Learning Kolmogorov Backward Operators for Spectral Analysis and Forecasting: A Data-Driven Variational Galerkin Framework

For a time-homogeneous Markov process, the fixed-lag solution operator \( (T_τf)(x)=\mathbb E[f(X_{t+τ})\mid X_t=x] \) of the Kolmogorov backward equation evolves observables linearly and deterministically, even when the dynamics are stochastic. We approximate this operator using a two-stage method where we first approximate this operator directly from data, without knowledge of the governing equation (e.g., Fourier Neural Operator), and then compress the Galerkin projection onto a finite-dimensional observable space that is fixed in advance or selected variationally. The error analysis traces the learning error of the estimator, the sampling error of the Galerkin inner products, and the invariance and reconstruction properties of the observable space to the computed spectra and forecasts. The sampling error enters only through a multiplicative factor. Near an isolated spectral cluster, the invariance defect of a subspace controls its distance from the target invariant subspace. The forecast error is controlled by the one-step error of the reduced model and the projection error of the observable; when the measure is invariant and the reduced model preserves constants and is stable on mean-zero observables, the bound is uniform in the horizon. For deterministic dynamics, a learning rate for Fourier neural operators (FNOs) gives one explicit bound for the learning error. Numerical experiments illustrate spectral approximation and forecasting, including long-rollout stability and a comparison with extended dynamic mode decomposition (EDMD) when fewer transition pairs than basis functions are observed.