Learning dynamical systems with hit-and-run random feature maps.

Area of Science:

• Dynamical Systems and Chaos Theory
• Machine Learning
• Computational Science

Background:

• Forecasting complex dynamical systems is challenging due to their inherent nonlinearity and chaotic behavior.
• Traditional methods often require extensive hyperparameter tuning and struggle with high-dimensional data.
• Random feature maps offer a promising alternative for modeling complex systems.

Purpose of the Study:

• To develop and evaluate a novel forecasting method for chaotic dynamical systems using modified random feature maps.
• To demonstrate the efficacy of data-driven weight selection and architectural enhancements for improved forecasting skill.
• To compare the performance of the proposed method against existing techniques like reservoir computing.

Main Methods:

• Utilized random feature maps with a tanh activation function and data-driven internal weight selection.
• Introduced skip connections to create deep variants of random feature maps.
• Incorporated localization and conditional independence to mitigate the curse of dimensionality.
• Applied the method to chaotic dynamical systems with dimensions up to 512.

Main Results:

• Achieved excellent forecasting skill for both single trajectories and long-time statistical properties.
• Demonstrated superior performance on a range of chaotic dynamical systems.
• Showcased the method's ability to handle high-dimensional systems (up to 512 dimensions).
• Required tuning of only a single hyperparameter, significantly reducing computational effort.

Conclusions:

• Modified random feature maps provide a powerful and efficient tool for forecasting chaotic dynamical systems.
• The proposed method achieves state-of-the-art forecasting skill with significantly reduced network size and hyperparameter tuning.
• This approach offers a scalable and effective solution for complex system modeling and prediction.