mlsanityData to 7 Oct 2026

Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks

arXiv 2012.03133 ↗First posted 5 Dec 2020cs.LGcited by 37

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Abstract (arXiv)

We propose the Poisson neural networks (PNNs) to learn Poisson systems and trajectories of autonomous systems from data. Based on the Darboux-Lie theorem, the phase flow of a Poisson system can be written as the composition of (1) a coordinate transformation, (2) an extended symplectic map and (3) the inverse of the transformation. In this work, we extend this result to the unknotted trajectories of autonomous systems. We employ structured neural networks with physical priors to approximate the three aforementioned maps. We demonstrate through several simulations that PNNs are capable of handling very accurately several challenging tasks, including the motion of a particle in the electromagnetic potential, the nonlinear Schr"odinger equation, and pixel observations of the two-body problem.

Key facts. Cited by 37 papers in the map (data through 7 Oct 2026). First posted on arXiv 5 Dec 2020.

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Citing papers per year: 2021: 7; 2022: 7; 2023: 8; 2024: 4; 2025: 5; 2026: 6.

37 papers.

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  1. 2021GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systemscited by 48
  2. 2021Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learningcited by 37
  3. 2023Physics-informed neural networks for solving forward and inverse problems in complex beam systemscited by 24
  4. 2022Pseudo-Hamiltonian Neural Networks with State-Dependent External Forcescited by 17
  5. 2023Constructing Custom Thermodynamics Using Deep Learningcited by 16
  6. 2023On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulatorscited by 13
  7. 2025Stable Port-Hamiltonian Neural Networkscited by 13
  8. 2022FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantitiescited by 12
  9. 2021Physics perception in sloshing scenes with guaranteed thermodynamic consistencycited by 10
  10. 2022Constants of motion networkcited by 10
  11. 2023Pseudo-Hamiltonian neural networks for learning partial differential equationscited by 10
  12. 2021Locally-symplectic neural networks for learning volume-preserving dynamicscited by 9
25 more
  1. 2022SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problemscited by 9
  2. 2023Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetriescited by 9
  3. 2023Symmetry Preservation in Hamiltonian Systems: Simulation and Learningcited by 6
  4. 2021Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEscited by 5
  5. 2022Unifying physical systems' inductive biases in neural ODE using dynamics constraintscited by 5
  6. 2021Approximation capabilities of measure-preserving neural networkscited by 4
  7. 2022ModLaNets: Learning Generalisable Dynamics via Modularity and Physical Inductive Biascited by 4
  8. 2024Symplectic Neural Networks Based on Dynamical Systemscited by 4
  9. 2022Physically Consistent Neural ODEs for Learning Multi-Physics Systemscited by 3
  10. 2023Deep Learning for Structure-Preserving Universal Stable Koopman-Inspired Embeddings for Nonlinear Canonical Hamiltonian Dynamicscited by 3
  11. 2021KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Losscited by 2
  12. 2024Designing Poisson Integrators Through Machine Learningcited by 2
  13. 2025Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approachcited by 2
  14. 2025Neural non-canonical Hamiltonian dynamics for long-time simulationscited by 2
  15. 2023Pseudo-Hamiltonian system identificationcited by 1
  16. 2025Time-adaptive SympNets for separable Hamiltonian systemscited by 1
  17. 2026Nonlinear GENERIC-Embedded Neural Networks (N-GENNs): Learning GENERIC dynamics with non-quadratic dissipation potentialscited by 1
  18. 2024Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEscited by 0
  19. 2024Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domainscited by 0
  20. 2025Variational Neural Networks for Observable Thermodynamics (V-NOTS)cited by 0
  21. 2026An Invariant Compiler for Neural ODEs in AI-Accelerated Scientific Simulationcited by 0
  22. 2026Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory datacited by 0
  23. 2026Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamicscited by 0
  24. 2026CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamicscited by 0
  25. 2026Structure-preserving uncertainty quantification for GENERIC dynamicscited by 0