We evaluate some methods designed for tensor- (or data-) based multivariate model construction (approximation and compression). To this aim, a collection of multivariate functions and an evaluation methodology are suggested. First, these functions, with varying complexity (e.g., number and degree of the variables) and nature (e.g., rational, irrational, differentiable or not, symmetric, etc.) are used to build $n$-dimensional tensors, each of different dimension and memory size. Second, grounded on this tensor, we evaluate the performances of different methods and implementations leading to different types of surrogate models (e.g., rational functions, networks). The accuracy, the computational time, the parameter tuning impact, etc. are monitored and reported. One objective is to evaluate the different available strategies to guide users on the prospects, advantages, and limits of the various tools. The contributions are twofold: (i) to suggest a comprehensive benchmark collection together with a methodology for tensor approximation with a surrogate model and, in addition, (ii) to provide a digest and additional details of the multivariate Loewner Framework (mLF) approach [Antoulas et al., 2025], as well as detailed examples and code.

翻译：我们评估了一些为基于张量（或数据）的多元模型构建（逼近与压缩）而设计的方法。为此，我们提出了一组多元函数及相应的评估方法。首先，利用这些在复杂度（如变量数量与阶数）和性质（如有理、无理、可微或不可微、对称性等）上各异的函数，构建了多个$n$维张量，每个张量具有不同的维度和内存大小。其次，基于这些张量，我们评估了不同方法及其实施方案的性能，这些方法可生成不同类型的代理模型（如有理函数、网络等）。我们监测并报告了精度、计算时间、参数调优影响等指标。目标之一是评估现有的不同策略，以指导用户了解各种工具的前景、优势与局限。本文的贡献包括两个方面：(i) 提出了一套全面的基准测试集及相应的张量逼近代理模型方法；(ii) 提供了多元Loewner框架（mLF）方法[Antoulas等人，2025]的概要说明与补充细节，以及详细的示例和代码。