克里洛夫可观测量的信息内容：一种机器学习方法
The Information Content of Krylov Observables: A Machine Learning Approach
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中文总结 AI 辅助
该研究用机器学习量化克里洛夫空间可观测量的信息，训练小型残差网络等，发现其能确定热场温度，无法重构精细谱形式因子，二阶矩信息盈余是混沌标志，还推导了相关机制和界限。
中文摘要 AI 辅助
我们采用机器学习来量化三个克里洛夫空间可观测量所携带的信息：展宽复杂度$\mathcal{C}(t)$、离散维格纳负性$N(t)$以及归一化负性$\chi(t)=N(t)/|S(t)|$，其中$S(t)$为生存振幅，最近被提议作为二阶矩下落探针（arXiv:2607.04065）。在跨越高斯酉系综（GUE）、高斯正交系综（GOE）和泊松系综、可积的$SL(2,\mathbb{R})$/共形场论（CFT）扇区以及混沌插值$H(\varepsilon)=H_{SL(2,\mathbb{R})}+\varepsilon R_0 W_{GUE}$的约57000个有标签演化的一半数据上训练小型残差网络（16 - 32个神经元）和提升树。任意一个矩都能在$R^2\simeq 0.999$时确定热场温度。两者都无法重构精细谱形式因子（每个系综中$R^2\simeq 0.18$）；对目标进行平滑处理也无法修复此问题，且宽窗口虽有助于消除凹陷 - 斜坡物理现象本身，但谱形式因子严格细化了两个矩。不过在$R^2 = 0.861$时能恢复粗略的$e^S$平台，主要来自$\mathcal{C}(t)$的前20%。一条曲线能以高达98%的准确率识别对称类。在可积扇区，可观测量在信息上是等价的，正如精确的负二项从属要求的那样，而负性最能解决$(h,\alpha)$简并（$N\to h$：0.999）。沿着插值，$\chi$相对于$\mathcal{C}$的不对称差距随着混沌开启，随着能级统计过渡到GUE从 +0.33增长到 +0.77，而原始$N$差距衰减到零。因此，二阶矩信息盈余是混沌的一个标志，由归一化负性专门携带，并且我们为此推导了一种分析机制和一个定量界限。
英文摘要
We employ machine learning to quantify the information carried by three Krylov-space observables: the spread complexity $\mathcal{C}(t)$, the discrete Wigner negativity $N(t)$, and the normalized negativity $χ(t)=N(t)/|S(t)|$, with $S(t)$ the survival amplitude, recently proposed as a second-moment infall probe (arXiv:2607.04065). Small residual networks (16-32 neurons) and boosted trees are trained on half of $\sim 57{,}000$ labeled evolutions spanning the GUE, GOE and Poisson ensembles, the integrable $SL(2,\mathbb{R})$/CFT sector, and the chaos interpolation $H(\varepsilon)=H_{SL(2,\mathbb{R})}+\varepsilon R_0 W_{GUE}$. Either moment determines the thermofield temperature at $R^2\simeq 0.999$. Neither reconstructs the fine spectral form factor ($R^2\simeq 0.18$ in every ensemble); smoothing the target does not repair this, and windows wide enough to help erase the dip-ramp physics itself: the SFF strictly refines both moments. The coarse $e^S$ plateau is nevertheless recovered at $R^2=0.861$, mostly from the first 20% of $\mathcal{C}(t)$. A single curve identifies the symmetry class at up to 98% accuracy. In the integrable sector the observables are informationally equivalent, as exact negative-binomial slaving demands, while the negativity best resolves the $(h,α)$ degeneracy ($N\to h$: 0.999). Along the interpolation the asymmetry gap of $χ$ over $\mathcal{C}$ switches on with chaos, growing from +0.33 to +0.77 as the level statistics cross to GUE, while the raw-$N$ gap decays to zero. The second-moment informational surplus is therefore a signature of chaos, carried specifically by the normalized negativity, and we derive an analytical mechanism and a quantitative bound for it.
