A Function Space Approach to Spectral Related Problems

SynopsisA function space approach is employed to obtain bifurcation functions for which the zeros correspond to the occurrence of periodic or aperiodic solutions near heteroclinic or homoclinic cycles. The bifurcation function for the existence of homoclinic solutions is the limiting case where the period is infinite. Examples include generalisations of Silnikov's main theorems and a retreatment of a singularly perturbed delay differential equation.

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Function space approach to a class of stochastic optimal control problems

Journal of Optimization Theory and Applications ◽

10.1007/bf00928298 ◽

1970 ◽

Vol 5 (2) ◽

pp. 97-113

Author(s):

J. Y. S. Luh ◽

M. P. Lukas

Keyword(s):

Optimal Control ◽

Function Space ◽

Stochastic Optimal Control ◽

Optimal Control Problems ◽

Control Problems ◽

Space Approach

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A Function Estimation Approach to Sequential Learning with Neural Networks

Neural Computation ◽

10.1162/neco.1993.5.6.954 ◽

1993 ◽

Vol 5 (6) ◽

pp. 954-975 ◽

Cited By ~ 339

Author(s):

Visakan Kadirkamanathan ◽

Mahesan Niranjan

Keyword(s):

Function Space ◽

Time Series Prediction ◽

Approximation Error ◽

Lms Algorithm ◽

Sequential Learning ◽

Superior Performance ◽

Function Estimation ◽

Minimal Network ◽

Growth Criterion ◽

Space Approach

In this paper, we investigate the problem of optimal sequential learning, viewed as a problem of estimating an underlying function sequentially rather than estimating a set of parameters of the neural network. First, we arrive at a suboptimal solution to the sequential estimate that can be mapped by a growing gaussian radial basis function (GaRBF) network. This network adds hidden units for each observation. The function space approach in which the estimates are represented as vectors in a function space is used in developing a growth criterion to limit its growth. A simplification of the criterion leads to two joint criteria on the distance of the present pattern and the existing unit centers in the input space and on the approximation error of the network for the given observation to be satisfied together. This network is similar to the resource allocating network (RAN) (Platt 1991a) and hence RAN can be interpreted from a function space approach to sequential learning. Second, we present an enhancement to the RAN. The RAN either allocates a new unit based on the novelty of an observation or adapts the network parameters by the LMS algorithm. The function space interpretation of the RAN lends itself to an enhancement of the RAN in which the extended Kalman filter (EKF) algorithm is used in place of the LMS algorithm. The performance of the RAN and the enhanced network are compared in the experimental tasks of function approximation and time-series prediction demonstrating the superior performance of the enhanced network with fewer number of hidden units. The approach adopted here has led us toward the minimal network required for a sequential learning problem.

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