| name | predictable-mean-field-chaos-rnn |
| description | Predictable Mean-Field Chaos in Random Recurrent Networks methodology — Krylov state space analysis revealing latent determinism in mean-field dynamics for analytic nonlinearities with fast Fourier decay. Demonstrates that microscopic sensitivity and predictive complexity are distinct aspects of chaos. Use when: analyzing RNN chaos, mean-field theory, Krylov complexity, Lyapunov exponents, Hamiltonian chaotic dynamics, classical dissipative systems, or prediction theory in recurrent networks. |
Predictable Mean-Field Chaos in Random Recurrent Networks
Background
Dynamical mean-field theory recasts deterministic chaos in random recurrent networks as an effective stochastic process. This skill provides theoretical and computational tools for analyzing when this stochasticity is only apparent, and the mean-field trajectory is uniquely determined by its continuous past.
Core Innovation
For analytic nonlinearities with sufficiently fast Fourier decay, mean-field stochasticity is only apparent: the continuous past of a realized mean-field trajectory uniquely determines its future.
Key Theoretical Result
Mean-field theory is not merely an ensemble description, but a conditional prediction theory for individual trajectories.
Methodology
Krylov State Space Unfolding
Unfolding the power spectrum into a Krylov state space exposes how latent determinism is organized across an infinite hierarchy of temporal modes.
Complexity Bounds
The Krylov growth rate:
- Sets the complexity of finite-resolution prediction
- Upper-bounds the largest Lyapunov exponent in this class of networks
Distinction Framework
Microscopic sensitivity ≠ Predictive complexity
Two distinct aspects of mean-field chaos:
- Microscopic sensitivity: exponential divergence of nearby trajectories
