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：本文研究前馈结构的单隐层人工神经网络（亦称浅层或双层网络），其结构由神经元类型与数量决定。定义函数所需的参数（即训练过程）通过求解逼近问题实现，即基于特定节点集的插值约束进行参数确定。我们提出采用极端学习机（ELM）方法训练参数的方案，该方法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。重点研究当采样插值节点分别为等距节点、切比雪夫节点及随机节点时，网络在给定采样节点外的插值精度。该研究源于经典的钟形龙格实例，该实例表明全局插值多项式仅在使用合适节点（如切比雪夫节点）训练时才能保证精度。为评估插值节点数量增长时的行为特征，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具有不同正则性的经典测试函数进行验证。结果表明：全局多项式逼近精度仅在使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呈衰减趋势，且多数情况下其收敛性遵循多项式在切比雪夫节点上的收敛规律，无论训练时使用的是何种节点集。