On the Optimal Node Ratio between Hidden Layers: A Probabilistic Study

Two-hidden layer feedforward neural networks (TLFNs) have been shown to outperform single-hidden-layer neural networks (SLFNs) for function approximation in many cases. However, their added complexity makes them more difficult to find. Given a constant number of hidden nodes nh, this paper investigates how their allocation between the first and second hidden layers (n h = n1 + n2 ) affects the likelihood of finding the best generaliser. The experiments were carried out over a total of ten public domain datasets with nh = 8 and 16. The findings were that the heuristic n1 = 0.5nh + 1 has an average probability of at least 0.85 of finding a network with a generalisation error within 0.18% of the best generaliser. Furthermore, the worst case over all data sets was within 0.23% for nh = 8, and within 0.15% for nh = 16. These findings could be used to reduce the complexity of the search for TLFNs from quadratic to linear, or alternatively for ‘topology mapping’ between TLFNs and SLFNs, given the same number of hidden nodes, to compare their performance

Neural networks in control engineering

The purpose of this thesis is to investigate the viability of integrating neural networks into control structures. These networks are an attempt to create artificial intelligent systems with the ability to learn and remember. They mathematically model the biological structure of the brain and consist of a large number of simple interconnected processing units emulating brain cells. Due to the highly parallel and consequently computationally expensive nature of these networks, intensive research in this field has only become feasible due to the availability of powerful personal computers in recent years. Consequently, attempts at exploiting the attractive learning and nonlinear optimization characteristics of neural networks have been made in most fields of science and engineering, including process control. The control structures suggested in the literature for the inclusion of neural networks in control applications can be divided into four major classes. The first class includes approaches in which the network forms part of an adaptive mechanism which modulates the structure or parameters of the controller. In the second class the network forms part of the control loop and replaces the conventional control block, thus leading to a pure neural network control law. The third class consists of topologies in which neural networks are used to produce models of the system which are then utilized in the control structure, whilst the fourth category includes suggestions which are specific to the problem or system structure and not suitable for a generic neural network-based-approach to control problems. Although several of these approaches show promising results, only model based structures are evaluated in this thesis. This is due to the fact that many of the topologies in other classes require system estimation to produce the desired network output during training, whereas the training data for network models is obtained directly by sampling the system input(s) and output(s). Furthermore, many suggested structures lack the mathematical motivation to consider them for a general structure, whilst the neural network model topologies form natural extensions of their linear model based origins. Since it is impractical and often impossible to collect sufficient training data prior to implementing the neural network based control structure, the network models have to be suited to on-line training during operation. This limits the choice of network topologies for models to those that can be trained on a sample by sample basis (pattern learning) and furthermore are capable of learning even when the variation in training data is relatively slow as is the case for most controlled dynamic systems. A study of feedforward topologies (one of the main classes of networks) shows that the multilayer perceptron network with its backpropagation training is well suited to model nonlinear mappings but fails to learn and generalize when subjected to slow varying training data. This is due to the global input interpretation of this structure, in which any input affects all hidden nodes such that no effective partitioning of the input space can be achieved. This problem is overcome in a less flexible feedforward structure, known as regular Gaussian network. In this network, the response of each hidden node is limited to a -sphere around its center and these centers are fixed in a uniform distribution over the entire input space. Each input to such a network is therefore interpreted locally and only effects nodes with their centers in close proximity. A deficiency common to all feedforward networks, when considered as models for dynamic systems, is their inability to conserve previous outputs and states for future predictions. Since this absence of dynamic capability requires the user to identify the order of the system prior to training and is therefore not entirely self-learning, more advanced network topologies are investigated. The most versatile of these structures, known as a fully recurrent network, re-uses the previous state of each of its nodes for subsequent outputs. However, despite its superior modelling capability, the tests performed using the Williams and Zipser training algorithm show that such structures often fail to converge and require excessive computing power and time, when increased in size. Despite its rigid structure and lack of dynamic capability, the regular Gaussian network produces the most reliable and robust models and was therefore selected for the evaluations in this study. To overcome the network initialization problem, found when using a pure neural network model, a combination structure· _in which the network operates in parallel with a mathematical model is suggested. This approach allows the controller to be implemented without any prior network training and initially relies purely on the mathematical model, much like conventional approaches. The network portion is then trained during on-line operation in order to improve the model. Once trained, the enhanced model can be used to improve the system response, since model exactness plays an important role in the control action achievable with model based structures. The applicability of control structures based on neural network models is evaluated by comparing the performance of two network approaches to that of a linear structure, using a simulation of a nonlinear tank system. The first network controller is developed from the internal model control (IMC) structure, which includes a forward and inverse model of the system to be controlled. Both models can be replaced by a combination of mathematical and neural topologies, the network portion of which is trained on-line to compensate for the discrepancies between the linear model _ and nonlinear system. Since the network has no dynamic ·capacity, .former system outputs are used as inputs to the forward and inverse model. Due to this direct feedback, the trained structure can be tuned to perform within limits not achievable using a conventional linear system. As mentioned previously the IMC structure uses both forward and inverse models. Since the control law requires that these models are exact inverses, an iterative inversion algorithm has to be used to improve the values produced by the inverse combination model. Due to deadtimes and right-half-plane zeroes, many systems are furthermore not directly invertible. Whilst such unstable elements can be removed from mathematical models, the inverse network is trained directly from the forward model and can not be compensated. These problems could be overcome by a control structure for which only a forward model is required. The neural predictive controller (NPC) presents such a topology. Based on the optimal control philosophy, this structure uses a model to predict several future outputs. The errors between these and the desired output are then collected to form the cost function, which may also include other factors such as the magnitude of the change in input. The input value that optimally fulfils all the objectives used to formulate the cost function, can then be found by locating its minimum. Since the model in this structure includes a neural network, the optimization can not be formulated in a closed mathematical form and has to be performed using a numerical method. For the NPC topology, as for the neural network IMC structure, former system outputs are fed back to the model and again the trained network approach produces results not achievable with a linear model. Due to the single network approach, the NPC topology furthermore overcomes the limitations described for the neural network IMC structure and can be extended to include multivariable systems. This study shows that the nonlinear modelling capability of neural networks can be exploited to produce learning control structures with improved responses for nonlinear systems. Many of the difficulties described are due to the computational burden of these networks and associated algorithms. These are likely to become less significant due to the rapid development in computer technology and advances in neural network hardware. Although neural network based control structures are unlikely to replace the well understood linear topologies, which are adequate for the majority of applications, they might present a practical alternative where (due to nonlinearity or modelling errors) the conventional controller can not achieve the required control action

Impact of noise on a dynamical system: prediction and uncertainties from a swarm-optimized neural network

In this study, an artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey--Glass chaotic time series in the short-term $x(t+6)$. The performance prediction was evaluated and compared with another studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called {\it stochastic} hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute uncertainties of predictions for noisy Mackey--Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level ($\sigma_{N}$) from 0.01 to 0.1.Comment: 11 pages, 8 figure

New acceleration technique for the backpropagation algorithm

Artificial neural networks have been studied for many years in the hope of achieving human like performance in the area of pattern recognition, speech synthesis and higher level of cognitive process. In the connectionist model there are several interconnected processing elements called the neurons that have limited processing capability. Even though the rate of information transmitted between these elements is limited, the complex interconnection and the cooperative interaction between these elements results in a vastly increased computing power; The neural network models are specified by an organized network topology of interconnected neurons. These networks have to be trained in order them to be used for a specific purpose. Backpropagation is one of the popular methods of training the neural networks. There has been a lot of improvement over the speed of convergence of standard backpropagation algorithm in the recent past. Herein we have presented a new technique for accelerating the existing backpropagation without modifying it. We have used the fourth order interpolation method for the dominant eigen values, by using these we change the slope of the activation function. And by doing so we increase the speed of convergence of the backpropagation algorithm; Our experiments have shown significant improvement in the convergence time for problems widely used in benchmarKing Three to ten fold decrease in convergence time is achieved. Convergence time decreases as the complexity of the problem increases. The technique adjusts the energy state of the system so as to escape from local minima

Towards The Deep Semantic Learning Machine Neuroevolution Algorithm: An exploration on the CIFAR-10 problem task

Dissertation presented as the partial requirement for obtaining a Master's degree in Data Science and Advanced AnalyticsSelecting the topology and parameters of Convolutional Neural Network (CNN) for a given supervised machine learning task is a non-trivial problem. The Deep Semantic Learning Machine (Deep-SLM) deals with this problem by automatically constructing CNNs without the use of the Backpropagation algorithm. The Deep-SLM is a novel neuroevolution technique and functions as stochastic semantic hill-climbing algorithm searching over the space of CNN topologies and parameters. The geometric semantic properties of the Deep-SLM induce a unimodel error space and eliminate the existence of local optimal solutions. This makes the Deep-SLM potentially favorable in terms of search efficiency and effectiveness. This thesis provides an exploration of a variant of the Deep-SLM algorithm on the CIFAR-10 problem task, and a validation of its proof of concept. This specific variant only forms mutation node ! mutation node connections in the non-convolutional part of the constructed CNNs. Furthermore, a comparative study between the Deep-SLM and the Semantic Learning Machine (SLM) algorithms was conducted. It was observed that sparse connections can be an effective way to prevent overfitting. Additionally, it was shown that a single 2D convolution layer initialized with random weights does not result in well-generalizing features for the Deep-SLM directly, but, in combination with a 2D max-pooling down sampling layer, effective improvements in performance and generalization of the Deep-SLM could be achieved. These results constitute to the hypothesis that convolution and pooling layers can improve performance and generalization of the Deep-SLM, unless the components are properly optimized.Selecionar a topologia e os parâmetros da Rede Neural Convolucional (CNN) para uma tarefa de aprendizado automático supervisionada não é um problema trivial. A Deep Semantic Learning Machine (Deep-SLM) lida com este problema construindo automaticamente CNNs sem recorrer ao uso do algoritmo de Retro-propagação. A Deep-SLM é uma nova técnica de neuroevolução que funciona enquanto um algoritmo de escalada estocástico semântico na pesquisa de topologias e de parâmetros CNN. As propriedades geométrico-semânticas da Deep-SLM induzem um unimodel error space que elimina a existência de soluções ótimas locais, favorecendo, potencialmente, a Deep-SLM em termos de eficiência e eficácia. Esta tese providencia uma exploração de uma variante do algoritmo da Deep-SLM no problemo de CIFAR-10, assim como uma validação do seu conceito de prova. Esta variante específica apenas forma conexões nó de mutação!nó de mutação na parte non convolucional da CNN construída. Mais ainda, foi conduzido um estudo comparativo entre a Deep-SLM e o algoritmo da Semantic Learning Machine (SLM). Tendo sido observado que as conexões esparsas poderão tratar-se de uma forma eficiente de prevenir o overfitting. Adicionalmente, mostrou-se que uma singular camada de convolução 2D, iniciada com valores aleatórios, não resulta, directamente, em características generalizadas para a Deep-SLM, mas, em combinação com uma camada de 2D max-pooling, melhorias efectivas na performance e na generalização da Deep-SLM poderão ser concretizadas. Estes resultados constituem, assim, a hipótese de que as camadas de convolução e pooling poderão melhorar a performance e a generalização da Deep-SLM, a não ser que os componentes sejam adequadamente otimizados