  Surrogate models are used to alleviate the computational burden in
engineering tasks, which require the repeated evaluation of computationally
demanding models of physical systems, such as the efficient propagation of
uncertainties. For models that show a strongly non-linear dependence on their
input parameters, standard surrogate techniques, such as polynomial chaos
expansion, are not sufficient to obtain an accurate representation of the
original model response. Through applying a rational approximation instead, the
approximation error can be efficiently reduced for models whose non-linearity
is accurately described through a rational function. Specifically, our aim is
to approximate complex-valued models. A common approach to obtain the
coefficients in the surrogate is to minimize the sample-based error between
model and surrogate in the least-square sense. In order to obtain an accurate
representation of the original model and to avoid overfitting, the sample set
has be two to three times the number of polynomial terms in the expansion. For
models that require a high polynomial degree or are high-dimensional in terms
of their input parameters, this number often exceeds the affordable
computational cost. To overcome this issue, we apply a sparse Bayesian learning
approach to the rational approximation. Through a specific prior distribution
structure, sparsity is induced in the coefficients of the surrogate model. The
denominator polynomial coefficients as well as the hyperparameters of the
problem are determined through a type-II-maximum likelihood approach. We apply
a quasi-Newton gradient-descent algorithm in order to find the optimal
denominator coefficients and derive the required gradients through application
of $\mathbb{CR}$-calculus.

Machine Learning;Machine Learning