mlsanityData to 7 Oct 2026

Symplectic Neural Flows for Modeling and Discovery

arXiv 2412.16787 ↗First posted 21 Dec 2024cs.LG physics.comp-ph physics.flu-dyncited by 4

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Abstract (arXiv)

Hamilton's equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose, but neural network-based methods that incorporate these principles remain underexplored. This work introduces SympFlow, a time-dependent symplectic neural network designed using parameterized Hamiltonian flow maps. This design allows for backward error analysis and ensures the preservation of the symplectic structure. SympFlow allows for two key applications: (i) providing a time-continuous symplectic approximation of the exact flow of a Hamiltonian system purely based on the differential equations it satisfies, and (ii) approximating the flow map of an unknown Hamiltonian system relying on trajectory data. We demonstrate the effectiveness of SympFlow on diverse problems, including chaotic and dissipative systems, showing improved energy conservation compared to general-purpose numerical methods and accurate approximations from sparse irregular data. We also provide a thorough theoretical analysis of SympFlow, showing it can approximate the flow of any time-dependent Hamiltonian system, and providing an a-posteriori error estimate in terms of energy conservation.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 44 of its 76 references that are in the map. First posted on arXiv 21 Dec 2024.

Builds on

44 of 76 references.

References · 44

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32 more
  1. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  2. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  3. 2020Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learningcited by 77
  4. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  5. 2021Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systemscited by 66
  6. 2023Towards Stability of Autoregressive Neural Operatorscited by 54
  7. 2022Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separatelycited by 46
  8. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  9. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  10. 2021Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Mapscited by 41
  11. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  12. 2020Hamiltonian neural networks for solving equations of motioncited by 33
  13. 2023Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networkscited by 32
  14. 2020Nonseparable Symplectic Neural Networkscited by 31
  15. 2022Thermodynamics-informed graph neural networkscited by 31
  16. 2020Structure preserving deep learningcited by 30
  17. 2020Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systemscited by 24
  18. 2023Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approachcited by 24
  19. 2022Unravelling the Performance of Physics-informed Graph Neural Networks for Dynamical Systemscited by 23
  20. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 23
  21. 2020Sparse Symplectically Integrated Neural Networkscited by 20
  22. 2022Improved Training of Physics-Informed Neural Networks with Model Ensemblescited by 19
  23. 2021Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Designcited by 18
  24. 2023Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbertcited by 18
  25. 2022Learning Hamiltonians of constrained mechanical systemscited by 14
  26. 2022Dynamical systems' based neural networkscited by 10
  27. 2024Biophysics Informed Pathological Regularisation for Brain Tumour Segmentationcited by 7
  28. 2023Hamiltonian Neural Networks with Automatic Symmetry Detectioncited by 6
  29. 2022Learning Trajectories of Hamiltonian Systems with Neural Networkscited by 4
  30. 2024Symplectic Neural Networks Based on Dynamical Systemscited by 4
  31. 2024Hamiltonian Matching for Symplectic Neural Integratorscited by 2
  32. 2024A time-dependent symplectic network for non-convex path planning problems with linear and nonlinear dynamicscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 3; 2026: 1.

4 papers.

  1. 2025Generalized Lie Symmetries in Physics-Informed Neural Operatorscited by 2
  2. 2025Geometric Contact Flows: Contactomorphisms for Dynamics and Controlcited by 2
  3. 2025Learning Physical Systems: Symplectification via Gauge Fixing in Dirac Structurescited by 1
  4. 2026CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamicscited by 0