学习神经收缩动力学：扩展线性化与全局保证
Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees
AI总结：
提出首个可在任意度量下获得全局收缩保证的神经网络动力学系统 ELCD，并通过潜空间微分同胚实现数据空间的全局收缩。
摘要：
学习得到的动力学系统中的全局稳定性和鲁棒性保证，对于确保系统在不确定性面前表现良好至关重要。我们提出了扩展线性化收缩动力学（Extended Linearized Contracting Dynamics，ELCD），这是首个在任意度量下具有全局收缩性保证的基于神经网络的动力学系统。ELCD 的关键特征是对非线性向量场的扩展线性化进行参数化。在其最基本形式中，ELCD 被保证：（i）全局指数稳定，（ii）平衡点收缩，以及（iii）相对于某种度量全局收缩。为了允许在数据空间中相对于更一般的度量进行收缩，我们训练数据空间与潜空间之间的微分同胚，并在潜空间中强制满足收缩性，从而确保数据空间中的全局收缩性。我们在高维 LASA、多连杆摆和 Rosenbrock 数据集上展示了 ELCD 的性能。
英文摘要：
Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. We present Extended Linearized Contracting Dynamics (ELCD), the first neural network-based dynamical system with global contractivity guarantees in arbitrary metrics. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable, (ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets.
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