  The AAA algorithm has become a popular tool for data-driven rational
approximation of single variable functions, such as transfer functions of a
linear dynamical system. In the setting of parametric dynamical systems
appearing in many prominent applications, the underlying (transfer) function to
be modeled is a multivariate function. With this in mind, we develop the AAA
framework for approximating multivariate functions where the approximant is
constructed in the multivariate barycentric form. The method is data-driven, in
the sense that it does not require access to full state-space model and
requires only function evaluations. We discuss an extension to the case of
matrix-valued functions, i.e., multi-input/multi-output dynamical systems, and
provide a connection to the tangential interpolation theory. Several numerical
examples illustrate the effectiveness of the proposed approach.
