Deep learning-based state prediction of the Lorenz system with control parameters.

Area of Science:

• Computational physics
• Artificial intelligence
• Nonlinear dynamics

Background:

• Shallow neural networks struggle to model complex nonlinear dynamical systems.
• Parameter-dependent systems like the Lorenz system present significant modeling challenges.

Purpose of the Study:

• To develop a deep learning model capable of simultaneously learning diverse solutions (fixed-point, periodic, chaotic) of nonlinear systems.
• To investigate the effectiveness of very deep neural networks for modeling the parameter-dependent Lorenz system.

Main Methods:

• A very deep neural network architecture with numerous identical linear layers was employed.
• Residual connections were incorporated to facilitate information flow.
• A large dataset was utilized for training the deep learning model.

Main Results:

• The deep learning model accurately forecasted chaotic solutions for several Lyapunov times.
• Long-term predictions were successfully achieved for periodic solutions.
• Dynamical characteristics, including bifurcation diagrams and largest Lyapunov exponents, were well recovered.

Conclusions:

• Very deep neural networks offer superior nonlinear mapping capabilities for complex dynamical systems.
• The proposed deep learning approach enables accurate long-term prediction and characterization of nonlinear dynamics.
• The study highlights the potential of deep learning in advancing the understanding and prediction of chaotic systems.