用少量低频数据同时学系统动力和稳定性证书，更准更可靠。
Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems
- 结合柯普曼生成器与物理信息神经网络，从稀疏数据中同步学习系统向量场与李雅普诺夫函数。
- 学习到的李雅普诺夫函数经SMT验证，区域吸引域估计比现有方法更精确。
- 适合对非线性系统稳定性分析有高要求的研究者，尤其在数据稀缺场景下。
近年来，柯普曼算子理论因其能将离散时间非线性系统嵌入无限维线性空间而受到广泛关注。然而，在相对较低的观测频率下，基于柯普曼的学习框架在学习连续时间动力学的同时提供稳定性保证仍面临挑战。为解决此问题，我们提出一种算法框架，利用有限状态空间采样数据及沿轨迹的低频采样，同时学习未知非线性系统的向量场与李雅普诺夫函数。该框架基于近期发展的高精度柯普曼生成器学习方法，以捕捉系统瞬态演化，并结合物理信息神经网络训练李雅普诺夫函数。我们证明所学李雅普诺夫函数可通过满足性模理论（SMT）求解器进行形式化验证，并提供比现有方法更紧致的吸引域估计。
原文摘要 · Abstract (English)
Koopman operator theory has gained significant attention in recent years for identifying discrete-time nonlinear systems by embedding them into an infinite-dimensional linear vector space. However, providing stability guarantees while learning the continuous-time dynamics, especially under conditions of relatively low observation frequency, remains a challenge within the existing Koopman-based learning frameworks. To address this challenge, we propose an algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency. The proposed framework builds upon recently developed high-accuracy Koopman generator learning for capturing transient system transitions and physics-informed neural networks for training Lyapunov functions. We show that the learned Lyapunov functions can be formally verified using a satisfiability modulo theories (SMT) solver and provide less conservative estimates of the region of attraction compared to existing methods.
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