Grading your models: Assessing dynamics learning of models using persistent homology.

Area of Science:

• Complex Systems
• Dynamical Systems Theory
• Topological Data Analysis

Background:

• Assessing model accuracy for chaotic systems is challenging.
• Traditional methods using Lyapunov exponents and correlation dimensions have limitations.
• These limitations include lack of uniqueness, noise sensitivity, and inability to detect manifold density errors.

Purpose of the Study:

• To explore persistent homology for evaluating model accuracy in complex systems.
• To introduce a novel topological data analysis metric for assessing model performance.
• To address the limitations of existing dynamical invariant-based assessment methods.

Main Methods:

• Applied persistent homology to uniformly sampled trajectories from Lorenz system reservoir models.
• Utilized topological data analysis to analyze the geometric and topological features of model dynamics.
• Developed a new metric, "conformance," based on persistent homology summaries.

Main Results:

• Persistent homology successfully identified models exhibiting "dynamics learning."
• The "conformance" metric detected discrepancies in manifold density distribution, a pathological error.
• The technique proved robust in assessing model quality beyond traditional dynamical invariants.

Conclusions:

• Persistent homology offers a powerful new approach for assessing complex system models.
• The "conformance" metric provides a robust and sensitive measure of model fidelity.
• This method enhances the ability to detect subtle model errors and validate dynamics learning.