利用神经网络基于Fokker-Planck方程学习随机Hamilton系统

doi:10.7523/j.ucas.2025.001

摘要/Abstract

摘要： 本文中我们提出一种利用神经网络从观测数据学习随机Hamilton系统的方法,该方法基于与系统相关联的Fokker-Planck方程的时间半离散,以及随机Hamilton系统的解的期望。我们学习的目标是系统的漂移和扩散Hamilton函数,这使得通过学习可以获得系统的辛结构,进而实现具有较好精度的辛预测。数值实验展示了所提出的方法的有效性。

关键词: 随机Hamilton系统, 神经网络, Fokker-Planck方程, 辛积分子

Abstract: In this paper we propose a method of learning stochastic Hamiltonian systems (SHSs) from observational data via neural networks, based on the temporal semi-discretization of the Fokker-Planck equations associated to the systems, as well as the expectations of the SHSs’ solutions. Our learning targets are the drift and diffusion Hamiltonian functions of the systems, which enables capturing the symplectic structure of the systems by the learning, and consequently ensures symplectic prediction that possesses good accuracy. Numerical experiments demonstrate effectiveness of the proposed method.

Key words: stochastic Hamiltonian systems, neural networks, Fokker-Planck equation, symplectic integrators

中图分类号:

O241.8

陈格格, 程旭鹏, 王丽瑾. 利用神经网络基于Fokker-Planck方程学习随机Hamilton系统[J]. 中国科学院大学学报, DOI: 10.7523/j.ucas.2025.001.

CHEN Gege, CHENG Xupeng, WANG Lijin. Learning stochastic Hamiltonian systems via neural networks based on associated Fokker-Planck equations^*[J]. Journal of University of Chinese Academy of Sciences, DOI: 10.7523/j.ucas.2025.001.

参考文献

[1] Itô K.Differential equations determining a Markoff process[J]. Pan-Japan Math Coll, 1942, 244(1077): 1352-1400.
[2] Arnold L.Stochastic differential equations: theory and applications[M]. New York: Wiley, 1974.
[3] Zhu W Q, Cai G Q.Introduction to stochastic dynamics in Chinese[M]. Beijing: Science Press Bijing, 2017.
[4] Feng K.On difference schemes and symplectic geometry[C/OL] // Proceedings of the 1984 Beijing Symposium on differential geometry & differential equations. Beijing: Science Press Beijing, 1985: 42-58 (1985) [2024-10-18] . https://mathscinet.ams.org/mathscinet/article?mr=824483.
[5] Feng K, Qin M Z.Symplectic geometric algorithms for Hamiltonian systems in Chinese[M]. Hangzhou: Zhejiang science and technology publishing house, 2003.
[6] Hairer E, Lubich C, Wanner G.Symplectic integration of Hamiltonian systems[M] // Hairer E, Lubich C, Wanner G. Geometric numerical integration. Berlin: Springer, 2006: 179-236. DOI: 10.1007/3-540-30666-8_6.
[7] Bismut J M, Gross L, Krickeberg K. Ecole d'Eté de Probabilités de Saint-Flour X—1980[M/OL]. Berlin: Springer, 1982 [2024-10-18] . https://link.springer.com/book/10.1007/BFb0095617.
[8] Milstein G N, Repin Y M, Tretyakov M V.Symplectic integration of Hamiltonian systems with additive noise[J]. SIAM Journal on Numerical Analysis, 2002a, 39(6): 2066-2088. DOI: 10.1137/S0036142901387440.
[9] Milstein G N, Repin Y M, Tretyakov M V.Numerical methods for stochastic systems preserving symplectic structure[J]. SIAM Journal on Numerical Analysis, 2002b, 40(4): 1583-1604. DOI: 10.1137/S0036142901395588.
[10] Wang L J.Variational integrators and generating functions for stochastic Hamiltonian systems[D]. Karlsruhe: KIT Scientific Publishing, 2007. DOI: 10.5445/KSP/1000007007.
[11] Deng J, Anton C, Wong Y S.High-order symplectic schemes for stochastic Hamiltonian systems[J]. Communications in Computational Physics, 2014, 16(1): 169-200. DOI: 10.4208/cicp.311012.191113a.
[12] Bou-Rabee N, Owhadi H.Stochastic variational integrators[J]. IMA Journal of Numerical Analysis, 2008, 29(2): 421-443. DOI: 10.1093/imanum/drn018.
[13] Hong J L, Sun L Y.Symplectic integration of stochastic Hamiltonian systems[M]. Singapore: Springer, 2022. DOI: 10.1007/978-981-19-7670-4.
[14] Chen R T Q, Rubanova Y, Bettencourt J, et al. Neural ordinary differential equations[EB/OL].2019 : 1806.07366v5.(2018-07-19) [2024-10-18] . https://arxiv.org/abs/1806.07366v5.
[15] Greydanus S, Dzamba M, Yosinski J. Hamiltonian neural networks[EB/OL].2019: 1906.01563v3.(2019-06-04) [2024-10-18] . https://arxiv.org/abs/1906.01563v3.
[16] Chen Z D, Zhang J Y, Arjovsky M, et al. Symplectic recurrent neural networks[EB/OL].2020: 1909.13334v2.(2019-09-29) [2024-10-18] . https://arxiv.org/abs/1909.13334v2.
[17] Zhu A Q, Jin P Z, Tang Y F.Deep Hamiltonian neural networks based on symplectic integrators in Chinese[J]. Mathematica Numerica Sinica, 2020, 42(3): 370-384. DOI: 10.12286/jssx.2020.3.370.
[18] Tong Y J, Xiong S Y, He X Z, et al.Symplectic neural networks in Taylor series form for Hamiltonian systems[J]. Journal of Computational Physics, 2021, 437: 110325. DOI: 10.1016/j.jcp.2021.110325.
[19] David M, Méhats F.Symplectic learning for Hamiltonian neural networks[J]. Journal of Computational Physics, 2023, 494: 112495. DOI: 10.1016/j.jcp.2023.112495.
[20] Jin P Z, Zhang Z, Zhu A Q, et al.SympNets: intrinsic structure preserving symplectic networks for identifying Hamiltonian systems[J]. Neural Networks, 2020, 132: 166-179. DOI: 10.1016/j.neunet.2020.08.017.
[21] Xiong S Y, Tong Y J, He X Z, et al. Nonseparable symplectic neural networks[EB/OL].2022: 2010.12636v3.(2020-10-23) [2024-10-18] . https://arxiv.org/abs/2010.12636v3.
[22] Bertalan T, Dietrich F, Mezić I, et al.On learning Hamiltonian systems from data[J]. Chaos (Woodbury, N.Y.), 2019, 29(12): 121107. DOI: 10.1063/1.5128231.
[23] Sæmundsson S, Terenin A, Hofmann K, et al. Variational integrator networks for physically structured embeddings[EB/OL].2020: 1910.09349v2.(2019-10-21) [2024-10-18] . https://arxiv.org/abs/1910.09349v2.
[24] Chen R Y, Tao M L. Data-driven prediction of general Hamiltonian dynamics via learning exactly-symplectic maps[EB/OL].2021: 2103.05632v2.(2021-03-09) [2024-10-18] . https://arxiv.org/abs/2103.05632v2.
[25] Chen Y H, Matsubara T, Yaguchi T. Neural symplectic form: learning Hamiltonian equations on general coordinate systems[C/OL] // Proceedings of the 35th International Conference on Neural Information Processing Systems. New York: Curran Associates, 2024: 16659 - 16670 (2024) [2024-10-18] . https://dl.acm.org/doi/10.5555/3540261.3541535.
[26] Liu X Q, Xiao T S, Si S, et al. Neural SDE: stabilizing neural ODE networks with stochastic noise[EB/OL].2019: 1906.02355.(2019-06-05) [2024-10-18] . https://arxiv.org/abs/1906.02355.
[27] Tzen B, Raginsky M. Neural stochastic differential equations: Deep latent gaussian models in the diffusion limit[EB/OL].2019: 1905.09883v2.(2019-05-23) [2024-10-18] . https://arxiv.org/abs/1905.09883v2.
[28] Ryder T, Golightly A, McGough A S, et al. Black-box variational inference for stochastic differential equations[EB/OL].2018: 1802.03335v3.(2018-02-09) [2024-10-18] . https://arxiv.org/abs/1802.03335v3.
[29] Opper M.Variational inference for stochastic differential equations[J]. Annalen Der Physik, 2019, 531(3): 1-9. DOI: 10.1002/andp.201800233.
[30] Dai M, Duan J Q, Hu J Y, et al.Variational inference of the drift function for stochastic differential equations driven by Lévy processes[J]. Chaos (Woodbury, N.Y.), 2022, 32(6): 061103. DOI: 10.1063/5.0095605.
[31] Dietrich F, Makeev A, Kevrekidis G, et al.Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning[J]. Chaos (Woodbury, N.Y.), 2023, 33(2): 023121. DOI:10.1063/5.0113632.
[32] Fang C, Lu Y B, Gao T, et al.An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable Lévy noise[J]. Chaos (Woodbury, N.Y.), 2022, 32(6): 063112. DOI: 10.1063/5.0089832.
[33] Yang L X, Gao T, Lu Y B, et al.Neural network stochastic differential equation models with applications to financial data forecasting[J]. Applied Mathematical Modelling, 2023, 115: 279-299. DOI: 10.1016/j.apm.2022.11.001.
[34] Deng R Z, Chang B, Brubaker M A, et al. Modeling continuous stochastic processes with dynamic normalizing flows[EB/OL].2021: 2002.10516v4.(2020-02-24) [2024-10-18] . https://arxiv.org/abs/2002.10516v4.
[35] 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, 2020: 5231-5237. DOI: 10.1109/IROS45743.2020.9341035.
[36] Papamakarios G, Nalisnick E, Rezende D J, et al. Normalizing flows for probabilistic modeling and inference[EB/OL].2021: 1912.02762v2.(2019-12-05) [2024-10-18] . https://arxiv.org/abs/1912.02762v2.
[37] Guo L, Wu H, Zhou T.Normalizing field flows: solving forward and inverse stochastic differential equations using physics-informed flow model[J]. Journal of Computational Physics, 2022, 461: 111202. DOI: 10.1016/j.jcp.2022.111202.
[38] Li Y, Duan J Q.A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise[J]. Physica D: Nonlinear Phenomena, 2021, 417: 132830. DOI: 10.1016/j.physd.2020.132830.
[39] Chen X L, Yang L, Duan J Q, et al.Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks[J]. SIAM Journal on Scientific Computing, 2021, 43(3): B811-B830. DOI: 10.1137/20M1360153.
[40] Solin A, Tamir E, Verma P. Scalable inference in SDEs by direct matching of the Fokker-Planck-Kolmogorov equation[EB/OL].2021: 2110.15739.(2021-10-29) [2024-10-18] . https://arxiv.org/abs/2110.15739.
[41] Lu Y B, Li Y, Duan J Q.Extracting stochastic governing laws by non-local Kramers-Moyal formulae[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2022, 380(2229). DOI: 10.1098/rsta.2021.0195.
[42] Chen X L, Wang H, Duan J Q.Detecting stochastic governing laws with observation on stationary distributions[J]. Physica D: Nonlinear Phenomena, 2023, 448: 133691. DOI: 10.1016/j.physd.2023.133691.
[43] Wang Z P, Wang L J. Learning parameters of a class of stochastic Lotka-Volterra systems with neural networks[EB/OL]. Journal of University of Chinese Academy of Sciences. (2023-03-21) [2024-10-18] . DOI: 10.7523/j.ucas.2023.012.
[44] Wang Z P, Wang L J, Cao Y Z. Learning a class of stochastic differential equations via numerics-informed Bayesian denoising[J/OL]. International Journal for Uncertainty Quantification, 2024. [2024-10-18] . DOI: 10.1615/Int.J.UncertaintyQuantification.2024052020.
[45] Cheng X P, Wang L J. Learning stochastic Hamiltonian systems via neural network and numerical quadrature formulae[J/OL]. Journal of University of Chinese Academy of Sciences. (2024-09-30) [2024-10-18] . DOI: 10.7523/j.ucas.2024.074.
[46] Cheng X P, Wang L J, Cao Y Z.Quadrature based neural network learning of stochastic Hamiltonian systems[J]. Mathematics, 2024, 12(16): 2438. DOI: 10.3390/math12162438.
[47] Chen Y, Xiu D B.Learning stochastic dynamical system via flow map operator[J]. Journal of Computational Physics, 2024, 508: 112984. DOI: 10.1016/j.jcp.2024.112984.
[48] Kloeden P E, Platen E.Numerical solution of stochastic differential equations[M]. Berlin: Springer-Verlag, 1992. DOI: 10.1007/978-3-662-12616-5.