Krylov Mean-Field Chaos theory for random recurrent networks — demonstrating that deterministic chaos has latent predictability through Krylov state space decomposition. Extends Hamiltonian chaos concepts to classical dissipative systems.
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Krylov Mean-Field Chaos theory for random recurrent networks — demonstrating that deterministic chaos has latent predictability through Krylov state space decomposition. Extends Hamiltonian chaos concepts to classical dissipative systems.
Predictable Mean-Field Chaos in Random Recurrent Networks
authors
["Dynamical Systems Research"]
published
2026-06-07T00:00:00.000Z
Predictable Mean-Field Chaos in Random Recurrent Networks
Overview
This framework demonstrates that deterministic chaos in random recurrent networks with analytic nonlinearities has latent predictability — the continuous past uniquely determines the future. By unfolding the power spectrum into a Krylov state space, we expose how this hidden determinism is organized across an infinite hierarchy of temporal modes.
Revolutionary Finding
Key Discovery: For networks with analytic nonlinearities and sufficient Fourier decay:
Chaos is apparently stochastic but fundamentally deterministic
Continuous past trajectory uniquely predicts future
Mean-field theory becomes conditional prediction theory for individual trajectories
Krylov growth rate sets prediction complexity
Microscopic sensitivity ≠ predictive complexity (they are distinct)
In mean-field limit (N → ∞):
$$
\frac{dX}{dt} = -X + g \cdot \langle \phi(X) \rangle_W
$$
Krylov Decomposition
Step 1: Power Spectrum Analysis
defcompute_power_spectrum(x_trajectory):
"""
x_trajectory: [x(t₀), x(t₁), ..., x(tT)]
Returns: P(ω) for all frequencies
"""
x_hat = fft(x_trajectory)
P = np.abs(x_hat)**2return P
Step 2: Spectral Decay Characterization
defcheck_fourier_decay(P, threshold):
"""
Verify sufficient Fourier decay
Required for Krylov predictability
"""# P(k) should decay faster than 1/k^α# for some α > thresholdpass
Step 3: Krylov Mode Construction
defbuild_krylov_basis(P, N_modes):
"""
Unfold power spectrum into Krylov modes
ψ₀: slowest temporal mode
ψ₁: next frequency band
...
ψ_N: Nth mode
"""
krylov_modes = []
for n inrange(N_modes):
# Extract frequency band [ω_n, ω_{n+1}]# Construct Krylov mode ψ_npassreturn krylov_modes
Step 4: Prediction from Krylov Modes
defpredict_future(krylov_modes, growth_rate, T_future):
"""
Given past Krylov representation
Predict future trajectory
Error bounded by growth_rate × truncation level
"""# Extrapolate each mode# Reconstruct full trajectorypass
Lyapunov vs Krylov
Lyapunov Exponent (λ_max)
Measures microscopic sensitivity
Nearby trajectories diverge exponentially
Rate: |Δx(t)| ~ exp(λ_max·t)
Krylov Growth Rate (κ)
Measures prediction complexity
Finite-resolution prediction difficulty
Upper bound: κ ≥ λ_max
Key Insight
Different aspects of chaos:
λ_max: how fast nearby trajectories diverge
κ: how hard to predict from past history
κ bounds λ_max but they're distinct
Validation & Examples
Synthetic Networks
# Test case 1: Analytic φ
φ(x) = tanh(x) # Analytic, fast Fourier decay
→ Predictable chaos confirmed
# Test case 2: Non-analytic φ
φ(x) = ReLU(x) # Not analytic
→ Prediction breaks down
# Test case 3: Slow Fourier decay
φ(x) = some_slow_decay_function
→ Higher Krylov growth rate
Numerical Verification
Generate random W matrix (Gaussian)
Evolve dynamics for long time T
Compute power spectrum P(ω)
Check Fourier decay rate
Construct Krylov modes
Test prediction: past → future
Implications for Neuroscience
1. Neural Network Dynamics
Recurrent networks in brain: cortex, hippocampus
Chaotic dynamics: observed in neural recordings
Predictability: May be more predictable than thought
Krylov modes: Temporal organization of neural chaos
2. Prediction in Biological Systems
State prediction: From neural history to future activity
Information processing: Krylov hierarchy as computation
Memory: Past encoded in Krylov representation
Computation: Chaos as deterministic information flow
3. Chaos vs Randomness
Apparent stochasticity: Seemingly random neural dynamics
Hidden determinism: Structured information in chaos
Biological advantage: Predictability from structure
Encoding: Past trajectory as information carrier
4. Network Architecture Effects
Analytic activations: Predictability preserved
Non-analytic activations: Predictability breaks
Implication: Biological networks may use analytic nonlinearities
ReLU networks: Might sacrifice predictability
Comparison: Hamiltonian vs Dissipative
Hamiltonian Chaos (Previous Work)
Conservative systems (energy preserved)
Krylov concepts developed
Quantum/classical chaos connection
Dissipative Chaos (This Work)
Energy dissipating systems
Random recurrent networks
Extension of Krylov theory to classical dissipative case
Key Difference
Aspect
Hamiltonian
Dissipative (This Work)
Energy
Conserved
Dissipated
Dynamics
Reversible
Irreversible
Chaos type
Conservative
Dissipative
Application
Quantum systems
Neural networks
Theoretical Significance
1. Chaos Theory Revision
Chaos not purely unpredictable
Latent determinism in chaotic systems
Prediction possible with continuous history
2. Mean-Field Theory Upgrade
From ensemble statistics to individual prediction
Conditional on past trajectory
Practical prediction framework
3. Krylov Methods Extension
From Hamiltonian to dissipative systems
Classical neural dynamics
Infinite hierarchy organization
4. Complexity Measures
Lyapunov: Sensitivity (how fast divergence)
Krylov growth: Predictability (how hard to predict)
Separation: Different aspects of chaos
Implementation Guide
Requirements
import numpy as np
from scipy.fft import fft, ifft
from scipy.integrate import odeint
# Network parameters:
N = 1000# Network size (mean-field: N → ∞)
g = 1.5# Coupling strength (chaos regime: g > 1)
defanalyze_spectrum(x_trajectory):
"""
Compute power spectrum and check decay
"""# FFT of trajectory
x_hat = fft(x_trajectory[:, 0]) # Analyze one neuron# Power spectrum
P = np.abs(x_hat)**2# Frequency bins
freqs = np.fft.fftfreq(len(x_hat), d=T[1]-T[0])
# Check decay rate:# P(k) should decay exponentially or as power law# with sufficient ratereturn P, freqs
Krylov Mode Extraction
defkrylov_decomposition(P, freqs, N_modes=10):
"""
Unfold spectrum into Krylov modes
Divide frequency range into bands
Each band = one Krylov mode
"""# Sort by frequency magnitude
sorted_idx = np.argsort(np.abs(freqs))
# Divide into N_modes bands
band_size = len(P) // N_modes
krylov_modes = []
for n inrange(N_modes):
# Extract band [n*band_size, (n+1)*band_size]
band_indices = sorted_idx[n*band_size:(n+1)*band_size]
# Construct mode
psi_n = np.sum(P[band_indices]) # Energy in band
krylov_modes.append(psi_n)
return krylov_modes
Krylov Growth Rate Calculation
defcompute_growth_rate(krylov_modes):
"""
Growth rate from Krylov mode decay
κ = lim_{n→∞} log(ψ_n)/n
"""# Fit exponential decay
n = np.arange(len(krylov_modes))
log_psi = np.log(krylov_modes)
# Linear regression: log(ψ_n) = κ·n
kappa = np.polyfit(n, log_psi, 1)[0]
return kappa
Prediction Test
deftest_prediction(x_trajectory, T_train, T_test):
"""
Test if past predicts future
Train on T_train, predict T_test
"""# Split trajectory
x_train = x_trajectory[:len(T_train)]
x_test = x_trajectory[len(T_train):len(T_train)+len(T_test)]
# Compute Krylov from training
P_train, freqs = analyze_spectrum(x_train)
krylov_train = krylov_decomposition(P_train, freqs)
kappa = compute_growth_rate(krylov_train)
# Predict test from Krylov extrapolation# (Implementation depends on specific method)# Compare prediction vs actual
error = np.linalg.norm(x_pred - x_test)
return error, kappa
Key Results
For analytic φ with fast Fourier decay:
Past uniquely predicts future
Krylov growth rate finite
Prediction complexity bounded
For non-analytic or slow decay:
Prediction breaks down
Higher Krylov growth rate
More chaotic (larger κ)
Lyapunov exponent:
Upper bounded by Krylov growth rate
κ ≥ λ_max always
Sometimes κ >> λ_max (distinct aspects)
Applications
Neural Dynamics Analysis
Compute Krylov modes from neural recordings
Assess predictability from spectrum decay
Identify networks with hidden determinism
Chaos Characterization
Separate sensitivity (Lyapunov) from predictability (Krylov)
