Bivariate rational approximations of the general temperature integral

Aghili, Alireza; Sukhorukova, Nadezda; Ugon, Julien

doi:10.1007/s10910-021-01273-z

Cited by 5 publications

(4 citation statements)

References 33 publications

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“…Currently, the most popular optimisation methods for rational approximation are the bisection method for quasiconvex functions and the differential correction method. The advantage of these two methods is that these methods can be easily extended to the case of non-monomial cases (generalised rational approximation) [8,26] and to multivariate settings [2]. Overall, the differential correction method has a quadratic convergence [4], while the bisection method converges linearly [7].…”

Section: Rational Approximationmentioning

confidence: 99%

“…Our activation function is a rational function of degree (3,2): the rational approximation is the ratio of two polynomials, the degree of the numerator is 3 and the degree of the denominator is 2.…”

Section: Neural Network With Rational Approximation To Relu Activationmentioning

confidence: 99%

“…In this case, our activation function is a rational function of degree (3,2). The coefficients of the rational activation function are now a part of the parameter set.…”

Section: Neural Network With Rational Activationmentioning

confidence: 99%

“…Figure 28 The approximation appears to be accurate even around the "difficult point" when the number of epochs is 200. In this section, our activation function is a rational function of degree (3,2). The coefficients are now a part of the parameter set.…”

Section: A13 Neural Network With Rational Activationmentioning

confidence: 99%

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A comparison of rational and neural network based approximations

Peiris¹,

Millán²,

Sukhorukova³

et al. 2023

Preprint

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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.

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Section: Rational Approximationmentioning

confidence: 99%

Section: Neural Network With Rational Approximation To Relu Activationmentioning

confidence: 99%

“…In this case, our activation function is a rational function of degree (3,2). The coefficients of the rational activation function are now a part of the parameter set.…”

Section: Neural Network With Rational Activationmentioning

confidence: 99%

Section: A13 Neural Network With Rational Activationmentioning

confidence: 99%

See 2 more Smart Citations

A comparison of rational and neural network based approximations

Peiris¹,

Millán²,

Sukhorukova³

et al. 2023

Preprint

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Uniqueness of solutions in multivariate Chebyshev approximation problems

Roshchina,

Sukhorukova,

Ugon

2023

Optim Lett

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We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension of the solution set and show that nonuniqueness is generic for ill-posed problems on discrete domains. Moreover, given a prescribed set of points of minimal and maximal deviation we construct a function for which the dimension of the set of best approximating polynomials is maximal for any choice of domain. We also present several examples that illustrate the aforementioned phenomena, demonstrate practical application of our results and propose a number of open questions.

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