利用神经网络学习习一一类随机Lotka-Volterra系统的参数

doi:10.7523/j.ucas.2023.012

摘要/Abstract

摘要： 本文中,我们提出一种利用神经网络来学习一类随机Lotka-Volterra系统参数的方法。我们利用随机微分方程的Euler-Maruyama离散来近似推导出观测变量的期望和协方差矩阵,并在此基础上建立损失函数。在训练网络中我们使用了随机梯度下降方法。数值实验展示了我们算法的有效性。

关键词: 随机Lotka-Volterra系统, 神经网络, Euler-Maruyama格式, 参数估计

Abstract: In this paper, we propose a neural network approach to learn the parameters of a class of stochastic Lotka-Volterra systems. Approximations of the mean and covariance matrix of the observational variable are obtained from the Euler-Maruyama discretization of the underlying SDE, based on which the loss function is built. Stochastic gradient decent method is applied in the neural network training. Numerical experiments demonstrate the effectiveness of our method.

Key words: stochastic Lotka-Volterra systems, neural networks, Euler-Maruyama scheme, parameter estimation

中图分类号:

O241.8

汪展鹏, 王丽瑾. 利用神经网络学习习一一类随机Lotka-Volterra系统的参数[J]. 中国科学院大学学报, DOI: 10.7523/j.ucas.2023.012.

WANG Zhanpeng, WANG Lijin. Learning Parameters of a Class of Stochastic Lotka-Volterra Systems with Neural Networks^*[J]. Journal of University of Chinese Academy of Sciences, DOI: 10.7523/j.ucas.2023.012.

参考文献

[1] Jiang D Q, Ji C Y, Li X Y, et al.Analysis of autonomous Lotka-Volterra competition systems with random perturbation[J]. Journal of Mathematical Analysis and Applications, 2012, 390(2): 582-595. DOI:10.1016/j.jmaa.2011.12.049.
[2] Mao X R, Marion G, Renshaw E.Environmental Brownian noise suppresses explosions in population dynamics[J]. Stochastic Processes and Their Applications, 2002, 97(1): 95-110. DOI:10.1016/S0304-4149(01)00126-0.
[3] Zhou L N, Pan S M, Wang J W, et al.Machine learning on big data: Opportunities and challenges[J]. Neurocomputing, 2017, 237: 350-361. DOI:10.1016/j.neucom.2017.01.026.
[4] Raissi M, Perdikaris P, Karniadakis G E. Multistep neural networks for data-driven discovery of nonlinear dynamical systems[EB/OL].2018: arXiv: 1801.01236.(2018-01-04) [2023-01-11]. https://arxiv.org/abs/1801.01236.
[5] Chen R T Q, Rubanova Y, Bettencourt J, et al. Neural ordinary differential equations[EB/OL].2018: arXiv: 1806.07366.(2019-12-14) [2023-01-11]. https://arxiv.org/abs/1806.07366.
[6] Gholami A, Keutzer K, Biros G.ANODE: Unconditionally accurate memory-efficient gradients for neural ODEs[C]//Proceedings of the 28th International Joint Conference on Artificial Intelligence. August 10-16, 2019, Macao, China. New York: ACM, 2019: 730-736. DOI: 10.24963/ijcai.2019/103.
[7] Urain J, Ginesi M, Tateo D, et al.ImitationFlow: learning deep stable stochastic dynamic systems by normalizing flows[C]//2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). October 24 - January 24, 2021, Las Vegas, NV, USA. IEEE, 2021: 5231-5237. DOI:10.1109/IROS45743.2020.9341035.
[8] Kong L K, Sun J M, Zhang C. SDE-net: Equipping deep neural networks with uncertainty estimates[EB/OL].2020: arXiv: 2008.10546.(2020-08-24) [2023-01-11]. https://arxiv.org/abs/2008.10546.
[9] Yildiz C, Heinonen M, Intosalmi J, et al.Learning stochastic differential equations with Gaussian processes without gradient matching[C]//2018 IEEE 28th International Workshop on Machine Learning for Signal Processing (MLSP). September 17-20, 2018, Aalborg, Denmark. IEEE, 2018: 1-6. DOI:10.1109/MLSP.2018.8516991.
[10] Bento J, Ibrahimi M, Montanari A.Learning networks of stochastic differential equations[C]//Proceedings of the 23rd International Conference on Neural Information Processing Systems - Volume 1. December 6 - 9, 2010, Vancouver, British Columbia, Canada. New York: ACM, 2010: 172-180. DOI: 10.48550/arXiv.1011.0415.
[11] Tzen B, Raginsky M. Neural stochastic differential equations: Deep latent gaussian models in the diffusion Limit[EB/OL].2019: arXiv: 1905.09883.(2019-10-28) [2023-01-11]. https://arxiv.org/abs/1905.09883.
[12] Kloppers P H, Greeff J C.Lotka-Volterra model parameter estimation using experiential data[J]. Applied Mathematics and Computation, 2013, 224: 817-825. DOI:10.1016/j.amc.2013.08.093.
[13] Wu L F, Wang Y N.Estimation the parameters of Lotka-Volterra model based on grey direct modelling method and its application[J]. Expert Systems With Applications, 2011, 38(6): 6412-6416. DOI:10.1016/j.eswa.2010.09.013.
[14] Wu L F, Liu S F, Wang Y N.Grey Lotka-Volterra model and its application[J]. Technological Forecasting and Social Change, 2012, 79(9): 1720-1730. DOI:10.1016/j.techfore.2012.04.020.
[15] Xu L, Liu J Y, Zhang G.Pattern formation and parameter inversion for a discrete Lotka-Volterra cooperative system[J]. Chaos, Solitons & Fractals, 2018, 110: 226-231. DOI:10.1016/j.chaos.2018.03.035.
[16] Xu L, Lou S S, Xu P Q, et al.Feedback control and parameter invasion for a discrete competitive Lotka-Volterra system[J]. Discrete Dynamics in Nature and Society, 2018, 2018: 1-8. DOI:10.1155/2018/7473208.
[17] Shakurov I R, Asadullin R M.Parameter identification for systems of nonlinear differential equations by the example of Lotka-Volterra model[J]. Biophysics, 2014, 59(2): 339-340.
[18] Waniewski J, Jedruch W.Individual based modeling and parameter estimation for a Lotka-Volterra system[J]. Mathematical Biosciences, 1999, 157(1/2): 23-36. DOI: 10.1016/S0025-5564(98)10075-5.
[19] Luus R.Parameter estimation of Lotka-Volterra problem by direct search optimization[J]. Hungarian Journal of Industrial Chemistry, 1998, 26(4): 287-292.
[20] Higham D J.An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM Review, 2001, 43(3): 525-546. DOI:10.1137/S0036144500378302.
[21] Feller W.An introduction to probability theory and its applications[M]. 3rd ed. New York: Wiley, 1968.
[22] Paszke A, Gross S, Massa F, et al. PyTorch: An imperative style, high-performance deep learning library[EB/OL].2019: arXiv: 1912.01703.(2019-12-14) [2023-01-11]. https://arxiv.org/abs/1912.01703.