mlsanityData to 7 Oct 2026

Symplectic Learning for Hamiltonian Neural Networks

arXiv 2106.11753 ↗First posted 22 Jun 2021cs.LG cs.NA math.NAcited by 35

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Abstract (arXiv)

Machine learning methods are widely used in the natural sciences to model and predict physical systems from observation data. Yet, they are often used as poorly understood "black boxes," disregarding existing mathematical structure and invariants of the problem. Recently, the proposal of Hamiltonian Neural Networks (HNNs) took a first step towards a unified "gray box" approach, using physical insight to improve performance for Hamiltonian systems. In this paper, we explore a significantly improved training method for HNNs, exploiting the symplectic structure of Hamiltonian systems with a different loss function. This frees the loss from an artificial lower bound. We mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn. This allows us to prove and numerically analyze the errors made by HNNs which, in turn, renders them fully explainable. Finally, we present a novel post-training correction to obtain the true Hamiltonian only from discretized observation data, up to an arbitrary order.

Key facts. Cited by 35 papers in the map (data through 7 Oct 2026). Builds on 18 of its 41 references that are in the map. First posted on arXiv 22 Jun 2021.

Builds on

18 of 41 references.

References · 18

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  3. 2018Neural Ordinary Differential Equationscited by 3,733
  4. 2019Hamiltonian Neural Networkscited by 702
  5. 2020Lagrangian Neural Networkscited by 388
  6. 2019Machine Learning for Fluid Mechanicscited by 380
  7. 2019Symplectic ODE-Net: Learning Hamiltonian Dynamics with Controlcited by 183
  8. 2020Deep Network Approximation for Smooth Functionscited by 163
  9. 2019Symplectic Recurrent Neural Networkscited by 148
  10. 2019Hamiltonian Generative Networkscited by 136
  11. 2018Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flowcited by 97
  12. 2017Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledgecited by 95
6 more
  1. 2021On the approximation of functions by tanh neural networkscited by 81
  2. 2021Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Mapscited by 41
  3. 2020Nonseparable Symplectic Neural Networkscited by 31
  4. 2020Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systemscited by 24
  5. 2020Sparse Symplectically Integrated Neural Networkscited by 20
  6. 2019EM-like Learning Chaotic Dynamics from Noisy and Partial Observationscited by 11

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 7; 2023: 7; 2024: 8; 2025: 9; 2026: 4.

35 papers.

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  1. 2022Pseudo-Hamiltonian Neural Networks with State-Dependent External Forcescited by 17
  2. 2022Learning Hamiltonians of constrained mechanical systemscited by 14
  3. 2022Thermodynamics of learning physical phenomenacited by 10
  4. 2023Pseudo-Hamiltonian neural networks for learning partial differential equationscited by 10
  5. 2023Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetriescited by 9
  6. 2022Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systemscited by 7
  7. 2022Approximation of nearly-periodic symplectic maps via structure-preserving neural networkscited by 6
  8. 2023Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applicationscited by 6
  9. 2022Learning Trajectories of Hamiltonian Systems with Neural Networkscited by 4
  10. 2023Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integratorscited by 4
  11. 2024Symplectic Neural Networks Based on Dynamical Systemscited by 4
  12. 2024Noether's razor: Learning Conserved Quantitiescited by 4
23 more
  1. 2025Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraintscited by 4
  2. 2023Learning Hamiltonian Systems with Mono-Implicit Runge-Kutta Methodscited by 3
  3. 2024CLPNets: Coupled Lie-Poisson Neural Networks for Multi-Part Hamiltonian Systems with Symmetriescited by 3
  4. 2024Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflationcited by 3
  5. 2025Data-driven particle dynamics: Structure-preserving coarse-graining for emergent behavior in non-equilibrium systemscited by 3
  6. 2024Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modelingcited by 2
  7. 2024Learning Generalized Hamiltonians using fully Symplectic Mappingscited by 2
  8. 2024Training Hamiltonian neural networks without backpropagationcited by 2
  9. 2025Neural non-canonical Hamiltonian dynamics for long-time simulationscited by 2
  10. 2022Data-driven discovery of non-Newtonian astronomy via learning non-Euclidean Hamiltoniancited by 1
  11. 2023Pseudo-Hamiltonian system identificationcited by 1
  12. 2023What's Next? Predicting Hamiltonian Dynamics from Discrete Observations of a Vector Fieldcited by 1
  13. 2025Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modelingcited by 1
  14. 2025Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Networkcited by 1
  15. 2025Time-adaptive SympNets for separable Hamiltonian systemscited by 1
  16. 2025Time-adaptive H'enonNets for separable Hamiltonian systemscited by 1
  17. 2026CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systemscited by 1
  18. 2024Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical responsecited by 0
  19. 2025Improving regional weather forecasts with neural interpolationcited by 0
  20. 2025Hamiltonian Neural PDE Solvers through Functional Approximationcited by 0
  21. 2026Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory datacited by 0
  22. 2026Learning partially observed systems with neural Hamiltonian ordinary differential equationscited by 0
  23. 2026Gaussian process learning with flow map refinement for parameter estimation in dynamical systemscited by 0