Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.8.2026.3.07
Volume 8, Issue 3, 1 December 2026, Pages 439-463

 

Abstract. Neural networks (NNs) are popular techniques in modern science and engineering applications, where some type of approximation is required. These approaches are able to produce accurate approximations to nonsmooth and non-Lipschitz functions, including multivariate domain functions. Essentially, NNs are approximation tools, inspired by rigorous mathematical approaches and there are still many open problems of purely mathematical nature. Modern computer packages are designed in such way that the construction of approximations by NNs is straightforward for the users and they do not need to know the mathematics behind the scene. This seems to be convenient, but eventually most users would want to open the black-box and improve it. One possibility is to use rational approximation, which combines the high approximation accuracy and the simplicity of the optimisation tools: the algorithms are based on the repeated application of standard linear programming techniques that are part of most modern computer packages. In this paper, we compare the efficiency of function approximation using rational approximation, neural network and their combinations. 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, while for neural networks a higher number of decision variables is required. We demonstrate our findings in numerical examples, one of them is approximating solutions of the KdV (Korteweg-de Vries) equation which appears in fluid dynamics.

 

How to Cite this Article:
V. Peiris, N. Sukhorukova, R.D. Millán, J. Ugon, A comparison of rational approximations, neural networks, and their combinations in science and engineering applications, J. Appl. Numer. Optim. 8 (2026), 439-463.