Learning stochastic Hamiltonian systems via symplectic-constrained autoencoders&#x0002A;

CHEN Chen; WANG Lijin

doi:10.7523/j.ucas.2026.029

Journal of University of Chinese Academy of Sciences >

0 20260509

DOI: https://doi.org/10.7523/j.ucas.2026.029

Learning stochastic Hamiltonian systems via symplectic-constrained autoencoders^*

CHEN Chen ,
WANG Lijin

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School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 2026-01-05

Revised date: 2026-05-08

Online published: 2026-05-09

Supported by

^*National Key Research and Development Project of China 2024YFA1013101

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Abstract

Learning the phase flow mapping of stochastic Hamiltonian systems (SHSs) from observed data has been attracting growing attention in computational physics and machine learning fields in recent years. In this paper, we propose a symplectic-constrained autoencoder approach, extending the stochastic flow map learning (sFML) framework to the structure-preserving learning of the flow mapping of SHSs. The proposed method extracts latent random variables following a standard normal distribution via the encoder and reconstructs the state evolution mapping through the decoder, with the loss function integrating symplectic constraint innovatively to ensure the preservation of symplectic structure. Numerical experiments conducted on the Kubo oscillator validate the superiority of our approach in comparison with the benchmark sFML model.

Key words： stochastic Hamiltonian systems; autoencoder neural networks; symplectic integrators

Cite this article

CHEN Chen , WANG Lijin . Learning stochastic Hamiltonian systems via symplectic-constrained autoencoders^*[J]. Journal of University of Chinese Academy of Sciences, 0 : 20260509 . DOI: 10.7523/j.ucas.2026.029

References

[1] Feng K, Qin M Z.Symplectic Geometric Algorithms for Hamiltonian Systems[M]. Berlin, Heidelberg: Springer, 2010. DOI:10.1007/978-3-642-01777-3.
[2] Marsden J E, Ratiu T S.Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems[M]. New York, NY: Springer New York, 1999. DOI:10.1007/978-0-387-21792-5.
[3] Milstein G N, Repin Y M, Tretyakov M V.Symplectic integration of Hamiltonian systems with additive noise[J]. SIAM Journal on Numerical Analysis, 2002, 39(6): 2066-2088. DOI:10.1137/S0036142901387440.
[4] Milstein G N, Repin Y M, Tretyakov M V.Numerical methods for stochastic systems preserving symplectic structure[J]. SIAM Journal on Numerical Analysis, 2003, 40(4): 1583-1604. DOI:10.1137/S0036142901395588.
[5] Wang L J.Variational integrators and generating functions for stochastic Hamiltonian systems[D]. Karlsruhe: KIT Scientific Publishing, 2007. DOI:10.5445/KSP/1000007007.
[6] Hong J L, Scherer R, Wang L J.Predictor-corrector methods for a linear stochastic oscillator with additive noise[J]. Mathematical and Computer Modelling, 2007, 46(5/6): 738-764. DOI:10.1016/j.mcm.2006.12.009.
[7] 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.
[8] Bou-Rabee N, Owhadi H.Stochastic variational integrators[J]. IMA Journal of Numerical Analysis, 2008, 29(2): 421-443. DOI:10.1093/imanum/drn018.
[9] Hong J L, Sun L Y.Symplectic Integration of Stochastic Hamiltonian Systems[M]. Singapore: Springer Nature Singapore, 2022. DOI:10.1007/978-981-19-7670-4.
[10] Chen R T Q, Rubanova Y, Bettencourt J, et al. Neural ordinary differential equations[EB/OL].2018: arXiv: 1806.07366.(2018-06-19)[2025-12-18]. https://arxiv.org/abs/1806.07366.
[11] Greydanus S, Dzamba M, Yosinski J. Hamiltonian neural networks[EB/OL].2019: arXiv: 1906.01563.(2019-06-04)[2025-12-18]. https://arxiv.org/abs/1906.01563.
[12] Chen Z D, Zhang J Y, Arjovsky M, et al. Symplectic recurrent neural networks[EB/OL].2019: arXiv: 1909.13334.(2019-09-29)[2025-12-18]. https://arxiv.org/abs/1909.13334.
[13] Zhu A Q, Jin P Z, Tang Y F. Deep Hamiltonian networks based on symplectic integrators[EB/OL].2020: arXiv: 2004.13830.(2020-04-23)[2025-12-18]. https://arxiv.org/abs/2004.13830.
[14] 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.
[15] Chen R Y, Tao M L. Data-driven prediction of general Hamiltonian dynamics via learning exactly-symplectic maps[EB/OL].2021: arXiv: 2103.05632.(2021-03-09)[2025-12-18]. https://arxiv.org/abs/2103.05632.
[16] Saemundsson S, Terenin A, Hofmann K, et al. Variational integrator networks for physically structured embeddings[EB/OL].2019: arXiv: 1910.09349.(2019-10-21)[2025-12-18]. https://arxiv.org/abs/1910.09349.
[17] 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.
[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] Dietrich F, Makeev A, Kevrekidis G, et al.Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning[J]. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2023, 33(2): 023121. DOI:10.1063/5.0113632.
[20] Wang Z P, Wang L J.Learning 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. DOI:10.7523/j.ucas.2023.012.
[21] Wang Z P, Wang L J, Cao Y Z.Learning a class of stochastic differential equations via numerics-informed Bayesian denoising[J]. International Journal for Uncertainty Quantification, 2025, 15(3): 1-20. DOI: 10.1615/Int.J.UncertaintyQuantification.2024052020.
[22] 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.
[23] Xu Z S, Chen Y, Chen Q F, et al.Modeling unknown stochastic dynamical system via autoencoder[J]. Journal of Machine Learning for Modeling and Computing, 2024, 5(3): 87-112. DOI:10.1615/JMachLearnModelComput.2024055773.
[24] Tzen B, Raginsky M. Neural stochastic differential equations: Deep latent Gaussian models in the diffusion limit[EB/OL].2019: arXiv: 1905.09883.(2019-05-23)[2025-12-18]. https://arxiv.org/abs/1905.09883.
[25] Ryder T, Golightly A, McGough A S, et al. Black-box variational inference for stochastic differential equations[EB/OL].2018: arXiv: 1802.03335.(2018-02-09)[2025-12-18]. https://arxiv.org/abs/1802.03335.
[26] Opper M.Variational inference for stochastic differential equations[J]. Annalen der Physik, 2019, 531(3): 1800233. DOI:10.1002/andp.201800233.
[27] 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: An Interdisciplinary Journal of Nonlinear Science, 2022, 32(6): 061103. DOI:10.1063/5.0095605.
[28] 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: An Interdisciplinary Journal of Nonlinear Science, 2022, 32(6): 063112. DOI:10.1063/5.0089832.
[29] 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.
[30] Deng R Z, Chang B, Brubaker M A, et al. Modeling continuous stochastic processes with dynamic normalizing flows[EB/OL].2020: arXiv: 2002.10516.(2020-02-24)[2025-12-18]. https://arxiv.org/abs/2002.10516.
[31] 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 2020-January 24, 2021, Las Vegas, NV, USA. IEEE, 2020: 5231-5237. DOI:10.1109/IROS45743.2020.9341035.
[32] Papamakarios G, Nalisnick E, Rezende D J, et al. Normalizing flows for probabilistic modeling and inference[EB/OL].2019: arXiv: 1912.02762.(2019-12-05)[2024-10-18]. https://arxiv.org/abs/1912.02762.
[33] Guo L, Wu H, Zhou T.Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models[J]. Journal of Computational Physics, 2022, 461: 111202. DOI:10.1016/j.jcp.2022.111202.
[34] 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.
[35] 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.
[36] Solin A, Tamir E, Verma P. Scalable inference in SDEs by direct matching of the Fokker-Planck-Kolmogorov equation[EB/OL].2021: arXiv: 2110.15739.(2021-10-29)[2025-12-18]. https://arxiv.org/abs/2110.15739.
[37] 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): 20210195. DOI:10.1098/rsta.2021.0195.
[38] 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.
[39] Chen G G, Cheng X P, Wang L J.Learning stochastic Hamiltonian systems via neural networks based on associated Fokker-Planck equations[EB/OJ]. (2025-03-26)[2025-12-18]. Learning stochastic Hamiltonian systems via neural networks based on associated Fokker-Planck equations[EB/OJ]. (2025-03-26)[2025-12-18]. http://journal.ucas.ac.cn/CN/10.7523/j.ucas.2025.001.
[40] Hairer E, Wanner G, Lubich C.Symplectic integration of Hamiltonian systems[M]//Geometric Numerical Integration. Berlin/Heidelberg: Springer-Verlag, 2006: 179-236. DOI:10.1007/3-540-30666-8_6.
[41] D’Ambrosio R, Di Giovacchino S. Strong backward error analysis of symplectic integrators for stochastic Hamiltonian systems[J]. Applied Mathematics and Computation, 2024, 467: 128488. DOI:10.1016/j.amc.2023.128488.

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