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

Yuanchao Xu; Claire Valva; Ben Gao; Isao Ishikawa

math.DSOct 29, 2026

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

Authors: Yuanchao Xu, Claire Valva, Ben Gao, Isao Ishikawa

Organizations: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan · Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA · Laboratoire Hubert Curien, UMR 5516, CNRS, IOGS, INRIA, Université Jean Monnet, 18 Rue du Professeur Benoît Lauras, 42000 Saint-Étienne, France

Abstract

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.

Figures & tables

Figure 1: OU spectral approximation at

\tau=0.15

. Top: analytic generator rates and the BSO (learned dictionary) estimates, labeled “trained basis” in the legend, each matched to the nearest analytic rate. Bottom: postprocessed BSO (learned dictionary) eigenfunctions and the analytic Hermite modes on

|x|\leq 3

Figure 2: Triple-well spectral approximation at the lag

3\tau=0.9

. Top: leading BSO (learned dictionary) multipliers and finer-grid finite-difference references. Bottom: the stationary, left–right, and upper–lower modes, paired by one-to-one matching; the panel labels give the continuous-time rates

\lambda=(3\tau)^{-1}\log\mu

corresponding to the multipliers

\mu

of the top panel ( section 2.2 ).

Figure 3: Pooled relative

L^{2}

forecast error of BSO (fixed dictionary) and the autoregressive FNO over 64 held-out origins for Lorenz–63 (top) and Lorenz–96 (bottom). The short-term Lorenz–63 panel uses a logarithmic vertical axis; the remaining panels use linear axes with different vertical scales.

Figure 4: Global SST evaluation. Top: forecast skill of BSO (fixed dictionary), FNO, SFNO, and the three baselines over 96 held-out 2020–2022 origins, showing area-weighted RMSE and MAE through day 30 and RMSE and centered anomaly ACC through day 90. The SFNO curve ends at the first lead time for which scores are not available for all 96 origins. Bottom: area-weighted SST RMSE in the 540-day numerical-stability test over 48 origins, shown over the full and a zoomed logarithmic range.

Method	Error $E_{40}^{\rm EEG}$	Reduction with BSO
BSO	0.749	–
Direct FNO	0.874	$14.2\%$
EDMD-DL	0.941	$20.4\%$
Kernel EDMD	0.945	$20.7\%$
PCA–VAR	0.963	$22.2\%$

Table 1: EEG experiment: relative

L^{2}

error accumulated through step 40 of the four-second rollout,

E_{40}^{\rm EEG}

, and the relative reduction achieved by BSO with respect to each comparator, computed from unrounded errors.

Figure 5: Cumulative relative

L^{2}

error of BSO (learned dictionary) and the comparators for the four-second autonomous rollout on the S029R02 resting EEG record. Each step advances 0.1 seconds.

Figure 6: One-step prediction error on held-out transition pairs (relative

L^{2}

) versus the number

n_{\rm op}

of observed transition pairs for Lorenz–96 with 40 sites and

m=40

(vertical dashed line). BSO (fixed dictionary) is available for every

n_{\rm op}

shown. The dashed and dotted curves are the ridge-regularized and minimum-norm

m=40

EDMD fits, which coincide to within

10^{-4}

; the solid EDMD curve uses the first

\min\{n_{\rm op},m\}

dictionary coordinates. In the legend,

n

stands for

n_{\rm op}

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Field	Source (variable)	Level / depth	Unit
SST (target)	WB2 (ERA5 SST)	surface	K
Zonal wind stress	WB2 (from 10 m winds)	surface	N m -2
Meridional wind stress	WB2 (from 10 m winds)	surface	N m -2
Sea-surface height	GODAS ( sshg )	surface	m
Potential temperature	GODAS ( pottmp )	55 m	∘ C
Potential temperature	GODAS ( pottmp )	105 m	∘ C

Appendix

Table 2: Input fields of the SST experiment on the

32\times 64

grid. WB2: WeatherBench 2 archive of ERA5. Units are those of the fields as used, before standardization. Only SST is verified.

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Operator learning studies models that map functions or fields to functions, enabling fast prediction across scientific and dynamical systems.

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From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek

University of Chicago

Dynamical Systems Fourier Neural Operator PDE Operator Learning Neural Network Approximation Theory Neural Operators

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We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-

T

solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

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