A novel structure automatic-determined Fourier extreme learning machine for generalized Black–Scholes partial differential equation

摘要

This paper focuses on the numerical solutions of generalized Black–Scholes partial differential equations and proposes a novel structure automatic-determined Fourier extreme learning machine (SAD-FELM) to solve the partial differential equations. In the SAD-FELM, the Fourier extreme learning machine (FELM) is designed. The product of two Fourier polynomials is used to serve as basis functions in the hidden layer. The optimal sampling frequencies of two Fourier polynomials depend on the distribution of data samples. Then, the improved extreme learning machine algorithm is used to solve the linear equation systems derived from the generalized Black–Scholes partial differential equations. Furthermore, to determine the optimal sampling frequencies and the number of neurons, this paper designs a structure automatic-determined algorithm based on metaheuristic algorithms. The research conducts comparative experiments among the SAD-FELMs based on different metaheuristic algorithms. The experimental results show that the numerical solutions obtained by the SAD-FELM based on the hunger games search algorithm are the most accurate. Finally, compared with other methods, the superiority and feasibility of the SAD-FELM are proved.