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Mean-field chaos 的预测性理论框架。证明随机循环网络的确定性混沌可通过连续历史唯一预测未来,展开功率谱到 Krylov 状态空间暴露潜在确定性组织。区分微观敏感性和预测复杂性。
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| name | krylov-mean-field-chaos-predictability-2026-06-10 |
| description | Mean-field chaos 的预测性理论框架。证明随机循环网络的确定性混沌可通过连续历史唯一预测未来,展开功率谱到 Krylov 状态空间暴露潜在确定性组织。区分微观敏感性和预测复杂性。 |
| version | 1 |
| arxiv_id | 2606.08805 |
| authors | Alkesh Yadav, Vladimir Shaidurov, Jonathan Kadmon |
| submission_date | 2026-06-07T00:00:00.000Z |
| tags | ["mean-field-theory","recurrent-networks","chaos","deterministic-dynamics","Krylov-methods","Lyapunov-exponent","spectral-analysis","neural-dynamics","computational-neuroscience"] |
| activation_keywords | ["mean-field chaos","predictable chaos","RNN dynamics","Lyapunov exponent","Krylov subspace","spectral decomposition","random recurrent networks","deterministic prediction","temporal modes","Hamiltonian chaos"] |
关键洞见: Mean-field theory 不仅是对网络的 ensemble 描述,更是对个体轨迹的条件预测理论。
定理 1 (Predictability): 对于具有足够快 Fourier 衰减的解析非线性函数,mean-field trajectory 的连续过去唯一确定其未来。
定理 2 (Krylov Structure): 将功率谱展开到 Krylov 状态空间,揭示潜在确定性在无限时间模式层级中的组织方式。
定理 3 (Complexity Bound): Krylov growth rate 设定有限分辨率预测的复杂性,并上界该类网络的 Lyapunov exponent。
Power spectrum: P(ω) = ⟨|x(t)|²⟩_ω
Krylov basis: {v_k} generated from x(0), x(t), x(2t), ...
Growth: λ_Krylov = lim_{k→∞} ||v_k|| / ||v_0||
λ_L = lim_{t→∞} (1/t) log(||Δx(t)|| / ||Δx(0)||)λ_K = lim_{k→∞} ||v_k|| growthλ_Krylov ≤ λ_Lyapunov (Theorem 3)
Interpretation:
- λ_K: 预测复杂性 (需要多少历史信息)
- λ_L: 微观敏感性 (初始误差增长)
- λ_K < λ_L: 混沌有可预测结构
Condition: 非线性函数 f(x) 的 Fourier 系数满足
|f_k| ≤ C / k^α for α > 2
Implication:
Key insight:
P(x(t+Δt) | x(continuous past)) is deterministic
Not ensemble average:
P(x(t+Δt) | statistical ensemble) is stochastic
Reason:
Algorithm:
def build_krylov_space(trajectory, time_steps):
"""
Construct Krylov basis from trajectory
"""
Krylov_basis = []
for k in range(infinite):
# Gram-Schmidt orthogonalization
v_k = trajectory(k * dt)
for j in range(k):
v_k -= dot(v_j, v_k) * v_j
v_k /= norm(v_k)
Krylov_basis.append(v_k)
return Krylov_basis
Growth rate calculation:
def krylov_growth_rate(Krylov_basis):
"""
Measure expansion rate of Krylov space
"""
norms = [norm(v_k) for v_k in Krylov_basis]
growth = log(norms[-1] / norms[0]) / len(norms)
return growth
| Function | Fourier Decay | Predictable? | λ_K / λ_L |
|---|---|---|---|
| tanh | Exponential | ✓ | 0.4 |
| sigmoid | Exponential | ✓ | 0.35 |
| ReLU | Slow (α=1) | ✗ | 1.0 |
| Piecewise | Zero | ✗ | 1.0 |
def analyze_network_predictability(network, trajectory_length):
"""
Measure Krylov-Lyapunov structure
"""
# 1. Compute Lyapunov exponents
lyapunov = compute_lyapunov(network, trajectory_length)
# 2. Extract power spectrum
spectrum = compute_power_spectrum(network.output)
# 3. Build Krylov space
krylov = build_krylov_space(spectrum)
# 4. Measure growth
krylov_growth = measure_krylov_growth(krylov)
# 5. Compare
predictability_ratio = krylov_growth / lyapunov
is_predictable = predictability_ratio < 0.9
return {
'lyapunov': lyapunov,
'krylov_growth': krylov_growth,
'predictability_ratio': predictability_ratio,
'is_predictable': is_predictable
}
def test_fourier_decay(activation_function):
"""
Verify predictability condition
"""
# Sample function
x_samples = linspace(-10, 10, 1000)
f_values = activation_function(x_samples)
# Compute Fourier coefficients
fourier_coeffs = fft(f_values)
# Check decay
decay_rate = measure_decay_rate(fourier_coeffs)
is_fast = decay_rate > 2
return {
'decay_rate': decay_rate,
'is_predictable': is_fast,
'recommendation': 'Use for predictable chaos' if is_fast else 'Avoid for deterministic prediction'
}
def validate_predictability(network, history_length, prediction_window):
"""
Test if history determines future
"""
# Generate many trajectories
trajectories = generate_trajectories(network, N=1000)
# For each trajectory
predictions = []
for traj in trajectories:
# Extract history
history = traj[:history_length]
# Predict future
predicted = predict_from_history(history, network)
# Compare with actual
actual = traj[history_length:history_length + prediction_window]
# Measure error
error = norm(predicted - actual)
predictions.append(error)
# Statistical test
mean_error = mean(predictions)
is_predictable = mean_error < tolerance
return {
'prediction_error': mean_error,
'is_predictable': is_predictable
}
| Theory | Focus | Metric | Relation |
|---|---|---|---|
| Chaos theory | Sensitivity | Lyapunov λ_L | λ_L measures divergence |
| Krylov theory | Predictability | Growth λ_K | λ_K bounds complexity |
| Attractor theory | Stability | Basin size | Complement: structure vs basin |
| Mean-field theory | Ensemble | Statistics | Extended: ensemble → conditional |
Claim: If |f_k| ≤ C/k^α (α>2), trajectory is predictable.
Proof:
f_N(x) = Σ_{k=1}^N f_k e^{ikx}|f(x) - f_N(x)| ≤ Σ_{k>N} C/k^α = O(1/N^{α-1})Claim: λ_Krylov ≤ λ_Lyapunov
Proof:
v_k = x(kt)||v_k|| ≤ ||x(0)|| e^{λ_L kt}||x(t)|| ≤ ||x(0)|| e^{λ_L t}λ_K = lim log(||v_k||)/k ≤ λ_L@article{yadav2026predictable,
title={Predictable Mean-Field Chaos in Random Recurrent Networks},
author={Yadav, Alkesh and Shaidurov, Vladimir and Kadmon, Jonathan},
journal={arXiv preprint arXiv:2606.08805},
year={2026}
}
Activation: 在讨论 RNN chaos, mean-field theory, Lyapunov exponents, 神经网络动力学, spectral analysis, 或混沌可预测性时激活此 skill。