# Variational Integrator Networks for Physically Structured Embeddings

arXiv: https://arxiv.org/abs/1910.09349 · HTML: https://mlsanity.com/paper/1910.09349/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2019.

## Built on by (most cited first, 30 of 36)

- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics](https://mlsanity.com/paper/2210.06036/) (2022)
- [Differentiable Physics Models for Real-world Offline Model-based Reinforcement Learning](https://mlsanity.com/paper/2011.01734/) (2020)
- [Which priors matter? Benchmarking models for learning latent dynamics](https://mlsanity.com/paper/2111.05458/) (2021)
- [Combining Physics and Deep Learning to learn Continuous-Time Dynamics Models](https://mlsanity.com/paper/2110.01894/) (2021)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Learning Contact Dynamics using Physically Structured Neural Networks](https://mlsanity.com/paper/2102.11206/) (2021)
- [Four Principles for Physically Interpretable World Models](https://mlsanity.com/paper/2503.02143/) (2025)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems](https://mlsanity.com/paper/2211.16006/) (2022)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [PhysORD: A Neuro-Symbolic Approach for Physics-infused Motion Prediction in Off-road Driving](https://mlsanity.com/paper/2404.01596/) (2024)
- [Approximation of nearly-periodic symplectic maps via structure-preserving neural networks](https://mlsanity.com/paper/2210.05087/) (2022)
- [Forced Variational Integrator Networks for Prediction and Control of Mechanical Systems](https://mlsanity.com/paper/2106.02973/) (2021)
- [SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision](https://mlsanity.com/paper/2111.05986/) (2021)
- [From latent dynamics to meaningful representations](https://mlsanity.com/paper/2209.00905/) (2022)
- [Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery](https://mlsanity.com/paper/2211.10830/) (2022)
- [Projected Neural Differential Equations for Learning Constrained Dynamics](https://mlsanity.com/paper/2410.23667/) (2024)
- [Hamiltonian GAN](https://mlsanity.com/paper/2308.11216/) (2023)
- [Online Control-Informed Learning](https://mlsanity.com/paper/2410.03924/) (2024)
- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [Metriplectic Conditional Flow Matching for Dissipative Dynamics](https://mlsanity.com/paper/2509.19526/) (2025)
- [Structure-Preserving Learning Using Gaussian Processes and Variational Integrators](https://mlsanity.com/paper/2112.05451/) (2021)
- [Symplectic Momentum Neural Networks -- Using Discrete Variational Mechanics as a prior in Deep Learning](https://mlsanity.com/paper/2201.08281/) (2022)
- [Towards Cross Domain Generalization of Hamiltonian Representation via Meta Learning](https://mlsanity.com/paper/2212.01168/) (2022)
- [Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling](https://mlsanity.com/paper/2401.12476/) (2024)
- [CHyLL: Learning Continuous Neural Representations of Hybrid Systems](https://mlsanity.com/paper/2512.10117/) (2025)

## 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.