Using machine learning to predict extreme events in the Hénon map.

Area of Science:

• Dynamical Systems and Chaos Theory
• Machine Learning Applications
• Computational Physics

Background:

• Chaotic dynamical systems present significant challenges for prediction.
• Machine learning (ML) offers novel approaches for analyzing and forecasting such systems.
• The Hénon map is a classic model for studying chaotic behavior.

Purpose of the Study:

• To evaluate the performance of a machine learning algorithm for predicting extreme events in the 2D Hénon map.
• To understand the relationship between ML model parameters and the chaotic dynamics of the system.
• To establish a geometric interpretation for assessing ML model accuracy.

Main Methods:

• Utilized a machine learning algorithm to predict trajectories exceeding a threshold in the Hénon map.
• Analyzed the geometric interpretation of the prediction task within the Hénon map dynamics.
• Investigated the impact of prediction time (T), training samples (N_T), and network size (N_p) on prediction success rate.

Main Results:

• The success rate of ML models was analyzed concerning prediction time, training data size, and network architecture.
• Observed that to maintain accuracy, the number of training samples scales as N_T∝exp(2hT) and network size as N_p∝exp(hT).
• Topological entropy (h) was identified as a key factor relating chaotic dynamics to ML parameter requirements.

Conclusions:

• Machine learning algorithms demonstrate potential for forecasting extreme events in chaotic systems.
• The study reveals a quantitative relationship between chaotic properties (topological entropy) and the necessary ML model complexity and training data.
• These findings suggest that similar scaling relationships may apply to other chaotic dynamical systems.