# Lagrangian--Hamiltonian Flows for Video Prediction and Image Generation: A Symplectic Perspective

arXiv: https://arxiv.org/abs/2609.35710 · HTML: https://mlsanity.com/paper/2609.35710/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Video generation, one of 14 directions in Diffusion models.
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

- Diffusion models / [Video generation](https://mlsanity.com/diffusion/video-generation/)

## Builds on (most cited first, 16 of 16)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Unsupervised Learning of Video Representations using LSTMs](https://mlsanity.com/paper/1502.04681/) (2015)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [SimVP: Simpler yet Better Video Prediction](https://mlsanity.com/paper/2206.05099/) (2022)
- [PredRNN: A Recurrent Neural Network for Spatiotemporal Predictive Learning](https://mlsanity.com/paper/2103.09504/) (2021)
- [PredRNN++: Towards A Resolution of the Deep-in-Time Dilemma in Spatiotemporal Predictive Learning](https://mlsanity.com/paper/1804.06300/) (2018)
- [Memory In Memory: A Predictive Neural Network for Learning Higher-Order Non-Stationarity from Spatiotemporal Dynamics](https://mlsanity.com/paper/1811.07490/) (2018)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Efficient Video Prediction via Sparsely Conditioned Flow Matching](https://mlsanity.com/paper/2211.14575/) (2022)
- [SwinLSTM:Improving Spatiotemporal Prediction Accuracy using Swin Transformer and LSTM](https://mlsanity.com/paper/2308.09891/) (2023)
- [Video Prediction Transformers without Recurrence or Convolution](https://mlsanity.com/paper/2410.04733/) (2024)

## Abstract (arXiv)

We introduce LHFM, a geometric framework for learning image dynamics. Drawing on structures central to classical mechanics, symplectic geometry, and geometric quantization, LHFM represents each image as an exact Lagrangian graph and models its evolution through image-dependent Hamiltonian flows, which yield a transport--source parameterization of image velocities. Our primary application is deterministic video prediction: LHFM-V is a recurrent model that advances frames by integrating predicted transport and source fields, and achieves the lowest reported FLOP count among the compared recurrent models with similar prediction accuracy. The image variant, LHFM-I, shows that the same construction is compatible with flow matching: in a matched experiment, it attains a lower FID than the flow-matching baseline.