Learning Ergodic Dynamical Systems from a Finite Trajectory

2026-07-24 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, quick

Summary

This paper, published on 2026-07-24, addresses the challenge of learning from a single finite trajectory generated by an ergodic stochastic dynamical system, specifically discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. The authors focus on estimating the optimal one-step prediction function using nonlinear least squares, providing high-probability guarantees measured against the process's invariant measure. A key contribution is explicitly detailing how the non-independent and non-identically distributed nature of trajectory data impacts classical statistical learning analysis. The framework extends to higher-order systems, finite-state spaces, and learning Koopman operators, combining statistical learning theory with quantitative ergodic theory for Markov chains, particularly leveraging a concentration inequality for Hilbert-space-valued additive functionals.

Key takeaway

For research scientists developing models for stochastic dynamical systems, this work provides a robust framework for learning from limited, non-independent and non-identically distributed trajectory data. You should consider integrating nonlinear least squares with quantitative ergodic theory to derive high-probability guarantees, especially when extending to Koopman operators. This approach offers a principled way to handle the statistical challenges inherent in single-trajectory learning.

Key insights

This work details learning ergodic dynamical systems from single finite trajectories, accounting for non-IID data.

Principles

• Non-IID trajectory data alters classical statistical learning analysis.
• Least squares and concentration arguments extend to Koopman operators.

Method

The approach estimates optimal one-step prediction functions via nonlinear least squares, then extends this framework to higher-order systems, finite-state spaces, and Koopman operators.

In practice

• Applying statistical learning to ergodic system trajectories.
• Extending prediction functions to Koopman operator learning.

Topics

• Ergodic Dynamical Systems
• Stochastic Processes
• Statistical Learning Theory
• Markov Chains
• Koopman Operators
• Nonlinear Least Squares

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.