We present KeyCLD, a framework to learn Lagrangian dynamics from images. Learned keypoints represent semantic landmarks in images and can directly represent state dynamics. We show that interpreting this state as Cartesian coordinates, coupled with explicit holonomic constraints, allows expressing the dynamics with a constrained Lagrangian. KeyCLD is trained unsupervised end-to-end on sequences of images. Our method explicitly models the mass matrix, potential energy and the input matrix, thus allowing energy based control. We demonstrate learning of Lagrangian dynamics from images on the dm_control pendulum, cartpole and acrobot environments. KeyCLD can be learned on these systems, whether they are unactuated, underactuated or fully actuated. Trained models are able to produce long-term video predictions, showing that the dynamics are accurately learned. We compare with Lag-VAE, Lag-caVAE and HGN, and investigate the benefit of the Lagrangian prior and the constraint function. KeyCLD achieves the highest valid prediction time on all benchmarks. Additionally, a very straightforward energy shaping controller is successfully applied on the fully actuated systems. Please refer to our project page for code and additional results: https://rdaems.github.io/keycld/

We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative function distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression to determine optimal approximation coefficients. Furthermore, we propose the use of neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction,-an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.