Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks

 

 

 

Artificial neurons

The field of artificial neural networks has a long history of development, and is closely connected with the advancement of computer science and computers in general. A model of artificial neurons was first developed by McCulloch and Pitts in 1943 to study signal processing in the brain and has later been refined by others. The general idea is to mimic neural networks in the human brain, which is composed of billions of neurons that communicate with each other by sending electrical signals. Each neuron accumulates its incoming signals, which must exceed an activation threshold to yield an output. If the threshold is not overcome, the neuron remains inactive, i.e. has zero output.

This behaviour has inspired a simple mathematical model for an artificial neuron. $$ \begin{equation} y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u) \tag{1} \end{equation} $$ Here, the output \( y \) of the neuron is the value of its activation function, which have as input a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.

Conceptually, it is helpful to divide neural networks into four categories:

  1. general purpose neural networks for supervised learning,
  2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),
  3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and
  4. neural networks for unsupervised learning such as Deep Boltzmann Machines.
In natural science, DNNs and CNNs have already found numerous applications. In statistical physics, they have been applied to detect phase transitions in 2D Ising and Potts models, lattice gauge theories, and different phases of polymers, or solving the Navier-Stokes equation in weather forecasting. Deep learning has also found interesting applications in quantum physics. Various quantum phase transitions can be detected and studied using DNNs and CNNs, topological phases, and even non-equilibrium many-body localization. Representing quantum states as DNNs quantum state tomography are among some of the impressive achievements to reveal the potential of DNNs to facilitate the study of quantum systems.

In quantum information theory, it has been shown that one can perform gate decompositions with the help of neural.

The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine.