Rational and neural network based approximations are efficient tools in modern approximation. These approaches are able to produce accurate approximations to nonsmooth and non-Lipschitz functions, including multivariate domain functions. In this paper we compare the efficiency of function approximation using rational approximation, neural network and their combinations. It was found that rational approximation is superior to neural network based approaches with the same number of decision variables. Our numerical experiments demonstrate the efficiency of rational approximation, even when the number of approximation parameters (that is, the dimension of the corresponding optimisation problems) is small. Another important contribution of this paper lies in the improvement of rational approximation algorithms. Namely, the optimisation based algorithms for rational approximation can be adjusted to in such a way that the conditioning number of the constraint matrices are controlled. This simple adjustment enables us to work with high dimension optimisation problems and improve the design of the neural network. The main strength of neural networks is in their ability to handle models with a large number of variables: complex models are decomposed in several simple optimisation problems. Therefore the the large number of decision variables is in the nature of neural networks.

翻译：有理逼近和神经网络逼近是现代逼近中的高效工具。这些方法能够对非光滑和非Lipschitz函数（包括多元域函数）产生精确的逼近。本文比较了使用有理逼近、神经网络及其组合进行函数逼近的效率。研究发现，在决策变量数量相同的情况下，有理逼近优于基于神经网络的逼近方法。我们的数值实验证明了有理逼近的高效性，即使逼近参数（即相应优化问题的维度）数量较少时也是如此。本文的另一重要贡献在于对有理逼近算法的改进。具体而言，基于优化的有理逼近算法可通过调整约束矩阵的条件数进行控制。这一简单的调整使我们能够处理高维优化问题，并改进神经网络的设计。神经网络的主要优势在于其处理大量变量模型的能力：复杂模型可分解为若干个简单的优化问题。因此，大量的决策变量是神经网络的固有特性。