# Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems

arXiv: https://arxiv.org/abs/2606.14079 · HTML: https://mlsanity.com/paper/2606.14079/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 12 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Deep Kalman Filters](https://mlsanity.com/paper/1511.05121/) (2015)
- [Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces](https://mlsanity.com/paper/1905.07357/) (2019)
- [Learning Deep Features in Instrumental Variable Regression](https://mlsanity.com/paper/2010.07154/) (2020)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Learning interpretable continuous-time models of latent stochastic dynamical systems](https://mlsanity.com/paper/1902.04420/) (2019)
- [Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series](https://mlsanity.com/paper/2301.11308/) (2023)
- [Identifying Latent Stochastic Differential Equations](https://mlsanity.com/paper/2007.06075/) (2020)
- [Learning Dynamics Models with Stable Invariant Sets](https://mlsanity.com/paper/2006.08935/) (2020)
- [Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators](https://mlsanity.com/paper/2501.02721/) (2025)

## Abstract (arXiv)

We propose a spectral learning method for stochastic nonlinear dynamical systems represented with embedded latent transfer operators in deep feature spaces. We instantiate the method as Deep Spectral Encoder (DSE), an operator-based latent state-space model in which a time-invariant neural encoder implements learnable nonlinear feature maps from observations, and these features define Markovian latent states whose temporal evolution and observation mapping are described by the transfer and observation operators, respectively. Functional canonical correlation analysis in a learnable Galerkin-projected feature space provides state coordinates from past and future observations, and the two linear operators are estimated on the state coordinates as ridge-regularized closed-form solutions that coincide with Galerkin projections of the associated covariance operators. On this representation, we generalize sequential Bayesian filtering and Koopman spectral mode decomposition in feature space. Experiments on several scenarios show stable and superior performance with sequential Bayesian filtering and dynamic mode decomposition baselines even under noise and partial observability.