# VPNets: Volume-preserving neural networks for learning source-free dynamics

arXiv: https://arxiv.org/abs/2204.13843 · HTML: https://mlsanity.com/paper/2204.13843/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 46 references that are in the map.
- First posted on arXiv 29 Apr 2022.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [The Next Decade in AI: Four Steps Towards Robust Artificial Intelligence](https://mlsanity.com/paper/2002.06177/) (2020)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Building machines that adapt and compute like brains](https://mlsanity.com/paper/1711.04203/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Structure preserving deep learning](https://mlsanity.com/paper/2006.03364/) (2020)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data](https://mlsanity.com/paper/1905.10396/) (2019)
- [Discovery of Dynamics Using Linear Multistep Methods](https://mlsanity.com/paper/1912.12728/) (2019)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [Locally-symplectic neural networks for learning volume-preserving dynamics](https://mlsanity.com/paper/2109.09151/) (2021)
- [Approximation capabilities of measure-preserving neural networks](https://mlsanity.com/paper/2106.10911/) (2021)

## Built on by (most cited first, 2 of 2)

- [Learning Non-Linear Invariants for Unsupervised Out-of-Distribution Detection](https://mlsanity.com/paper/2407.04022/) (2024)
- [LLN: Learnable Lens Networks for Parameter-Efficient Long-Horizon Dynamical Prediction](https://mlsanity.com/paper/2609.34493/) (2026)

## Abstract (arXiv)

We propose volume-preserving networks (VPNets) for learning unknown source-free dynamical systems using trajectory data. We propose three modules and combine them to obtain two network architectures, coined R-VPNet and LA-VPNet. The distinct feature of the proposed models is that they are intrinsic volume-preserving. In addition, the corresponding approximation theorems are proved, which theoretically guarantee the expressivity of the proposed VPNets to learn source-free dynamics. The effectiveness, generalization ability and structure-preserving property of the VP-Nets are demonstrated by numerical experiments.