Problem
Research questions and friction points this paper is trying to address.
Video Prediction
Image Generation
Image Dynamics
Hamiltonian Flows
Lagrangian--Hamiltonian Flows for Video Prediction and Image Generation: A Symplectic Perspective
🤖 AI Summary
This study addresses the low computational efficiency and inadequate dynamics modeling in video prediction and image generation by proposing the Lagrangian–Hamiltonian Flow Matching (LHFM) geometric framework. This work is the first to incorporate classical mechanics structures into flow matching, leveraging symplectic geometry to represent images as Lagrangian submanifolds and characterizing their dynamic evolution via Hamiltonian flows. Furthermore, a transport-source parameterized model is constructed, combining geometric quantization with recurrent networks to enable efficient generation. Experimental results demonstrate that LHFM achieves significantly lower FLOPs than existing methods at comparable accuracy. Moreover, its image generation variant outperforms baseline models in FID on matching experiments, effectively unifying computational efficiency with high generative quality.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Lagrangian-Hamiltonian Flows
Symplectic Geometry
Video Prediction
Flow Matching
Image Generation
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