In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. The focus is then on the accuracy of the interpolation outside of the given sampling interpolation nodes when they are the equispaced, the Chebychev, and the randomly selected ones. The study is motivated by the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when growing the number of interpolation nodes, we raise the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge's function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training.

翻译：本文研究具有前馈架构的单隐层人工神经网络（也称为浅层网络或双层网络），其结构由神经元的数量和类型决定。通过求解逼近问题（即在特定节点集上施加插值条件）来完成函数参数（即训练）的确定。我们提出了一种采用极限学习机（ELM）训练参数的方案，该方案将问题转化为线性插值问题。在此假设下，人工神经网络插值函数的存在性得以保证。研究重点在于当采样插值节点为等距节点、切比雪夫节点以及随机选取节点时，插值在给定节点外的精度。该研究源于著名的钟形龙格示例——该例清楚地表明，全局插值多项式仅在适当选取的节点（如切比雪夫节点）上训练时才能实现高精度。为评估随插值节点数量增加时的行为，我们增加网络神经元数量并与插值多项式进行对比。采用龙格函数及其他不同正则性的经典示例进行测试。结果表明：全局多项式的逼近精度仅在采用切比雪夫节点时提升；而人工神经网络插值函数的误差始终衰减，且在多数情况下，其收敛特性与基于切比雪夫节点的多项式插值情形一致，尽管训练所采用的节点集可能不同。