Learning stochastic Hamiltonian systems via neural networks based on associated Fokker-Planck equations*

doi:10.7523/j.ucas.2025.001

Abstract

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

CLC Number:

O241.8

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.

References

[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.

Learning stochastic Hamiltonian systems via neural networks based on associated Fokker-Planck equations^*

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	WANG Yixuan, ZHANG Huai, SHI Yaolin, CHENG Shu. Medium-term prediction of earthquakes in Southern California using LSTM neural network [J]. Journal of University of Chinese Academy of Sciences, 2025, 42(2): 199-208.
[2]	HU Yinhan, GUO Tiande, HAN Congying. Solutions of cross-entropy loss with spectral decoupling regularization [J]. Journal of University of Chinese Academy of Sciences, 2025, 42(2): 268-275.
[3]	WANG Zhanpeng, WANG Lijin. Learning the parameters of a class of stochastic Lotka-Volterra systems with neural networks [J]. Journal of University of Chinese Academy of Sciences, 2025, 42(1): 20-25.
[4]	YUAN Ming, ZHAO Tong. A framework of graph classification with self-supervised heterogeneous graph neural network [J]. Journal of University of Chinese Academy of Sciences, 2024, 41(6): 830-841.
[5]	LI Xueyuan, HAN Congying. Solving quadratic assignment problem based on actor-critic framework [J]. Journal of University of Chinese Academy of Sciences, 2024, 41(2): 275-284.
[6]	LUO Zhongkai, ZHANG Libo. Learning path planning methods [J]. Journal of University of Chinese Academy of Sciences, 2024, 41(1): 11-27.
[7]	LI Yuan, ZHANG Qi, ZHU Jianming, JIAO Jianbin. Identification of core rumor spreaders in online social networks based on multi-stage deep model [J]. Journal of University of Chinese Academy of Sciences, 2024, 41(1): 136-144.
[8]	DONG Wenhao, ZHANG Huai. Lithology recognition of cuttings based on transfer learning [J]. Journal of University of Chinese Academy of Sciences, 2023, 40(6): 743-750.
[9]	LIU Zijian, JIANG Quanjiang, LIU Huijie. An adaptive limiting method in LEO satellite multi-beam system based on neural network [J]. Journal of University of Chinese Academy of Sciences, 2023, 40(5): 670-676.
[10]	ZHANG Meng, PAN Zhigang. SAR image change detection algorithm based on hierarchical fuzzy clustering and wavelet convolution neural network [J]. Journal of University of Chinese Academy of Sciences, 2023, 40(5): 637-646.
[11]	LIAN Huiqiang, LIU Bing, LI Pengyuan, YU Hua. A fuel price recommendation model based on the sliced recurrent neural network under sales constraints [J]. Journal of University of Chinese Academy of Sciences, 2023, 40(4): 566-576.
[12]	YAO Mufeng, ZAN Luyang, LI Baipeng, LI Qingting, CHEN Zhengchao. Building change detection from remote sensing images using CAR-Siamese net [J]. Journal of University of Chinese Academy of Sciences, 2023, 40(3): 380-387.
[13]	ZHUANG Zijun, YUAN Xiaobing, PEI Jun, WANG Guohui, LIU Jianpo. An unsupervised representation learning approach for modelling forest landform characteristics and fire susceptibility assessment [J]. Journal of University of Chinese Academy of Sciences, 2023, 40(2): 227-239.
[14]	HUO Xinyi, LI Yanlei, CHEN Longyong, ZHANG Fubo, SUN Wei. SAR few-sample target recognition method based on convolutional block attention module and capsule network [J]. Journal of University of Chinese Academy of Sciences, 2022, 39(6): 783-792.
[15]	CAO Zhe, FENG Shanshan, SUN Xian, HONG Wen. PolSAR terrain classification based on image segmentation and EM algorithm [J]. Journal of University of Chinese Academy of Sciences, 2022, 39(5): 639-647.