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The use of artificial neural networks as models of chaotic dynamics has been rapidly expanding. Still, a theoretical understanding of how neural networks learn chaos is lacking. Here, we employ a geometric perspective to show that neural…

Extrapolation -- the ability to make inferences that go beyond the scope of one's experiences -- is a hallmark of human intelligence. By contrast, the generalization exhibited by contemporary neural network algorithms is largely limited to…

While the basic laws of Newtonian mechanics are well understood, explaining a physical scenario still requires manually modeling the problem with suitable equations and associated parameters. In order to adopt such models for artificial…

A promising approach to improve climate-model simulations is to replace traditional subgrid parameterizations based on simplified physical models by machine learning algorithms that are data-driven. However, neural networks (NNs) often lead…

The process of training an artificial neural network involves iteratively adapting its parameters so as to minimize the error of the network's prediction, when confronted with a learning task. This iterative change can be naturally…

Artificial Neural Networks (ANNs) implement a specific form of multi-variate extrapolation and will generate an output for any input pattern, even when there is no similar training pattern. Extrapolations are not necessarily to be trusted,…

The links between optimal control of dynamical systems and neural networks have proved beneficial both from a theoretical and from a practical point of view. Several researchers have exploited these links to investigate the stability of…

Dynamical systems see widespread use in natural sciences like physics, biology, chemistry, as well as engineering disciplines such as circuit analysis, computational fluid dynamics, and control. For simple systems, the differential…

A procedure to characterize chaotic dynamical systems with concepts of complex networks is pursued, in which a dynamical system is mapped onto a network. The nodes represent the regions of space visited by the system, while edges represent…

Linear Regression and neural networks are widely used to model data. Neural networks distinguish themselves from linear regression with their use of activation functions that enable modeling nonlinear functions. The standard argument for…

We study the generalization of over-parameterized deep networks (for image classification) in relation to the convex hull of their training sets. Despite their great success, generalization of deep networks is considered a mystery. These…

With an increasing amount of observations on the dynamics of many complex systems, it is required to reveal the underlying mechanisms behind these complex dynamics, which is fundamentally important in many scientific fields such as climate,…

Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…

Many real-world complex systems, such as epidemic spreading networks and ecosystems, can be modeled as networked dynamical systems that produce multivariate time series. Learning the intrinsic dynamics from observational data is pivotal for…

Given a set of snapshots from a temporal network we develop, analyze, and experimentally validate a so-called network interpolation scheme. Our method allows us to build a plausible, albeit random, sequence of graphs that transition between…

Deciphering the underpinnings of the dynamical processes leading to information transmission, processing, and storing in the brain is a crucial challenge in neuroscience. An inspiring but speculative theoretical idea is that such dynamics…

Complex networks theory has commonly been used for modelling and understanding the interactions taking place between the elements composing complex systems. More recently, the use of generative models has gained momentum, as they allow…

Dynamical systems play a key role in modeling, forecasting, and decision-making across a wide range of scientific domains. However, variations in system parameters, also referred to as parametric variability, can lead to drastically…

Neural network realizes multi-parameter optimization and control by simulating certain mechanisms of the human brain. It can be used in many fields such as signal processing, intelligent driving, optimal combination, vehicle abnormality…

Statistical modeling of data sets by neural-network techniques is offered as an alternative to traditional semiempirical approaches to global modeling of nuclear properties. New results are presented to support the position that such novel…