发表机构
College of Computing, Mohammed VI Polytechnic University (UM6P)（计算学院，穆罕默德六世polytechnic大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出速度梯度张量非对角分量的符号成对图元离散编码，发现不平衡量Δn可稳健预测Q符号及PQR拓扑，虽丢弃幅度信息仍显著提升预测概率，且预测能力主要源于非对角贡献。
AI 中文摘要
本文引入速度梯度张量 $\bm{A}$ 的符号成对图元编码，作为其非对角结构的极简离散表示，并利用 $Re_\lambda \approx 433$ 的直接数值模拟数据，研究该表示与均匀各向同性湍流 PQR 拓扑之间的统计关系。每个非对角分量根据局部阈值被赋予三元标签，将阈值化后的非对角符号模式映射到 $3^6=729$ 个离散状态之一。旋转促进对与应变促进对之间的不平衡量 $\Delta n=n_W-n_S$ 被证明是 $Q$ 符号的稳健代理指标：$P(Q>0\mid\Delta n)$ 从 $\Delta n\leq-2$ 时的 $0\\%$ 单调递增至 $\Delta n=+3$ 时的 $100\\%$，且仅含单个旋转促进对（$\Delta n=+1$）的图元预测稳定焦点拉伸区域的概率约为 $65\\%$——尽管丢弃了所有幅度信息，该概率仍接近全局基准率 $36.5\\%$ 的两倍。这种预测能力主要归因于 $\bm{A}$ 的非对角贡献，在大多数流动状态中，该贡献主导了 $Q$ 的符号。图元与 PQR 的关联在多个 Kolmogorov 时间尺度上保持统计相关性，并在耗散尺度上表现出与 $Q$ 场相当的空间相干性，这为离散阈值化保留了速度梯度场固有的短程组织提供了一致性检验。进一步的样本外预测对数损失测试表明，$\Delta n$ 捕获了完整 $729$ 状态图元代码的大部分（但非全部）预测内容：完整代码在预测未来 PQR 拓扑时，相对于 $\Delta n$ 和连续不变量，保留了每次观测约 $10^{-3}$ 比特的小幅可复现增益，该增益在有限滞后时存在，在零滞后时不存在。
英文摘要
A signed pairwise graphlet encoding of the velocity gradient tensor $\bm{A}$ is introduced as a minimal discrete representation of its off-diagonal structure, and its statistical relationship with the PQR topology of homogeneous isotropic turbulence is investigated using direct numerical simulation data at $Re_λ\approx 433$. Each off-diagonal component is assigned a ternary label based on a local threshold, mapping the thresholded off-diagonal sign pattern to one of $3^6=729$ discrete states. The imbalance $Δn=n_W-n_S$ between rotation-promoting and strain-promoting pairs is shown to act as a robust proxy for the sign of $Q$: $P(Q>0\midΔn)$ increases monotonically from $0\%$ at $Δn\leq-2$ to $100\%$ at $Δn=+3$, and graphlets with a single rotation-promoting pair $(Δn=+1)$ predict the stable-focus-stretching region with probability approximately $65\%$ -nearly double the global base rate of $36.5\%$- despite discarding all magnitude information. This predictive power can be largely attributed to the off-diagonal contribution of $\bm{A}$, which dominates the sign of $Q$ in a majority of flow states. The graphlet--PQR association remains statistically correlated over several Kolmogorov time scales and exhibits spatial coherence comparable to that of the $Q$ field at dissipative scales, providing a consistency check that the discrete thresholding preserves the short-range organisation inherited from the velocity-gradient field. An out-of-sample predictive log-loss test further shows that $Δn$ captures most, but not all, of the predictive content of the complete $729$-state graphlet code: the full code retains a small, reproducible gain of order $10^{-3}$ bits per observation in predicting future PQR topology beyond $Δn$ and the continuous invariants, present at finite lag and absent at zero lag.
