Physical Networks Become What They Learn.

Area of Science:

• Complex Systems
• Network Science
• Computational Physics

Background:

• Physical networks can achieve desired functions through design, evolution, or learning.
• Learning in many physical networks involves minimizing a global quantity (Lyapunov function) alongside task-specific costs.
• This process represents a
• double optimization
• where task and physical objectives are minimized concurrently.

Purpose of the Study:

• To elucidate the relationship between physical constraints and learned functions in adaptable networks.
• To demonstrate how the learning process couples the cost and physical landscapes.
• To show that physical responses to perturbations reveal a network's learned function.

Main Methods:

• Formulating learning as a double optimization problem involving coupled variables.
• Analyzing the coupling between the cost and physical landscapes via Hessian matrices.
• Using electrical networks with adaptable resistors as an experimental model.

Main Results:

• The learning process establishes a direct link between the high-dimensional cost landscape and the physical landscape.
• Physical and cost Hessian matrices become coupled, reflecting the optimization process.
• Physical responses of trained networks to perturbations accurately indicate their tuned functions.

Conclusions:

• The study provides a unified framework for understanding learning in physical networks that minimize global quantities.
• Physical responses serve as a direct readout of learned functions in adaptable networks.
• The findings are applicable to a broad range of networks operating in the linear response regime.