# Latent Space Energy-based Neural ODEs

arXiv: https://arxiv.org/abs/2409.03845 · HTML: https://mlsanity.com/paper/2409.03845/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 43 references that are in the map.
- First posted on arXiv 5 Sep 2024.

## Builds on (most cited first, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [MoCoGAN: Decomposing Motion and Content for Video Generation](https://mlsanity.com/paper/1707.04993/) (2017)
- [dm_control: Software and Tasks for Continuous Control](https://mlsanity.com/paper/2006.12983/) (2020)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [Gaussian Process Prior Variational Autoencoders](https://mlsanity.com/paper/1810.11738/) (2018)
- [Latent Diffusion Energy-Based Model for Interpretable Text Modeling](https://mlsanity.com/paper/2206.05895/) (2022)
- [Trajectory Prediction with Latent Belief Energy-Based Model](https://mlsanity.com/paper/2104.03086/) (2021)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Learning Generative Vision Transformer with Energy-Based Latent Space for Saliency Prediction](https://mlsanity.com/paper/2112.13528/) (2021)
- [Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series](https://mlsanity.com/paper/2301.11308/) (2023)
- [Unsupervised Foreground Extraction via Deep Region Competition](https://mlsanity.com/paper/2110.15497/) (2021)
- [Learning Dynamic Generator Model by Alternating Back-Propagation Through Time](https://mlsanity.com/paper/1812.10587/) (2018)
- [On Out-of-distribution Detection with Energy-based Models](https://mlsanity.com/paper/2107.08785/) (2021)
- [Latent Space Energy-Based Model of Symbol-Vector Coupling for Text Generation and Classification](https://mlsanity.com/paper/2108.11556/) (2021)
- [Deep Temporal Sigmoid Belief Networks for Sequence Modeling](https://mlsanity.com/paper/1509.07087/) (2015)
- [Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equation](https://mlsanity.com/paper/2110.15739/) (2021)
- [Latent Plan Transformer for Trajectory Abstraction: Planning as Latent Space Inference](https://mlsanity.com/paper/2402.04647/) (2024)
- [Motion-Based Generator Model: Unsupervised Disentanglement of Appearance, Trackable and Intrackable Motions in Dynamic Patterns](https://mlsanity.com/paper/1911.11294/) (2019)
- [Learning Energy-Based Prior Model with Diffusion-Amortized MCMC](https://mlsanity.com/paper/2310.03218/) (2023)
- [Modulated Neural ODEs](https://mlsanity.com/paper/2302.13262/) (2023)
- [Molecule Design by Latent Space Energy-Based Modeling and Gradual Distribution Shifting](https://mlsanity.com/paper/2306.14902/) (2023)
- [Molecule Design by Latent Prompt Transformer](https://mlsanity.com/paper/2310.03253/) (2023)
- [Learning Hierarchical Features with Joint Latent Space Energy-Based Prior](https://mlsanity.com/paper/2310.09604/) (2023)
- [An Energy-Based Prior for Generative Saliency](https://mlsanity.com/paper/2204.08803/) (2022)

## Built on by (most cited first, 4 of 4)

- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [Latent Adaptive Planner for Dynamic Manipulation](https://mlsanity.com/paper/2505.03077/) (2025)
- [Uncertainty Quantification for Reduced-Order Surrogate Models Applied to Cloud Microphysics](https://mlsanity.com/paper/2511.04534/) (2025)
- [Learning to Plan from Random Exploration](https://mlsanity.com/paper/2609.38383/) (2026)

## Abstract (arXiv)

This paper introduces novel deep dynamical models designed to represent continuous-time sequences. Our approach employs a neural emission model to generate each data point in the time series through a non-linear transformation of a latent state vector. The evolution of these latent states is implicitly defined by a neural ordinary differential equation (ODE), with the initial state drawn from an informative prior distribution parameterized by an Energy-based model (EBM). This framework is extended to disentangle dynamic states from underlying static factors of variation, represented as time-invariant variables in the latent space. We train the model using maximum likelihood estimation with Markov chain Monte Carlo (MCMC) in an end-to-end manner. Experimental results on oscillating systems, videos and real-world state sequences (MuJoCo) demonstrate that our model with the learnable energy-based prior outperforms existing counterparts, and can generalize to new dynamic parameterization, enabling long-horizon predictions.