Related papers: Generalized saddle-node ghosts and their composite…

Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analyzed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations…

Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. We here derive an expression for the number of attractors in…

Characterizing and understanding the dynamics of stochastic gradient descent (SGD) around saddle points remains an open problem. We first show that saddle points in neural networks can be divided into two types, among which the Type-II…

The study of complex many-body systems via analysis of the trajectories of the units that dynamically move and interact within them is a non-trivial task. The workflow for extracting meaningful information from the raw trajectory data is…

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…

Probabilistic approaches for handling count-valued time sequences have attracted amounts of research attentions because their ability to infer explainable latent structures and to estimate uncertainties, and thus are especially suitable for…

Close to a saddle-node bifurcation, when two invariant solutions collide and disappear, the behavior of a dynamical system can closely resemble that of a solution which is no longer present at the chosen parameter value. For bifurcating…

We analyze the input-output behavior of residual networks from a dynamical system point of view by disentangling the residual dynamics from the output activities before the classification stage. For a network with simple skip connections…

This paper addresses the problem of finding cycles in the state transition graphs of synchronous Boolean networks. Synchronous Boolean networks are a class of deterministic finite state machines which are used for the modeling of gene…

The understanding of Boolean automata networks dynamics takes an important place in various domains of computer science such as computability, complexity and discrete dynamical systems. In this paper, we make a step further in this…

The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose…

We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit,…

Strange nonchaotic attractors (SNAs) in noise driven systems are investigated. Before the transition to chaos, due to the effect of noise, a typical trajectory will wander between the periodic attractor and its nearby chaotic saddle in an…

Learning dynamical systems through operator-theoretic representations provides a powerful framework for analyzing complex dynamics, as spectral quantities such as eigenvalues and invariant structures encode characteristic time scales and…

Stochastic gradient descent (SGD) is central to deep learning, yet the dynamical origin of its preference for flatter, more generalizable solutions remains unclear. Here, by analyzing SGD learning dynamics, we identify a nonequilibrium…

In this work, we reveal a strong implicit bias of stochastic gradient descent (SGD) that drives overly expressive networks to much simpler subnetworks, thereby dramatically reducing the number of independent parameters, and improving…

The characterization of the "most connected" nodes in static or slowly evolving complex networks has helped in understanding and predicting the behavior of social, biological, and technological networked systems, including their robustness…

Motion prediction, recently popularized as world models, refers to the anticipation of future agent states or scene evolution, which is rooted in human cognition, bridging perception and decision-making. It enables intelligent systems, such…

The superior performance of deep convolutional networks over high-dimensional problems have made them very popular for several applications. Despite their wide adoption, their underlying mechanisms still remain unclear with their…

Turbulent flows present rich dynamics originating from non-trivial energy fluxes across scales, non-stationary forcings and geometrical constraints. This complexity manifests in non-hyperbolic chaos, randomness, state-dependent persistence…