Learning transfer operators by kernel density estimation

Area of Science:

• Dynamical Systems and Ergodic Theory
• Statistical Inference
• Numerical Analysis

Background:

• Transfer operator inference is crucial for analyzing dynamical systems.
• The Ulam method, specifically the Ulam-Galerkin approach, is a conventional technique.
• This method can be viewed as density estimation using histograms.

Purpose of the Study:

• To recast the inference of transfer operators within statistical density estimation.
• To enable rigorous analysis of bias, variance, and mean square error.
• To evaluate the performance of different density estimation techniques.

Main Methods:

• Formulating transfer operator inference as a statistical density estimation problem.
• Applying histogram density estimation (HDE) and kernel density estimation (KDE).
• Analyzing bias-variance trade-offs and mean square error.

Main Results:

• Kernel density estimation (KDE) generally shows higher accuracy than histogram density estimation (HDE).
• KDE demonstrates limitations near boundary points and discontinuities.
• The study validates the effectiveness of density estimation for Frobenius-Perron operator eigenvector estimation.

Conclusions:

• Statistical density estimation provides a powerful framework for transfer operator inference.
• KDE is a promising, though not perfect, method for this task.
• Future research should explore other density estimation methods and high-dimensional applications.