We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be $\tau$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $\tau$ seconds (resp. $k$ steps). We show that under mild assumptions a $\tau$-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allow them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.

翻译：我们考虑在无显式动力学模型条件下，学习渐近稳定平衡点吸引域(ROA)内近似的问题。不同于利用具有有界不确定性的近似模型来寻找ROA中的（鲁棒）不变集，我们提出学习满足更宽松包含关系（称为循环性）的集合。我们将集合定义为$\tau$-循环（或$k$-循环），若从该集合内出发的每条轨迹在最多$\tau$秒（或$k$步）后返回该集合。我们证明在温和假设下，包含稳定平衡点的$\tau$-循环集必定是其吸引域的子集。进而利用该性质开发算法，通过采样有限长度轨迹获得的循环反例来计算吸引域的内近似。我们的算法可顺序处理样本，这使得即使完成初始离线训练阶段后仍能持续执行。我们进一步给出了算法所使用反例数量的上界以及几乎必然收敛性保证。