  Highly accurate simulations of complex phenomena governed by partial
differential equations (PDEs) typically require intrusive methods and entail
expensive computational costs, which might become prohibitive when
approximating steady-state solutions of PDEs for multiple combinations of
control parameters and initial conditions. Therefore, constructing efficient
reduced order models (ROMs) that enable accurate but fast predictions, while
retaining the dynamical characteristics of the physical phenomenon as
parameters vary, is of paramount importance. In this work, a data-driven,
non-intrusive framework which combines ROM construction with reduced dynamics
identification, is presented. Starting from a limited amount of full order
solutions, the proposed approach leverages autoencoder neural networks with
parametric sparse identification of nonlinear dynamics (SINDy) to construct a
low-dimensional dynamical model. This model can be queried to efficiently
compute full-time solutions at new parameter instances, as well as directly fed
to continuation algorithms. These aim at tracking the evolution of periodic
steady-state responses as functions of system parameters, avoiding the
computation of the transient phase, and allowing to detect instabilities and
bifurcations. Featuring an explicit and parametrized modeling of the reduced
dynamics, the proposed data-driven framework presents remarkable capabilities
to generalize with respect to both time and parameters. Applications to
structural mechanics and fluid dynamics problems illustrate the effectiveness
and accuracy of the proposed method.
