# Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators

arXiv: https://arxiv.org/abs/2501.02721 · HTML: https://mlsanity.com/paper/2501.02721/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 58 references that are in the map.
- First posted on arXiv 6 Jan 2025.

## Builds on (most cited first, 9 of 9)

- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces](https://mlsanity.com/paper/1905.07357/) (2019)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1807.09331/) (2018)

## Built on by (most cited first, 2 of 2)

- [Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems](https://mlsanity.com/paper/2606.14079/) (2026)
- [Structured Noise Adaptation for Sequential Bayesian Filtering with Embedded Latent Transfer Operators](https://mlsanity.com/paper/2606.14195/) (2026)

## Abstract (arXiv)

We consider an operator-based latent Markov representation of a stochastic nonlinear dynamical system, where the stochastic evolution of the latent state embedded in a reproducing kernel Hilbert space is described with the corresponding transfer operator, and develop a spectral method to learn this representation based on the theory of stochastic realization. The embedding may be learned simultaneously using reproducing kernels, for example, constructed with feed-forward neural networks. We also address the generalization of sequential state-estimation (Kalman filtering) in stochastic nonlinear systems, and of operator-based eigen-mode decomposition of dynamics, for the representation. Several examples with synthetic and real-world data are shown to illustrate the empirical characteristics of our methods, and to investigate the performance of our model in sequential state-estimation and mode decomposition.