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Variational Integrator Networks for Physically Structured Embeddings

arXiv 1910.09349 ↗First posted 21 Oct 2019stat.ML cs.LGcited by 36

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Abstract (arXiv)

Learning workable representations of dynamical systems is becoming an increasingly important problem in a number of application areas. By leveraging recent work connecting deep neural networks to systems of differential equations, we propose variational integrator networks, a class of neural network architectures designed to preserve the geometric structure of physical systems. This class of network architectures facilitates accurate long-term prediction, interpretability, and data-efficient learning, while still remaining highly flexible and capable of modeling complex behavior. We demonstrate that they can accurately learn dynamical systems from both noisy observations in phase space and from image pixels within which the unknown dynamics are embedded.

Key facts. Cited by 36 papers in the map (data through 7 Oct 2026). First posted on arXiv 21 Oct 2019.

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Citing papers per year: 2020: 1; 2021: 10; 2022: 12; 2023: 2; 2024: 5; 2025: 4; 2026: 2.

36 papers.

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  1. 2021Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modelingcited by 59
  2. 2021One-Shot Transfer Learning of Physics-Informed Neural Networkscited by 49
  3. 2022Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamicscited by 30
  4. 2020Differentiable Physics Models for Real-world Offline Model-based Reinforcement Learningcited by 27
  5. 2021Which priors matter? Benchmarking models for learning latent dynamicscited by 26
  6. 2021Combining Physics and Deep Learning to learn Continuous-Time Dynamics Modelscited by 20
  7. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  8. 2021Learning Contact Dynamics using Physically Structured Neural Networkscited by 10
  9. 2025Four Principles for Physically Interpretable World Modelscited by 10
  10. 2021Structure-preserving Gaussian Process Dynamicscited by 9
  11. 2022Learning Neural Hamiltonian Dynamics: A Methodological Overviewcited by 8
  12. 2022A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacousticcited by 7
24 more
  1. 2022Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systemscited by 7
  2. 2024PhysORD: A Neuro-Symbolic Approach for Physics-infused Motion Prediction in Off-road Drivingcited by 7
  3. 2022Approximation of nearly-periodic symplectic maps via structure-preserving neural networkscited by 6
  4. 2023Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applicationscited by 6
  5. 2021Forced Variational Integrator Networks for Prediction and Control of Mechanical Systemscited by 5
  6. 2021SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Visioncited by 5
  7. 2022From latent dynamics to meaningful representationscited by 5
  8. 2022Discrete Lagrangian Neural Networks with Automatic Symmetry Discoverycited by 5
  9. 2024Projected Neural Differential Equations for Learning Constrained Dynamicscited by 5
  10. 2023Hamiltonian GANcited by 4
  11. 2025Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamicscited by 4
  12. 2021Structure-Preserving Learning Using Gaussian Processes and Variational Integratorscited by 3
  13. 2022Symplectic Momentum Neural Networks -- Using Discrete Variational Mechanics as a prior in Deep Learningcited by 3
  14. 2022Towards Cross Domain Generalization of Hamiltonian Representation via Meta Learningcited by 3
  15. 2024Online Control-Informed Learningcited by 3
  16. 2025Metriplectic Conditional Flow Matching for Dissipative Dynamicscited by 3
  17. 2024Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modelingcited by 2
  18. 2021Physical Constraint Embedded Neural Networks for inference and noise regulationcited by 1
  19. 2022Symplectically Integrated Symbolic Regression of Hamiltonian Dynamical Systemscited by 1
  20. 2025CHyLL: Learning Continuous Neural Representations of Hybrid Systemscited by 1
  21. 2022KeyCLD: Learning Constrained Lagrangian Dynamics in Keypoint Coordinates from Imagescited by 0
  22. 2024Unsupervised Learning of Hybrid Latent Dynamics: A Learn-to-Identify Frameworkcited by 0
  23. 2026Learned Lagrangian Models of PDEs via Euler-Lagrange Residual Minimizationcited by 0
  24. 2026Learning Deterministic and Stochastic Forced Hamiltonian Systemscited by 0