Inferring bifurcation diagrams with transformers.

Area of Science:

• Dynamical Systems and Chaos Theory
• Machine Learning Applications in Science
• Nonlinear Dynamics

Background:

• Bifurcation diagrams are crucial for understanding nonlinear dynamical systems.
• Estimating these diagrams is challenging, especially when only data, not equations, are available.
• Existing methods struggle with noisy or incomplete data.

Purpose of the Study:

• To develop a data-driven method for directly estimating bifurcation diagrams.
• To overcome the limitations of traditional methods when dealing with arbitrary dynamical systems and limited data.
• To create a robust technique for analyzing nonlinear systems from observational data.

Main Methods:

• A transformer-based deep learning architecture was employed.
• The model was trained to predict the number of segments, their positions, shapes, and stability.
• The method directly processes noisy data from dynamical systems without requiring system equations.

Main Results:

• The transformer model accurately estimated bifurcation diagrams for 1D and 2D systems.
• The method demonstrated reliability even with limited trajectories (as few as 30).
• The approach proved robust against noise in both state variables and system parameters.

Conclusions:

• A powerful, data-driven tool for analyzing nonlinear dynamical systems has been developed.
• This method significantly advances the ability to study complex systems where underlying equations are unknown or difficult to derive.
• The transformer-based approach offers a robust and efficient alternative for bifurcation analysis.