  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.
