Deep Spectral Encoder Improves Stochastic Dynamical System Modeling

A new spectral learning method, the Deep Spectral Encoder (DSE), has been proposed for modeling stochastic nonlinear dynamical systems. This approach utilizes embedded latent transfer operators within deep feature spaces to represent complex system dynamics. At its core, DSE incorporates a time-invariant neural encoder that learns nonlinear feature maps from raw observations, which then define Markovian latent states. The method leverages functional canonical correlation analysis (FCCA) in a learnable Galerkin-projected feature space to derive state coordinates from past and future observations. Subsequently, two linear operators—the transfer and observation operators—are estimated on these state coordinates using ridge-regularized closed-form solutions, which align with Galerkin projections of associated covariance operators. This representation allows for the generalization of sequential Bayesian filtering and Koopman spectral mode decomposition within the learned feature space. Experimental evaluations across various scenarios demonstrate that DSE achieves stable and superior performance compared to traditional sequential Bayesian filtering and dynamic mode decomposition baselines, even when faced with significant noise and partial observability in the system.