Gaussian Radial Basis Function Neural Networks on Time Scales
TL;DR
A radial basis function neural network for dynamic equations on time scales using a Gaussian-type activation generated by the time-scale exponential to establish two-sided exponential bounds, uniform convergence to the classical Gaussian as the graininess tends to zero, and a localisation–admissibility condition arising from positive regressivity.
Abstract
We develop a radial basis function neural network (RBFNN) for dynamic equations on time scales using a Gaussian-type activation generated by the time-scale exponential. The activation ϕα,c=e⊖pα,c(·,c), with pα,c(τ)=α(τ−c), provides a natural time-scale analogue of the classical Gaussian. We establish two-sided exponential bounds, uniform convergence to the classical Gaussian as the graininess tends to zero, and a localisation–admissibility condition arising from positive regressivity. Training is based on residual minimisation with positive width parametrisation, an analytic Jacobian, and variable projection. Numerical experiments on quantum, clustering, and hybrid time scales, together with two-point boundary value problems and joint parameter identification, demonstrate high accuracy and reliable training. The clearest improvement over the classical Gaussian is observed on hybrid domains. The framework is also applied to irregularly sampled orange-tree growth and theophylline pharmacokinetic data, yielding parameter estimates consistent with recurrence-based and published continuous-model values. The proposed method provides a unified time-scale-adapted global surrogate with admissible-width control and a computable a posteriori error certificate.