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Learning Deterministic and Stochastic Forced Hamiltonian Systems

arXiv 2608.19688 ↗First posted 20 Aug 2026math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a Cr convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 23 of its 91 references that are in the map. First posted on arXiv 20 Aug 2026.

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23 of 91 references.

References · 23

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  2. 2018Neural Ordinary Differential Equationscited by 3,779
  3. 2019Hamiltonian Neural Networkscited by 712
  4. 2020Lagrangian Neural Networkscited by 398
  5. 2019Symplectic ODE-Net: Learning Hamiltonian Dynamics with Controlcited by 184
  6. 2019Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noisecited by 101
  7. 2020Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learningcited by 79
  8. 2020SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimatescited by 75
  9. 2021Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systemscited by 67
  10. 2022Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separatelycited by 47
  11. 2021Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learningcited by 39
  12. 2019Variational Integrator Networks for Physically Structured Embeddingscited by 36
11 more
  1. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  2. 2023Modeling Unknown Stochastic Dynamical System via Autoencodercited by 12
  3. 2021Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama schemecited by 10
  4. 2024Generalized Lagrangian Neural Networkscited by 10
  5. 2024DynGMA: a robust approach for learning stochastic differential equations from datacited by 8
  6. 2022Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systemscited by 7
  7. 2025Learning mechanical systems from real-world data using discrete forced Lagrangian dynamicscited by 7
  8. 2021Forced Variational Integrator Networks for Prediction and Control of Mechanical Systemscited by 5
  9. 2023Symplectic Autoencoders for Model Reduction of Hamiltonian Systemscited by 5
  10. 2025Neural non-canonical Hamiltonian dynamics for long-time simulationscited by 2
  11. 2025Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Networkcited by 1

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