# Neural Canonical Transformation with Symplectic Flows

arXiv: https://arxiv.org/abs/1910.00024 · HTML: https://mlsanity.com/paper/1910.00024/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2019.

## Built on by (most cited first, 16 of 16)

- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Flow-matching -- efficient coarse-graining of molecular dynamics without forces](https://mlsanity.com/paper/2203.11167/) (2022)
- [Deconstructing the Inductive Biases of Hamiltonian Neural Networks](https://mlsanity.com/paper/2202.04836/) (2022)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [Machine-Learning Non-Conservative Dynamics for New-Physics Detection](https://mlsanity.com/paper/2106.00026/) (2021)
- [Machine Learning in Nuclear Physics](https://mlsanity.com/paper/2112.02309/) (2021)
- [Learning Non-linear Wavelet Transformation via Normalizing Flow](https://mlsanity.com/paper/2101.11306/) (2021)
- [Embedded-model flows: Combining the inductive biases of model-free deep learning and explicit probabilistic modeling](https://mlsanity.com/paper/2110.06021/) (2021)
- [KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss](https://mlsanity.com/paper/2102.11923/) (2021)
- [Learning Integrable Dynamics with Action-Angle Networks](https://mlsanity.com/paper/2211.15338/) (2022)
- [Geometric Contact Flows: Contactomorphisms for Dynamics and Control](https://mlsanity.com/paper/2506.17868/) (2025)
- [Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning models](https://mlsanity.com/paper/2604.09769/) (2026)
- [Information bottleneck for learning the phase space of dynamics from high-dimensional experimental data](https://mlsanity.com/paper/2604.24662/) (2026)
- [CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems](https://mlsanity.com/paper/2607.03339/) (2026)

## Abstract (arXiv)

Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. We construct flexible and powerful canonical transformations as generative models using symplectic neural networks. The model transforms physical variables towards a latent representation with an independent harmonic oscillator Hamiltonian. Correspondingly, the phase space density of the physical system flows towards a factorized Gaussian distribution in the latent space. Since the canonical transformation preserves the Hamiltonian evolution, the model captures nonlinear collective modes in the learned latent representation. We present an efficient implementation of symplectic neural coordinate transformations and two ways to train the model. The variational free energy calculation is based on the analytical form of physical Hamiltonian. While the phase space density estimation only requires samples in the coordinate space for separable Hamiltonians. We demonstrate appealing features of neural canonical transformation using toy problems including two-dimensional ring potential and harmonic chain. Finally, we apply the approach to real-world problems such as identifying slow collective modes in alanine dipeptide and conceptual compression of the MNIST dataset.