Related papers: Descriptive complexity for neural networks via Boo…

We utilize a connection between compositional kernels and branching processes via Mehler's formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation…

Neural networks have proved an effective means of learning control policies for autonomous systems, but these learned policies are difficult to understand due to the black-box nature of neural networks. This lack of interpretability makes…

The irreducible complexity of natural phenomena has led Graph Neural Networks to be employed as a standard model to perform representation learning tasks on graph-structured data. While their capacity to capture local and global patterns is…

Neural networks are increasingly employed to model, analyze and control non-linear dynamical systems ranging from physics to biology. Owing to their universal approximation capabilities, they regularly outperform state-of-the-art…

A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely…

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…

Boolean circuits abstract away from physical details to focus on the logical structure and computational behaviour of digital components. Although such circuits have been studied for many decades, compositionality has been widely ignored or…

An artificial neural network is presented based on the idea of connections between units that are only active for a specific range of input values and zero outside that range (and so are not evaluated outside the active range). The…

Convolutional Neural Networks (CNNs) have been widely applied. But as the CNNs grow, the number of arithmetic operations and memory footprint also increase. Furthermore, typical non-linear activation functions do not allow associativity of…

Continuous-time systems with switch-like behaviour occur in chemical kinetics, gene regulatory networks and neural networks. Networks with hard switching, as a limiting case of smooth sigmoidal switching, retain the richest possible range…

We study sign structures of the ground states of spin-$1/2$ magnetic systems using the methods of Boolean Fourier analysis. Previously it was shown that the sign structures of frustrated systems are of complex nature: specifically, neural…

Boolean networks may be viewed as idealizations of biological genetic networks, where each node is represented by an on-off switch which is a function of the binary output from some other nodes. We evolve connectivity in a single Boolean…

Artificial neural networks (ANN), typically referred to as neural networks, are a class of Machine Learning algorithms and have achieved widespread success, having been inspired by the biological structure of the human brain. Neural…

Despite substantial efforts, neural network interpretability remains an elusive goal, with previous research failing to provide succinct explanations of most single neurons' impact on the network output. This limitation is due to the…

In the past decade, deep learning became the prevalent methodology for predictive modeling thanks to the remarkable accuracy of deep neural networks in tasks such as computer vision and natural language processing. Meanwhile, the structure…

A long standing open problem in the theory of neural networks is the development of quantitative methods to estimate and compare the capabilities of different architectures. Here we define the capacity of an architecture by the binary…

In pioneering work from 2019, Barcel\'o and coauthors identified logics that precisely match the expressive power of constant iteration-depth graph neural networks (GNNs) relative to properties definable in first-order logic. In this…

A Boolean network (BN) is a transformation of the set of Boolean configurations of a given length. A trapspace of a BN is a subcube invariant by the BN; a principal trapspace is the smallest trapspace containing a given configuration; a…

Nonlinear network dynamics are notoriously difficult to understand. Here we study a class of recurrent neural networks called combinatorial threshold-linear networks (CTLNs) whose dynamics are determined by the structure of a directed…

Polynomial functions have plenty of useful analytical properties, but they are rarely used as learning models because their function class is considered to be restricted. This work shows that when trained properly polynomial functions can…