Miguel Atencia

Hopfield neural networks for optimization : Study of the different dynamics

This work summarizes a tutorial on the main aspects of the application of Hopfield networks to optimization. The main formulations of the dynamics are studied, and the particular problems that arise in their application to optimization are brought to light. As a particular engineering problem, systems identification is formulated as an optimization problem and the Hopfield methodology is adapted to its solution.

Dynamical Analysis of Continuous Higher-Order Hopfield Networks for Combinatorial Optimization

In this letter, the ability of higher-order Hopfield networks to solve combinatorial optimization problems is assessed by means of a rigorous analysis of their properties. The stability of the continuous network is almost completely clarified: (1) hyperbolic interior equilibria, which are unfeasible, are unstable; (2) the state cannot escape from the unitary hypercube; and (3) a Lyapunov function exists. Numerical methods used to implement the continuous equation on a computer should be designed with the aim of preserving these favorable properties. The case of nonhyperbolic fixed points, which occur when the Hessian of the target function is the null matrix, requires further study. We prove that these nonhyperbolic interior fixed points are unstable in networks with three neurons and order two. The conjecture that interior equilibria are unstable in the general case is left open.In this letter, the ability of higher-order Hopfield networks to solve combinatorial optimization problems is assessed by means of a rigorous analysis of their properties. The stability of the continuous network is almost completely clarified: (1) hyperbolic interior equilibria, which are unfeasible, are unstable; (2) the state cannot escape from the unitary hypercube; and (3) a Lyapunov function exists. Numerical methods used to implement the continuous equation on a computer should be designed with the aim of preserving these favorable properties. The case of nonhyperbolic fixed points, which occur when the Hessian of the target function is the null matrix, requires further study. We prove that these nonhyperbolic interior fixed points are unstable in networks with three neurons and order two. The conjecture that interior equilibria are unstable in the general case is left open.

Parametric identification of robotic systems with stable time-varying Hopfield networks

In this work, a novel method for on-line identification of non-linear systems is proposed based upon the optimisation methodology with Hopfield neural networks. The original Hopfield model is adapted so that the weights of the resulting network are time-varying. A rigorous analytical study proves that, under mild assumptions, the estimations provided by the method converge to the actual parameter values in the case of constant parameters, or to a bounded neighbourhood of the parameters when these are time-varying. Time-varying parameters, often appearing in mechanical systems, are dealt with by the neural estimator in a more natural way than by least squares techniques. Both sudden and slow continuous variations are considered. Besides, in contrast to the gradient method, the neural estimator does not critically depend on the adjustment of the gain. The proposed method is applied to the identification of a robotic system with a flexible link. A reduced output prediction error and an accurate estimation of parameters are observed in simulation results.

FPGA implementation of a systems identification module based upon Hopfield networks

The aim of this contribution is to implement a hardware module that performs parametric identification of dynamical systems. The design is based upon the methodology of optimization with Hopfield neural networks, leading to an adapted version of these networks. An outstanding feature of this modified Hopfield network is the existence of weights that vary with time. Since weights can no longer be stored in read-only memories, these dynamic weights constitute a significant challenge for digital circuits, in addition to the usual issues of area occupation, fixed-point arithmetic and nonlinear functions computations. The implementation, which is accomplished on FPGA circuits, achieves modularity and flexibility, due to the usage of parametric VHDL to describe the network. In contrast to software simulations, the natural parallelism of neural networks is preserved, at a limited cost in terms of circuitry cost and processing time. The functional simulation and the synthesis show the viability of the design. In particular, the FPGA implementation exhibits a reasonably fast convergence, which is required to produce accurate parameter estimations. Current research is oriented towards integrating the estimator within an embedded adaptive controller for autonomous systems.

Hopfield Neural Networks for Parametric Identification of Dynamical Systems

In this work, a novel method, based upon Hopfield neural networks, is proposed for parameter estimation, in the context of system identification. The equation of the neural estimator stems from the applicability of Hopfield networks to optimization problems, but the weights and the biases of the resulting network are time-varying, since the target function also varies with time. Hence the stability of the method cannot be taken for granted. In order to compare the novel technique and the classical gradient method, simulations have been carried out for a linearly parameterized system, and results show that the Hopfield network is more efficient than the gradient estimator, obtaining lower error and less oscillations. Thus the neural method is validated as an on-line estimator of the time-varying parameters appearing in the model of a nonlinear physical system.

Identification of noisy dynamical systems with parameter estimation based on Hopfield neural networks

An algorithm for estimating time-varying parameters of dynamical systems is proposed, within the large family of prediction error methods. The algorithm is based on the ability of Hopfield neural networks to solve optimisation problems, since its formulation can be summarized as minimisation of the prediction error by means of a continuous Hopfield network. In previous work, it was proved, under mild assumptions, that the estimates converge towards the actual values of parameters and the estimation error remains asymptotically bounded in the presence of measurement noise. The novelty of this work is the advance in the robustness analysis, by considering deterministic disturbances, which do not fulfil the usual statistical hypothesis such as normality and uncorrelatedness. A model of HIV epidemics in Cuba is used as suitable benchmark, which is confirmed by the computation of the sensitivity matrix. The results show a promising performance, in comparison to the conventional Least Squares Estimator. Indeed, the estimation error is almost always lower in the proposed method that in least squares, and it is never significantly higher. Further, from a qualitative point of view, the estimate provided by the Hopfield estimator is smoother, with no overshoot that could eventually destabilize a closed control loop. A significant finding is the fact that the form of the perturbation affects critically the dynamical behaviour and magnitude of the estimation, since the estimation error asymptotically vanishes when the disturbances are additive, but not when they are multiplicative. To summarize, we can conclude that the proposed estimator is an efficient and robust method to estimate time-varying parameters of dynamical systems.

Modelling the HIV-AIDS Cuban Epidemics with Hopfield Neural Networks

In this work, Hopfield neural networks are applied to estimation of parameters in a dynamical model of Cuban HIV-AIDS epidemics. The time-varying weights are derived, and its formulation is adapted to the discrete case. The method is tested on a data sequence obtained from numerical solution of the model. Simulation results show that the proposed technique quickly reduces the output prediction error, and it adapts well to parameter changes. Results concerning estimation error are poor, and some directions to deal with this issue are proposed.

Associating arbitrary-order energy functions to an artificial neural network Implications concerning the resolution of optimization problems

Abstract We have studied the restrictions that a first order asynchronous feedback neural network must fulfill to be associated to an arbitrary order energy function of the kind described by Kobuchi [6], i.e., the network evolution is related to the descent to a minimum of such a function. These restrictions do not avoid the association of the even order energy functions to a first order network. However, for the odd order energy functions, most of the weights of each neuron must be zero. This result discards using first order neural networks for the solution of optimization problems associated with an odd order function, justifying in this way the use of high order neural networks. For these ones, we have obtained a general expression of their possible energy functions, which includes, as a special case, the high order generalization of Hopfields energy functions until now used, for example, in [5], [8].We have studied the restrictions that a first order asynchronous feedback neural network must fulfill to be associated to an arbitrary order energy function of the kind described by Kobuchi [6], i.e., the network evolution is related to the descent to a minimum of such a function. These restrictions do not avoid the association of the even order energy functions to a first order network. However, for the odd order energy functions, most of the weights of each neuron must be zero. This result discards using first order neural networks for the solution of optimization problems associated with an odd order function, justifying in this way the use of high order neural networks. For these ones, we have obtained a general expression of their possible energy functions, which includes, as a special case, the high order generalization of Hopfield’s energy functions until now used, for example, in [5], [8].

A Learning Rule to Model the Development of Orientation Selectivity in Visual Cortex

Abstract.This paper presents a learning rule, CBA, to develop oriented receptive fields similar to those founded in cat striate cortex. The inherent complexity of the development of selectivity in visual cortex has led most authors to test their models by using a restricted input environment. Only recently, some learning rules (the PCA and the BCM rules) have been studied in a realistic visual environment. For these rules, which are based upon Hebbian learning, single neuron models have been proposed in order to get a better understanding of their properties and dynamics. These models suffered from unbounded growing of synaptic strength, which is remedied by a normalization process. However, normalization seems biologically implausible, given the non-local nature of this process. A detailed stability analysis of the proposed rule proves that the CBA attains a stable state without any need for normalization. Also, a comparison among the results achieved in different types of visual environments by the PCA, the BCM and the CBA rules is provided. The final results show that the CBA rule is appropriate for studying the biological process of receptive field formation and its application in image processing and artificial vision tasks.

Estimation of the rate of detection of infected individuals in an epidemiological model

This paper presents a method for estimation of parameters in dynamical systems, applied to a model of the HIV-AIDS epidemics in Cuba. This estimation technique, based upon artificial neural networks, has been successfully applied to robotic systems, whereas the application to epidemiological models is challenged by the possible uncertainty of the model; besides, a state variable exists that is not directly measurable. With regard to the first limitation, a model provided by experts, previously validated by statistical techniques, has been used; with respect to the second drawback, an evaluation of the unknown variable has been carried out from comparisons with other models of the development of the disease. Among the parameters that intervene in the model, three important factors have been considered: the detection rate of the disease, through the contact tracing program; the detection rate through other methods; and the rate of transition to AIDS of previously undetected infected individuals. Results are plausible, according to experts, and they support both the estimation method and the model.