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.
翻译:超值模型用于减轻工程任务的计算负担,这需要反复评估计算要求很高的物理系统模型,例如不确定性的有效传播。对于显示对其输入参数高度非线性依赖的非线性模型,标准代金技术,例如多元混乱扩展,不足以准确反映原始模型响应。通过应用合理近似,对于非线性通过合理函数得到准确描述的模型来说,近似错误可以有效减少。具体地说,我们的目标是接近复杂估价的模型。在代金系统中获取系数的一个共同办法是将模型和超值参数之间的基于样本的错误最小化。为了获得原始模型的准确代表性和避免超值,样板的代金技术不足以准确反映原始模型的响应。通过应用合理近似差值近似差差数,对于需要高超度或高度输入参数的模型来说,这一数字往往超过可承受的计算成本。为了克服这个问题,我们应用一个基于样本和超值的基于样本的比值差性标准差差差差差差值方法,我们应用了原始模型的比值方法,作为原始模型的比值模型的比值模型的比值模型的比值法,我们先研算。