Search

Direction-dependent scaling, shaping, and rotation of Gaussian basis functions are introduced for maximal trend sensing with minimal parameter representations for input output approximation. It is shown that shaping and rotation of the radial basis functions helps in reducing the total number of function units required to approximate any given input-output data, while improving accuracy. Several alternate formulations that enforce minimal parameterization of the most general radial basis functions are presented. A novel "directed graph" based algorithm is introduced to facilitate intelligent direction based learning and adaptation of the parameters appearing in the radial basis function network. Further, a parameter estimation algorithm is incorporated to establish starting estimates for the model parameters using multiple windows of the input-output data. The efficacy of direction-dependent shaping and rotation in function approximation is evaluated by modifying the minimal resource allocating network and considering different test examples. The examples are drawn from recent literature to benchmark the new algorithm versus existing methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the stationary Fokker–Planck equation (FPE) is solved for nonlinear dynamical systems using a local numerical technique based on the meshless partition of unity finite element method (PUFEM). The method is applied to stationary FPE for 2-, 3- and 4-D systems and is argued to be an excellent candidate for higher dimensional problems and the transient problem. Local refinement is applied by introducing higher order polynomials in selected subdomains (local -refinement) to keep the problem size small while ensuring high approximation accuracy. Various local approximations are blended using novel pasting functions that provide any desired order of continuity. Results are compared with existing global and local techniques. Local -refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of FPE. [Copyright &y& Elsevier]