Artificial Neural Networks – ANNs

In Machine Learning - ML and Cognitive Science, Artificial Neural Networks (ANNs) are a family of models inspired by biological neural networks (the central nervous systems of animals, in particular the brain) which are used to estimate or approximate functions that can depend on a large number of inputs and are generally unknown. Artificial neural networks are typically specified using three things:

  • Architecture specifies what variables are involved in the network and their topological relationships—for example the variables involved in a neural network might be the weights of the connections between the neurons, along with activities of the neurons
  • Activity Rule Most neural network models have short time-scale dynamics: local rules define how the activities of the neurons change in response to each other. Typically the activity rule depends on the weights (the parameters) in the network.
  • Learning Rule The learning rule specifies the way in which the neural network's weights change with time. This learning is usually viewed as taking place on a longer time scale than the time scale of the dynamics under the activity rule. Usually the learning rule will depend on the activities of the neurons. It may also depend on the values of the target values supplied by a teacher and on the current value of the weights.

For example, a neural network for handwriting recognition is defined by a set of input neurons which may be activated by the pixels of an input image. After being weighted and transformed by a function (determined by the network's designer), the activations of these neurons are then passed on to other neurons. This process is repeated until finally, the output neuron that determines which character was read is activated.

Like other machine learning methods – systems that learn from data – neural networks have been used to solve a wide variety of tasks, like computer vision and speech recognition, that are hard to solve using ordinary rule-based programming.


Background

Examinations of humans' central nervous systems inspired the concept of artificial neural networks. In an artificial neural network, simple artificial nodes, known as "neurons", "neurodes", "processing elements" or "units", are connected together to form a network which mimics a biological neural network.

There is no single formal definition of what an artificial neural network is. However, a class of statistical models may commonly be called "neural" if it possesses the following characteristics:

  1. contains sets of adaptive weights, i.e. numerical parameters that are tuned by a learning algorithm, and
  2. is capable of approximating non-linear functions of their inputs.

The adaptive weights can be thought of as connection strengths between neurons, which are activated during training and prediction.

Artificial neural networks are similar to biological neural networks in the performing by its units of functions collectively and in parallel, rather than by a clear delineation of subtasks to which individual units are assigned. The term "neural network" usually refers to models employed in statistics, cognitive psychology and artificial intelligence. Neural network models which command the central nervous system and the rest of the brain are part of theoretical Neuroscience and Computational Neuroscience.

In modern software implementations of artificial neural networks, the approach inspired by biology has been largely abandoned for a more practical approach based on statistics and signal processing. In some of these systems, neural networks or parts of neural networks (like artificial neurons) form components in larger systems that combine both adaptive and non-adaptive elements. While the more general approach of such systems is more suitable for real-world problem solving, it has little to do with the traditional, artificial intelligence connectionist models. What they do have in common, however, is the principle of non-linear, distributed, parallel and local processing and adaptation. Historically, the use of neural network models marked a directional shift in the late eighties from high-level (symbolic) artificial intelligence, characterized by expert systems with knowledge embodied in if-then rules, to low-level (sub-symbolic) machine learning, characterized by knowledge embodied in the parameters of a dynamical system.


Models

Neural network models in artificial intelligence are usually referred to as artificial neural networks (ANNs); these are essentially simple mathematical models defining a function f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} \textstyle f:X\rightarrow Y or a distribution over X {\displaystyle \textstyle X} \textstyle X or both X {\displaystyle \textstyle X} \textstyle X and Y {\displaystyle \textstyle Y} \textstyle Y, but sometimes models are also intimately associated with a particular learning algorithm or learning rule. A common use of the phrase "ANN model" is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons or their connectivity).

Network Function

The word network in the term 'artificial neural network' refers to the inter–connections between the neurons in the different layers of each system. An example system has three layers. The first layer has input neurons which send data via synapses to the second layer of neurons, and then via more synapses to the third layer of output neurons. More complex systems will have more layers of neurons, some having increased layers of input neurons and output neurons. The synapses store parameters called "weights" that manipulate the data in the calculations.

An ANN is typically defined by three types of parameters:

  1. The interconnection pattern between the different layers of neurons
  2. The learning process for updating the weights of the interconnections
  3. The activation function that converts a neuron's weighted input to its output activation.

Mathematically, a neuron's network function f ( x ) {\displaystyle \textstyle f(x)} \textstyle f(x) is defined as a composition of other functions g i ( x ) {\displaystyle \textstyle g_{i}(x)} \textstyle g_{i}(x), which can further be defined as a composition of other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between variables. A widely used type of composition is the nonlinear weighted sum, where f ( x ) = K ( ∑ i w i g i ( x ) ) {\displaystyle \textstyle f(x)=K\left(\sum {i}w{i}g_{i}(x)\right)} \textstyle f(x)=K\left(\sum _{i}w_{i}g_{i}(x)\right), where K {\displaystyle \textstyle K} \textstyle K (commonly referred to as the activation function[31]) is some predefined function, such as the hyperbolic tangent. It will be convenient for the following to refer to a collection of functions g i {\displaystyle \textstyle g_{i}} \textstyle g_{i} as simply a vector g = ( g 1 , g 2 , … , g n ) {\displaystyle \textstyle g=(g_{1},g_{2},\ldots ,g_{n})} \textstyle g=(g_{1},g_{2},\ldots ,g_{n}).

 
ANN dependency graph

This figure depicts such a decomposition of f {\displaystyle \textstyle f} \textstyle f, with dependencies between variables indicated by arrows. These can be interpreted in two ways.

The first view is the functional view: the input x {\displaystyle \textstyle x} \textstyle x is transformed into a 3-dimensional vector h {\displaystyle \textstyle h} \textstyle h, which is then transformed into a 2-dimensional vector g {\displaystyle \textstyle g} \textstyle g, which is finally transformed into f {\displaystyle \textstyle f} \textstyle f. This view is most commonly encountered in the context of optimization.

The second view is the probabilistic view: the random variable F = f ( G ) {\displaystyle \textstyle F=f(G)} \textstyle F=f(G) depends upon the random variable G = g ( H ) {\displaystyle \textstyle G=g(H)} \textstyle G=g(H), which depends upon H = h ( X ) {\displaystyle \textstyle H=h(X)} \textstyle H=h(X), which depends upon the random variable X {\displaystyle \textstyle X} \textstyle X. This view is most commonly encountered in the context of graphical models.

The two views are largely equivalent. In either case, for this particular network architecture, the components of individual layers are independent of each other (e.g., the components of g {\displaystyle \textstyle g} \textstyle g are independent of each other given their input h {\displaystyle \textstyle h} \textstyle h). This naturally enables a degree of parallelism in the implementation.

 
Two separate depictions of the recurrent ANN dependency graph

Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where f {\displaystyle \textstyle f} \textstyle f is shown as being dependent upon itself. However, an implied temporal dependence is not shown.

Learning

What has attracted the most interest in neural networks is the possibility of learning. Given a specific task to solve, and a class of functions F {\displaystyle \textstyle F} \textstyle F, learning means using a set of observations to find f ∗ ∈ F {\displaystyle \textstyle f^{*}\in F} \textstyle f^{*}\in F which solves the task in some optimal sense.

This entails defining a cost function C : F → R {\displaystyle \textstyle C:F\rightarrow \mathbb {R} } \textstyle C:F\rightarrow \mathbb {R} such that, for the optimal solution f ∗ {\displaystyle \textstyle f^{}} \textstyle f^{*}, C ( f ∗ ) ≤ C ( f ) {\displaystyle \textstyle C(f^{})\leq C(f)} \textstyle C(f^{*})\leq C(f) ∀ f ∈ F {\displaystyle \textstyle \forall f\in F} \textstyle \forall f\in F – i.e., no solution has a cost less than the cost of the optimal solution (see mathematical optimization).

The cost function C {\displaystyle \textstyle C} \textstyle C is an important concept in learning, as it is a measure of how far away a particular solution is from an optimal solution to the problem to be solved. Learning algorithms search through the solution space to find a function that has the smallest possible cost.

For applications where the solution is dependent on some data, the cost must necessarily be a function of the observations, otherwise we would not be modelling anything related to the data. It is frequently defined as a statistic to which only approximations can be made. As a simple example, consider the problem of finding the model f {\displaystyle \textstyle f} \textstyle f, which minimizes C = E [ ( f ( x ) − y ) 2 ] {\displaystyle \textstyle C=E\left[(f(x)-y)^{2}\right]} \textstyle C=E\left[(f(x)-y)^{2}\right], for data pairs ( x , y ) {\displaystyle \textstyle (x,y)} \textstyle (x,y) drawn from some distribution D {\displaystyle \textstyle {\mathcal {D}}} \textstyle {\mathcal {D}}. In practical situations we would only have N {\displaystyle \textstyle N} \textstyle N samples from D {\displaystyle \textstyle {\mathcal {D}}} \textstyle {\mathcal {D}} and thus, for the above example, we would only minimize C ^ = 1 N ∑ i = 1 N ( f ( x i ) − y i ) 2 {\displaystyle \textstyle {\hat {C}}={\frac {1}{N}}\sum {i=1}^{N}(f(x{i})-y_{i})^{2}} \textstyle {\hat {C}}={\frac {1}{N}}\sum _{i=1}^{N}(f(x_{i})-y_{i})^{2}. Thus, the cost is minimized over a sample of the data rather than the entire distribution generating the data.

When N → ∞ {\displaystyle \textstyle N\rightarrow \infty } \textstyle N\rightarrow \infty some form of online machine learning must be used, where the cost is partially minimized as each new example is seen. While online machine learning is often used when D {\displaystyle \textstyle {\mathcal {D}}} \textstyle {\mathcal {D}} is fixed, it is most useful in the case where the distribution changes slowly over time. In neural network methods, some form of online machine learning is frequently used for finite datasets.

Choosing a Cost Function

While it is possible to define some arbitrary ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties (such as convexity) or because it arises naturally from a particular formulation of the problem (e.g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost). Ultimately, the cost function will depend on the desired task. An overview of the three main categories of learning tasks is provided below:

Learning Paradigms

There are three major learning paradigms, each corresponding to a particular abstract learning task. These are supervised learning, unsupervised learning and reinforcement learning.

Supervised Learning

In supervised learning, we are given a set of example pairs ( x , y ) , x ∈ X , y ∈ Y {\displaystyle \textstyle (x,y),x\in X,y\in Y} \textstyle (x,y),x\in X,y\in Y and the aim is to find a function f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} \textstyle f:X\rightarrow Y in the allowed class of functions that matches the examples. In other words, we wish to infer the mapping implied by the data; the cost function is related to the mismatch between our mapping and the data and it implicitly contains prior knowledge about the problem domain.

A commonly used cost is the mean-squared error, which tries to minimize the average squared error between the network's output, f ( x ) {\displaystyle \textstyle f(x)} \textstyle f(x), and the target value y {\displaystyle \textstyle y} \textstyle y over all the example pairs. When one tries to minimize this cost using gradient descent for the class of neural networks called multilayer perceptrons (MLP), one obtains the common and well-known backpropagation algorithm for training neural networks.

Tasks that fall within the paradigm of supervised learning are pattern recognition (also known as classification) and regression (also known as function approximation). The supervised learning paradigm is also applicable to sequential data (e.g., for speech and gesture recognition). This can be thought of as learning with a "teacher", in the form of a function that provides continuous feedback on the quality of solutions obtained thus far.

Unsupervised Learning

In unsupervised learning, some data x {\displaystyle \textstyle x} \textstyle x is given and the cost function to be minimized, that can be any function of the data x {\displaystyle \textstyle x} \textstyle x and the network's output, f {\displaystyle \textstyle f} \textstyle f.

The cost function is dependent on the task (what we are trying to model) and our a priori assumptions (the implicit properties of our model, its parameters and the observed variables).

As a trivial example, consider the model f ( x ) = a {\displaystyle \textstyle f(x)=a} \textstyle f(x)=a where a {\displaystyle \textstyle a} \textstyle a is a constant and the cost C = E [ ( x − f ( x ) ) 2 ] {\displaystyle \textstyle C=E[(x-f(x))^{2}]} \textstyle C=E[(x-f(x))^{2}]. Minimizing this cost will give us a value of a {\displaystyle \textstyle a} \textstyle a that is equal to the mean of the data. The cost function can be much more complicated. Its form depends on the application: for example, in compression it could be related to the mutual information between x {\displaystyle \textstyle x} \textstyle x and f ( x ) {\displaystyle \textstyle f(x)} \textstyle f(x), whereas in statistical modeling, it could be related to the posterior probability of the model given the data (note that in both of those examples those quantities would be maximized rather than minimized).

Tasks that fall within the paradigm of unsupervised learning are in general estimation problems; the applications include clustering, the estimation of statistical distributions, compression and filtering.

Reinforcement learning

In reinforcement learning, data x {\displaystyle \textstyle x} \textstyle x are usually not given, but generated by an agent's interactions with the environment. At each point in time t {\displaystyle \textstyle t} \textstyle t, the agent performs an action y t {\displaystyle \textstyle y_{t}} \textstyle y_{t} and the environment generates an observation x t {\displaystyle \textstyle x_{t}} \textstyle x_{t} and an instantaneous cost c t {\displaystyle \textstyle c_{t}} \textstyle c_{t}, according to some (usually unknown) dynamics. The aim is to discover a policy for selecting actions that minimizes some measure of a long-term cost, e.g., the expected cumulative cost. The environment's dynamics and the long-term cost for each policy are usually unknown, but can be estimated.

More formally the environment is modeled as a Markov decision process (MDP) with states s 1 , . . . , s n ∈ S {\displaystyle \textstyle {s_{1},...,s_{n}}\in S} \textstyle {s_{1},...,s_{n}}\in S and actions a 1 , . . . , a m ∈ A {\displaystyle \textstyle {a_{1},...,a_{m}}\in A} \textstyle {a_{1},...,a_{m}}\in A with the following probability distributions: the instantaneous cost distribution P ( c t | s t ) {\displaystyle \textstyle P(c_{t}|s_{t})} \textstyle P(c_{t}|s_{t}), the observation distribution P ( x t | s t ) {\displaystyle \textstyle P(x_{t}|s_{t})} \textstyle P(x_{t}|s_{t}) and the transition P ( s t + 1 | s t , a t ) {\displaystyle \textstyle P(s_{t+1}|s_{t},a_{t})} \textstyle P(s_{t+1}|s_{t},a_{t}), while a policy is defined as the conditional distribution over actions given the observations. Taken together, the two then define a Markov chain (MC). The aim is to discover the policy (i.e., the MC) that minimizes the cost.

ANNs are frequently used in reinforcement learning as part of the overall algorithm. Dynamic programming has been coupled with ANNs (Neuro dynamic programming) by Bertsekas and Tsitsiklis and applied to multi-dimensional nonlinear problems such as those involved in vehicle routing, natural resources management or medicine because of the ability of ANNs to mitigate losses of accuracy even when reducing the discretization grid density for numerically approximating the solution of the original control problems.

Tasks that fall within the paradigm of reinforcement learning are control problems, games and other sequential decision making tasks.

Learning Algorithms

Training a neural network model essentially means selecting one model from the set of allowed models (or, in a Bayesian framework, determining a distribution over the set of allowed models) that minimizes the cost criterion. There are numerous algorithms available for training neural network models; most of them can be viewed as a straightforward application of optimization theory and statistical estimation.

Most of the algorithms used in training artificial neural networks employ some form of gradient descent, using backpropagation to compute the actual gradients. This is done by simply taking the derivative of the cost function with respect to the network parameters and then changing those parameters in a gradient-related direction. The backpropagation training algorithms are usually classified into three categories: steepest descent (with variable learning rate, with variable learning rate and momentum, resilient backpropagation), quasi-Newton (Broyden-Fletcher-Goldfarb-Shanno, one step secant), Levenberg-Marquardt and conjugate gradient (Fletcher-Reeves update, Polak-Ribiére update, Powell-Beale restart, scaled conjugate gradient).

The training problem is in fact an inverse problem where our goal is to estimate the parameters of the neural network model. The Levenberg-Marquardt algorithm is shown to be more reliable in obtaining appropriate solutions to inverse problems than Gauss-Newton and quasi-Newton methods.

Evolutionary methods, gene expression programming, simulated annealing, expectation-maximization, non-parametric methods and particle swarm optimization are some other methods for training neural networks.


Employing Artificial Neural Networks

Perhaps the greatest advantage of ANNs is their ability to be used as an arbitrary function approximation mechanism that 'learns' from observed data. However, using them is not so straightforward, and a relatively good understanding of the underlying theory is essential.

  • Choice of model: This will depend on the data representation and the application. Overly complex models tend to lead to challenges in learning.
  • Learning algorithm: There are numerous trade-offs between learning algorithms. Almost any algorithm will work well with the correct hyperparameters for training on a particular fixed data set. However, selecting and tuning an algorithm for training on unseen data require a significant amount of experimentation.
  • Robustness: If the model, cost function and learning algorithm are selected appropriately, the resulting ANN can be extremely robust.

With the correct implementation, ANNs can be used naturally in online learning and large data set applications. Their simple implementation and the existence of mostly local dependencies exhibited in the structure allows for fast, parallel implementations in hardware.


Applications

The utility of artificial neural network models lies in the fact that they can be used to infer a function from observations. This is particularly useful in applications where the complexity of the data or task makes the design of such a function by hand impractical.

Real Life Applications

The tasks artificial neural networks are applied to tend to fall within the following broad categories:

  • Function approximation, or regression analysis, including time series prediction, fitness approximation and modeling
  • Classification, including pattern and sequence recognition, novelty detection and sequential decision making
  • Data processing, including filtering, clustering, blind source separation and compression
  • Robotics, including directing manipulators, prosthesis.
  • Control, including Computer numerical control

Application areas include the system identification and control (vehicle control, trajectory prediction, process control, natural resources management), quantum chemistry, game-playing and decision making (backgammon, chess, poker), pattern recognition (radar systems, face identification, object recognition and more), sequence recognition (gesture, speech, handwritten text recognition), medical diagnosis, financial applications (e.g. automated trading systems), data mining (or knowledge discovery in databases, "KDD"), visualization and e-mail spam filtering.

Artificial neural networks have also been used to diagnose several cancers. An ANN based hybrid lung cancer detection system named HLND improves the accuracy of diagnosis and the speed of lung cancer radiology. These networks have also been used to diagnose prostate cancer. The diagnoses can be used to make specific models taken from a large group of patients compared to information of one given patient. The models do not depend on assumptions about correlations of different variables. Colorectal cancer has also been predicted using the neural networks. Neural networks could predict the outcome for a patient with colorectal cancer with more accuracy than the current clinical methods. After training, the networks could predict multiple patient outcomes from unrelated institutions.

Neural Networks and Neuroscience

Theoretical and computational Neuroscience is the field concerned with the theoretical analysis and the computational modeling of biological neural systems. Since neural systems are intimately related to cognitive processes and behavior, the field is closely related to cognitive and behavioral modeling.

The aim of the field is to create models of biological neural systems in order to understand how biological systems work. To gain this understanding, neuroscientists strive to make a link between observed biological processes (data), biologically plausible mechanisms for neural processing and learning (biological neural network models) and theory (statistical learning theory and information theory).

Types of Models

Many models are used in the field, defined at different levels of abstraction and modeling different aspects of neural systems. They range from models of the short-term behavior of individual neurons (e.g.), models of how the dynamics of neural circuitry arise from interactions between individual neurons and finally to models of how behavior can arise from abstract neural modules that represent complete subsystems. These include models of the long-term, and short-term plasticity, of neural systems and their relations to learning and memory from the individual neuron to the system level.

Memory Networks

Integrating external memory components with artificial neural networks has a long history dating back to early research in distributed representations and self-organizing maps. E.g. in sparse distributed memory the patterns encoded by neural networks are used as memory addresses for content-addressable memory, with "neurons" essentially serving as address encoders and decoders.

More recently deep learning was shown to be useful in semantic hashing where a deep graphical model of the word-count vectors[53] is obtained from a large set of documents. Documents are mapped to memory addresses in such a way that semantically similar documents are located at nearby addresses. Documents similar to a query document can then be found by simply accessing all the addresses that differ by only a few bits from the address of the query document.

Neural Turing Machines developed by Google DeepMind extend the capabilities of deep neural networks by coupling them to external memory resources, which they can interact with by attentional processes. The combined system is analogous to a Turing Machine but is differentiable end-to-end, allowing it to be efficiently trained with gradient descent. Preliminary results demonstrate that Neural Turing Machines can infer simple algorithms such as copying, sorting, and associative recall from input and output examples.

Memory Networks is another extension to neural networks incorporating long-term memory which was developed by Facebook research. The long-term memory can be read and written to, with the goal of using it for prediction. These models have been applied in the context of question answering (QA) where the long-term memory effectively acts as a (dynamic) knowledge base, and the output is a textual response.


Neural Network Software

Neural Network Software is used to simulate, research, develop and apply artificial neural networks, biological neural networks and, in some cases, a wider array of adaptive systems.


Types of Artificial Neural Networks

Artificial neural network types vary from those with only one or two layers of single direction logic, to complicated multi–input many directional feedback loops and layers. On the whole, these systems use algorithms in their programming to determine control and organization of their functions. Most systems use "weights" to change the parameters of the throughput and the varying connections to the neurons. Artificial neural networks can be autonomous and learn by input from outside "teachers" or even self-teaching from written-in rules.


Theoretical Properties

Computational Power

The multilayer perceptron is a universal function approximator, as proven by the universal approximation theorem. However, the proof is not constructive regarding the number of neurons required or the settings of the weights.

Work by Hava Siegelmann and Eduardo D. Sontag has provided a proof that a specific recurrent architecture with rational valued weights (as opposed to full precision real number-valued weights) has the full power of a Universal Turing Machine using a finite number of neurons and standard linear connections. Further, it has been shown that the use of irrational values for weights results in a machine with super-Turing power.

Capacity

Artificial neural network models have a property called 'capacity', which roughly corresponds to their ability to model any given function. It is related to the amount of information that can be stored in the network and to the notion of complexity.

Convergence

Nothing can be said in general about convergence since it depends on a number of factors. Firstly, there may exist many local minima. This depends on the cost function and the model. Secondly, the optimization method used might not be guaranteed to converge when far away from a local minimum. Thirdly, for a very large amount of data or parameters, some methods become impractical. In general, it has been found that theoretical guarantees regarding convergence are an unreliable guide to practical application.

Generalization and Statistics

In applications where the goal is to create a system that generalizes well in unseen examples, the problem of over-training has emerged. This arises in convoluted or over-specified systems when the capacity of the network significantly exceeds the needed free parameters. There are two schools of thought for avoiding this problem: The first is to use cross-validation and similar techniques to check for the presence of overtraining and optimally select hyperparameters such as to minimize the generalization error. The second is to use some form of regularization. This is a concept that emerges naturally in a probabilistic (Bayesian) framework, where the regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting.

Supervised neural networks that use a mean squared error (MSE) cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the confidence interval of the output of the network, assuming a normal distribution. A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified.

By assigning a softmax activation function, a generalization of the logistic function, on the output layer of the neural network (or a softmax component in a component-based neural network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is very useful in classification as it gives a certainty measure on classifications.

The softmax activation function is:

y i = e x i ∑ j = 1 c e x j {\displaystyle y_{i}={\frac {e^{x_{i}}}{\sum _{j=1}^{c}e^{x_{j}}}}} y_{i}={\frac {e^{x_{i}}}{\sum _{j=1}^{c}e^{x_{j}}}}

Classes and Types of Artificial Neural Networks - ANNs

  • Dynamic Neural Network
    • Feedforward neural network FNN
    • Recurrent neural network RNN
      • Hopfield network
      • Boltzmann machine
      • Simple recurrent networks
      • Echo state network
      • Long short term memory network
      • Bi-directional RNN
      • Hierarchical RNN
      • Stochastic neural networks
    • Kohonen Self-Organizing Maps
    • Autoencoder
    • Backpropagation
    • probabilistic neural network PNN
    • Time delay neural network TDNN
  • Static Neural Network
    • Neocognitron
    • McCulloch-Pitts cell
    • Radial basis function network RBF
    • Learning vector quantization
    • Perceptron
      • Adaline model
      • Convolutional neural network CNN
    • Modular neural networks
      • Committee of machines COM
      • Associative neural network ASNN
  • Memory Network [1]
    • Google / Deep Mind
    • facebook / MemNN
    • Holographic associative memory
    • One-shot associative memory
    • Neural Turing Machine
    • Adaptive resonance theory
    • Hierarchical temporal memory
  • Other types of networks
    • Instantaneously trained networks ITNN
    • Spiking neural networks SNN
      • Pulse Coded Neural Networks PCNN
    • Cascading neural networks
    • Neuro-fuzzy networks
    • Growing Neural Gas GNG
    • Compositional pattern-producing networks
    • Counterpropagation network
    • Oscillating neural network
    • Hybridization neural network
    • Physical neural network
      • Optical neural network

MTANN stands for: Massive Training Artificial Neural Network