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

2026-06-11

AI Summary

This work addresses the challenge of modeling and forecasting stochastic nonlinear dynamical systems under noisy and partially observable conditions by proposing a deep spectral learning framework. The method employs a learnable neural encoder to construct Markovian latent states in a feature space, whose dynamics and observations are governed by learned transition and observation operators. It uniquely unifies spectral learning, Bayesian filtering, and Koopman mode decomposition within an end-to-end trainable architecture. Efficient state estimation is achieved through functional canonical correlation analysis, Galerkin projection, and a closed-form ridge-regularized solution. Experimental results demonstrate that the proposed approach significantly outperforms baseline methods—including sequential Bayesian filters and dynamic mode decomposition—across diverse scenarios, exhibiting strong robustness to both observation noise and partial observability.

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.

Problem

• stochastic dynamical systems
• latent transfer operators
• spectral learning
• deep feature spaces
• partial observability

Innovation

• spectral learning
• latent transfer operators
• deep feature spaces
• Galerkin projection
• Koopman operator