Consistent Long-Term Forecasting of Ergodic Dynamical Systems

2023/12/20 by Prune Inzerilli, Vladimir Kostić, Inzerilli, Prune +7 · 5 citations
Computer Science · Decision Sciences · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2312.13426

openalex publication_date 2023/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We study the evolution of distributions under the action of an ergodic dynamical system, which may be stochastic in nature. By employing tools from Koopman and transfer operator theory one can evolve any initial distribution of the state forward in time, and we investigate how estimators of these operators perform on long-term forecasting. Motivated by the observation that standard estimators may fail at this task, we introduce a learning paradigm that neatly combines classical techniques of eigenvalue deflation from operator theory and feature centering from statistics. This paradigm applies to any operator estimator based on empirical risk minimization, making them satisfy learning bounds which hold uniformly on the entire trajectory of future distributions, and abide to the conservation of mass for each of the forecasted distributions. Numerical experiments illustrates the advantages of our approach in practice.

Citations

• Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
• Koopman spectra in reproducing kernel Hilbert spaces
• Statistical Inference for Generative Models with Maximum Mean Discrepancy
• Modern Koopman Theory for Dynamical Systems
• Optimal Rates for Regularized Conditional Mean Embedding Learning
• Sharp Spectral Rates for Koopman Operator Learning
• Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
• Koopman Kernel Regression
• Estimating Koopman operators with sketching to provably learn large scale dynamical systems

Cited by

• kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learning
• Sequence Modeling with Spectral Mean Flows
• Information Shapes Koopman Representation
• A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems
• Maximum-likelihood reprojections for reliable Koopman-based predictions and bifurcation analysis of parametric dynamical systems

Related

• Bridging data science and dynamical systems theory
• Data-driven spectral decomposition and forecasting of ergodic dynamical systems
• Data-driven spectral decomposition and forecasting of ergodic dynamical systems
• Learning the ergodic decomposition
• Learning Ergodic Dynamical Systems from a Finite Trajectory