TY - JOUR
T1 - Statistical solution to SDEs with α -stable Lévy noise via deep neural network
AU - Zhang, Hao
AU - Xu, Yong
AU - Li, Yongge
AU - Kurths, Jürgen
N1 - Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/12/1
Y1 - 2020/12/1
N2 - The probability density function of the stochastic differential equations with α-stable Lévy noise is determined by the deterministic fractional Fokker–Planck (FFP) equation with the Riesz fractional derivative. In this paper, to solve the FFP equation, a new algorithm which is an efficient numerical approach using the deep neural network is proposed in view of the mesh-dependence phenomenon in the traditional discrete scheme and the nonsmooth solution of the Monte Carlo method. Under the framework of the DL-FP algorithm, the “fractional centered derivative” approach is applied to approximate the Riesz fractional derivative of the output in the neural network, which is the major novelty of this approach. Numerical results are presented to demonstrate the accuracy of the approach. Comparing with the traditional discrete scheme and the Monte Carlo method, the proposed algorithm is mesh-less and smoother. Furthermore, the proposed technique performs well in dealing with the heavy-tail case through comparing with the analytical solution.
AB - The probability density function of the stochastic differential equations with α-stable Lévy noise is determined by the deterministic fractional Fokker–Planck (FFP) equation with the Riesz fractional derivative. In this paper, to solve the FFP equation, a new algorithm which is an efficient numerical approach using the deep neural network is proposed in view of the mesh-dependence phenomenon in the traditional discrete scheme and the nonsmooth solution of the Monte Carlo method. Under the framework of the DL-FP algorithm, the “fractional centered derivative” approach is applied to approximate the Riesz fractional derivative of the output in the neural network, which is the major novelty of this approach. Numerical results are presented to demonstrate the accuracy of the approach. Comparing with the traditional discrete scheme and the Monte Carlo method, the proposed algorithm is mesh-less and smoother. Furthermore, the proposed technique performs well in dealing with the heavy-tail case through comparing with the analytical solution.
KW - Deep neural network
KW - Fractional Fokker–Planck
KW - Fractional centered derivative
KW - Heavy-tail
KW - Riesz fractional derivative
UR - http://www.scopus.com/inward/record.url?scp=85090755852&partnerID=8YFLogxK
U2 - 10.1007/s40435-020-00677-0
DO - 10.1007/s40435-020-00677-0
M3 - 文章
AN - SCOPUS:85090755852
SN - 2195-268X
VL - 8
SP - 1129
EP - 1140
JO - International Journal of Dynamics and Control
JF - International Journal of Dynamics and Control
IS - 4
ER -