Related papers: Riddled basin geometry sets fundamental limits to …

Using a system of two FitzHugh-Nagumo units, we demonstrate the occurrence of riddled basins of attraction in delay-coupled systems as the coupling between the units is increased. We characterize the riddled basin using the uncertainty…

Neural network models have recently demonstrated impressive prediction performance in complex systems where chaos and unpredictability appear. In spite of the research efforts carried out on predicting future trajectories or improving their…

The basin of attraction is the set of initial points that will eventually converge to some attracting set. Its knowledge is important in understanding the dynamical behavior of a given dynamical system of interest. In this work, we address…

This research addresses the challenge of characterizing the complexity and unpredictability of basins within various dynamical systems. The main focus is on demonstrating the efficiency of convolutional neural networks (CNNs) in this field.…

Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great…

Modern neural networks exhibit a striking property: basins of attraction in the loss landscape are often connected by low-loss paths, yet optimization dynamics generally remain confined to a single convex basin and rarely explore…

In this paper, a two parameters family $F_{\beta_1,\beta_2}$ of maps of the plane living two different subspaces invariant is studied. We observe that, our model exhibits two chaotic attractors $A_i$, $i=0,1$, lying in these invariant…

The underspecification of most machine learning pipelines means that we cannot rely solely on validation performance to assess the robustness of deep learning systems to naturally occurring distribution shifts. Instead, making sure that a…

In the realm of fractal geometry, intricate structures emerge from simple iterative processes that partition parameter spaces into regions of stability and instability. Likewise, training large language models involves iteratively applying…

We consider dynamical and geometrical aspects of deep learning. For many standard choices of layer maps we display semi-invariant metrics which quantify differences between data or decision functions. This allows us, when considering random…

In Radhakrishnan et al. [2020], the authors empirically show that autoencoders trained with usual SGD methods shape out basins of attraction around their training data. We consider network functions of width not exceeding the input…

Randomized Neural Networks explore the behavior of neural systems where the majority of connections are fixed, either in a stochastic or a deterministic fashion. Typical examples of such systems consist of multi-layered neural network…

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This problem is rather common especially in population dynamics models. Precisely, a particular solution of a…

The multi-modal nature of neural loss landscapes is often considered to be the main driver behind the empirical success of deep ensembles. In this work, we probe this belief by constructing various "connected" ensembles which are restricted…

Advancements in artificial intelligence call for a deeper understanding of the fundamental mechanisms underlying deep learning. In this work, we propose a theoretical framework to analyze learning dynamics through the lens of dynamical…

Deep learning algorithms are responsible for a technological revolution in a variety of tasks including image recognition or Go playing. Yet, why they work is not understood. Ultimately, they manage to classify data lying in high dimension…

A recurrent neural network is considered that can retrieve a collection of patterns, as well as slightly perturbed versions of this `pure' set of patterns via fixed points of its dynamics. By replacing the set of dynamical constraints,…

We investigate the dependence of the approximation capacity of deep residual networks on its depth in a continuous dynamical systems setting. This can be formulated as the general problem of quantifying the minimal time-horizon required to…

Deep learning, in the form of artificial neural networks, has achieved remarkable practical success in recent years, for a variety of difficult machine learning applications. However, a theoretical explanation for this remains a major open…

The fitness landscape metaphor plays a central role on the modeling of optimizing principles in many research fields, ranging from evolutionary biology, where it was first introduced, to management research. Here we consider the ensemble of…