  The Loewner framework-(LF) in combination with Volterra series-(VS) offers a
non-intrusive approximation method that is capable of identifying bilinear
models from time-domain measurements. This method uses harmonic inputs which
establish a natural way for data acquisition. For the general class of
nonlinear problems with VS representation, the growing exponential approach
allows the derivation of the generalized kernels, namely symmetric generalized
frequency response functions - (GFRFs). In addition, the homogeneity of the
Volterra operator determines the accuracy in terms of how many kernels are
considered. For the weakly nonlinear setup, only a few kernels are needed to
obtain a good approximation. In this direction, the proposed adaptive scheme is
able to improve the estimations of the computationally non-zero kernels. The
Fourier transform associates these measurements with the derived GFRFs and the
LF makes the connection with system theory. In the linear case, the LF
associates the so-called S-parameters with the linear transfer function by
interpolating in the frequency domain. The goal of the proposed method is to
extend identification to the case of bilinear systems from time-domain
measurements and to approximate other general nonlinear systems (by means of
the Carleman bilinearizarion scheme). By identifying the linear contribution
with the LF, a considerable reduction is achieved by means of the SVD. The
fitted linear system has the same McMillan degree as the original linear
system. Then, the performance of the linear model is improved by augmenting a
special nonlinear structure. In a nutshell, we learn reduced-dimension bilinear
models directly from a potentially large-scale system that is simulated in the
time domain. This is done by fitting first a linear model, and afterwards, by
fitting the corresponding bilinear operator.
