Solving differential equations with Fourier series and Evolution Strategies

Chaquet, José M.; Carmona, Enrique J.

doi:10.1016/j.asoc.2012.05.014

Cited by 27 publications

(20 citation statements)

References 14 publications

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“…Considering the same notation as that of Jose [56], we can describe the above linear or nonlinear ordinary differential equations in the following general form:…”

Section: Higher-order Odes and Higher-order Sode Problemmentioning

confidence: 99%

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

2018

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This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden-Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time.

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“…Considering the same notation as that of Jose [56], we can describe the above linear or nonlinear ordinary differential equations in the following general form:…”

Section: Higher-order Odes and Higher-order Sode Problemmentioning

confidence: 99%

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

2018

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“…Neural networks, trained for local fluctuant data relations of actual weather conditions in the last few days, with reference to the numerical prognoses of the same data types, may stand for the standard post-processing methods; the proposed method revision results are presented in the Section 3. Differential equations can describe physical or natural systems which it is difficult to model by unique exact functions; the solutions can apply power [8] or wave series [9], genetic programming [10] and neural networks [11]. Polynomial networks may be extended to apply some mathematical principles (Section 2) to form and substitute for general differential equations [12].…”

Section: Introductionmentioning

confidence: 99%

Wind speed forecast correction models using polynomial neural networks

Zjavka

2015

Renewable Energy

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a b s t r a c tAccurate short-term wind speed forecasting is important for the planning of a renewable energy power generation and utilization, especially in grid systems. In meteorology it is usual to improve the forecasts by means of some post-processing methods using local measurements and weather prediction model outputs. Neural networks, trained with local real data observations can improve short-term wind speed forecasts with respect to meso-scale numerical meteorological model outcomes of the same data types in the majority of cases. Large-scale forecast models are based on the numerical integration of differential equation systems, which can describe atmospheric circulation processes on account of global meteorological observations. Several layer 3D complex models, which involve a large number of matrix variables, cannot exactly describe conditions near the ground, highly influenced by a landscape relief, coast, structure and other factors. Polynomial neural networks can form and solve general differential equations, which allow to model real complex systems by means of substitution derivative term sum series. The proposed adaptive method forms a correction function according to real observations and consequently applies forecasts to revise a desired prognosis in a selected locality.

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“…Differential equations are generally able to describe most of physical or natural complex systems, which it is difficult to model by unique explicit functions. The solutions can apply power [2] or wave series [6], fractional calculus [11] [9] or standard soft-computing techniques [4] [12]. These make use in common of straight computational composing methods, which operate only within absolute interval values of input variables, e.g.…”

Section: Introductionmentioning

confidence: 99%

"Aladin" Weather Model Local Revisions Using the Differential Polynomial Neural Network

Zjavka¹

2014

NNW

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This tutorial is based on modification of the professor nomination lecture presented two years ago in front of the Scientific Council of the Czech Technical University in Prague [16].It is devoted to the techniques for the models developing suitable for processes forecasting in complex systems. Because of the high sensitivity of the processes to the initial conditions and, consequently, due to our limited possibilities to forecast the processes for the long-term horizon, the attention is focused on the techniques leading to practical applications of the short term prediction models. The aim of this tutorial paper is to bring attention to possible difficulties which designers of the predicting models and their users meet and which have to be solved during the prediction model developing, validation, testing, and applications. The presented overview is not complete, it only reflects the author's experience with developing of the prediction models for practical tasks solving in banking, meteorology, air pollution and energy sector. The paper is completed by an example of the global solar radiation prediction which forms an important input for the electrical energy production forecast from renewable sources. The global solar radiation forecasting is based on numerical weather prediction models. The time-lagged ensemble technique for uncertainty quantification is demonstrated on a simple example.

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Solving differential equations with Fourier series and Evolution Strategies

Cited by 27 publications

References 14 publications

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

Wind speed forecast correction models using polynomial neural networks

"Aladin" Weather Model Local Revisions Using the Differential Polynomial Neural Network

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