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

Ryogo Tanaka, Yoshinobu Kawahara
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:6610-6630, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-tanaka26a, title = {Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems}, author = {Tanaka, Ryogo and Kawahara, Yoshinobu}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {6610--6630}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/tanaka26a/tanaka26a.pdf}, url = {https://proceedings.mlr.press/v337/tanaka26a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems %A Ryogo Tanaka %A Yoshinobu Kawahara %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-tanaka26a %I PMLR %P 6610--6630 %U https://proceedings.mlr.press/v337/tanaka26a.html %V 337 %X 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.
APA
Tanaka, R. & Kawahara, Y.. (2026). Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:6610-6630 Available from https://proceedings.mlr.press/v337/tanaka26a.html.

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