收缩流的瓦瑟斯坦稳定性：有效速率、欧拉自校正和噪声收紧
Wasserstein Stability of Contracting Flows: Effective Rates, Euler Self-Correction, and Noise Tightening
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中文总结 AI 辅助
研究收缩流的瓦瑟斯坦稳定性，通过用局部收缩率的位移加权分布平均值代替最坏情况速率得出更紧界，给出收缩下自校正欧拉离散化误差特征，证明非线性收缩漂移有更小平稳方差，确立其噪声抑制优势，结果在相关向量场验证。
中文摘要 AI 辅助
收缩理论保证了稳定非线性系统轨迹之间的指数收敛。当初始条件不确定并表示为概率分布时，如在集成控制、贝叶斯估计和生成建模中，这种保证通过瓦瑟斯坦距离扩展到分布层面。然而，经典分布界仅对线性系统是紧的；对于非线性动力学，它可能非常保守，因为它将空间变化的局部收缩率归结为单个最坏情况常数，完全丢弃了分布信息。我们解决了这种保守性的三个具体后果。首先，通过用局部收缩率的位移加权分布平均值代替最坏情况速率，我们得出了更紧的瓦瑟斯坦界，这对每个非线性收缩系统都严格改进了经典界。其次，我们给出了收缩下自校正欧拉离散化误差的第一个理论特征：误差分布是非单调的，在仅取决于收缩率的通用时间达到峰值，然后指数衰减，这是非收缩动力学中不存在的行为。第三，我们证明了非线性收缩漂移总是比具有相同最坏情况收缩率的线性系统实现严格更小的平稳方差，正式确立了非线性控制器的噪声抑制优势。所有结果都在一组具有代表性的一维和二维向量场上得到了验证。
英文摘要
Contraction theory guaranties exponential convergence between trajectories of a stable nonlinear system. When initial conditions are uncertain and represented as probability distributions, as in ensemble control, Bayesian estimation, and generative modeling, this guaranty extends to the distributional level via Wasserstein distance. However, the classical distributional bound is tight only for linear systems; for nonlinear dynamics, it can be significantly conservative because it collapses the spatially varying local contraction rate to a single worst-case constant, discarding distributional information entirely. We address three concrete consequences of this conservatism. First, we derive a tighter Wasserstein bound by replacing the worst-case rate with a displacement-weighted distributional average of the local contraction rate, which strictly improves upon the classical bound for every nonlinear contracting system. Second, we provide the first theoretical characterization of the self-correcting Euler discretization error under contraction: the error profile is non-monotone, peaks at a universal time that depends only on the contraction rate, and then decays exponentially, a behavior absent in non-contracting dynamics. Third, we prove that nonlinear contracting drifts always achieve strictly smaller stationary variance than a linear system sharing the same worst-case contraction rate, formally establishing the noise-rejection advantage of nonlinear controllers. All results are validated on a representative suite of one- and two-dimensional vector fields.
