentially fueling advances in neuroscience, machine learning, and beyond. By deepening our understanding of continuous attractors, this work contributes to bridging the gap between theoretical abstraction and practical application in the paper of neural information processing and artificial intelligence.