# Variational Inference for Continuous-Time Switching Dynamical Systems

arXiv: https://arxiv.org/abs/2109.14492 · HTML: https://mlsanity.com/paper/2109.14492/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 72 references that are in the map.
- First posted on arXiv 29 Sep 2021.

## Builds on (most cited first, 11 of 11)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Composing graphical models with neural networks for structured representations and fast inference](https://mlsanity.com/paper/1603.06277/) (2016)
- [On Sparse variational methods and the Kullback-Leibler divergence between stochastic processes](https://mlsanity.com/paper/1504.07027/) (2015)
- [Deep Generative Markov State Models](https://mlsanity.com/paper/1805.07601/) (2018)
- [Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations](https://mlsanity.com/paper/1610.06773/) (2016)
- [A variational approach to path estimation and parameter inference of hidden diffusion processes](https://mlsanity.com/paper/1508.00506/) (2015)
- [Moment-Based Variational Inference for Markov Jump Processes](https://mlsanity.com/paper/1905.05451/) (2019)
- [Particle Smoothing for Hidden Diffusion Processes: Adaptive Path Integral Smoother](https://mlsanity.com/paper/1605.00278/) (2016)

## Built on by (most cited first, 10 of 10)

- [Foundation Inference Models for Markov Jump Processes](https://mlsanity.com/paper/2406.06419/) (2024)
- [Neural Markov Jump Processes](https://mlsanity.com/paper/2305.19744/) (2023)
- [Variational Gaussian Process Diffusion Processes](https://mlsanity.com/paper/2306.02066/) (2023)
- [Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems](https://mlsanity.com/paper/2408.03330/) (2024)
- [SING: SDE Inference via Natural Gradients](https://mlsanity.com/paper/2506.17796/) (2025)
- [Markov Chain Monte Carlo for Continuous-Time Switching Dynamical Systems](https://mlsanity.com/paper/2205.08803/) (2022)
- [Bayesian Inference for Jump-Diffusion Approximations of Biochemical Reaction Networks](https://mlsanity.com/paper/2304.06592/) (2023)
- [MODE: Learning compositional representations of complex systems with Mixtures Of Dynamical Experts](https://mlsanity.com/paper/2510.09594/) (2025)
- [Efficient Inference for Coupled Hidden Markov Models in Continuous Time and Discrete Space](https://mlsanity.com/paper/2510.12916/) (2025)
- [Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals](https://mlsanity.com/paper/2610.04390/) (2026)

## Abstract (arXiv)

Switching dynamical systems provide a powerful, interpretable modeling framework for inference in time-series data in, e.g., the natural sciences or engineering applications. Since many areas, such as biology or discrete-event systems, are naturally described in continuous time, we present a model based on an Markov jump process modulating a subordinated diffusion process. We provide the exact evolution equations for the prior and posterior marginal densities, the direct solutions of which are however computationally intractable. Therefore, we develop a new continuous-time variational inference algorithm, combining a Gaussian process approximation on the diffusion level with posterior inference for Markov jump processes. By minimizing the path-wise Kullback-Leibler divergence we obtain (i) Bayesian latent state estimates for arbitrary points on the real axis and (ii) point estimates of unknown system parameters, utilizing variational expectation maximization. We extensively evaluate our algorithm under the model assumption and for real-world examples.