This paper provides answers to an open problem: given a nonlinear data-driven dynamical system model, e.g., kernel conditional mean embedding (CME) and Koopman operator, how can one propagate the ambiguity sets forward for multiple steps? This problem is the key to solving distributionally robust control and learning-based control of such learned system models under a data-distribution shift. Different from previous works that use either static ambiguity sets, e.g., fixed Wasserstein balls, or dynamic ambiguity sets under known piece-wise linear (or affine) dynamics, we propose an algorithm that exactly propagates ambiguity sets through nonlinear data-driven models using the Koopman operator and CME, via the kernel maximum mean discrepancy geometry. Through both theoretical and numerical analysis, we show that our kernel ambiguity sets are the natural geometric structure for the learned data-driven dynamical system models.

翻译：本文针对一个开放性问题给出了答案：在给定非线性数据驱动动力学系统模型（例如核条件均值嵌入和库普曼算子）的情况下，如何将模糊集向前传播多个步长？该问题是解决数据分布偏移下此类学习系统模型的分布鲁棒控制与基于学习的控制的关键。不同于先前使用静态模糊集（如固定Wasserstein球）或在已知分段线性（或仿射）动力学下使用动态模糊集的工作，我们提出了一种算法，该算法通过核最大均值差异几何结构，利用库普曼算子和条件均值嵌入，精确地通过非线性数据驱动模型传播模糊集。通过理论和数值分析，我们证明核模糊集是学习到的数据驱动动力学系统模型的自然几何结构。