## 1 Introduction  
Many engineering systems often appear with glaring features such as being hierarchical, heterogeneous, and having high-dimensionality. In addition, computationally expensive analysis/simulations make it more difficult to identify system-wide design solutions for problems. For example, a complex design problem may be decomposed into several sub-problems, each of which may have independent/dependent design variables with different goals such as disciplines and scales, among others. Typical routines of materials selection strategies are no longer conducive to ensuring system requirements while designing such complex design problems. Therefore, the nature of complex systems have necessitated the inductive multiscale design of materials and products. [PERSON]'s hierarchical concept [1] distinguished inductive and deductive decision paths in materials design. As shown in Figure 1, the deductive path corresponds to bottom-up mappings from the processing level to the performance level, whereas the inductive path is the performances-means (top-down) approach to tailor the materials structure or processing parameters to satisfy multiple performance requirements. However, there are numerous challenges due in large part to the presence of uncertainties associated with material models, design variables, and propagation across the design hierarchy. Therefore, robust designs should be considered rather than a single optimal solution to support decision-making, considering performance variation. In response to these issues, set-based design approaches have become more promising as they tend to attain design requirements by exploring a set of satisfactory design solutions. [PERSON] et al. [PERSON] _et al._ (2018) proposed a set-based collaborative design method that classifies the design space into satisfactory and unsatisfactory regions by employing Bayesian network classifiers. This method is further applied in the hierarchical design of negative stiffness meta-materials at micro-, meso-, and macro-scales ([PERSON] _et al._, 2016; [PERSON] and [PERSON], 2018; [PERSON] and [PERSON], 2018). The inductive design exploration method (IDEM) is also a multilevel, robust design method that identifies feasible ranges of design variables in a step-by-step, top-down (inductive) manner ([PERSON] _et al._, 2016; [PERSON] and [PERSON], 2018). In this method, feasible ranges (not an optimal point) are sequentially identified in each solution space from the top-most to the lowest level, considering the propagation of uncertainty through the model hierarchy. The detailed process in IDEM can be found in the literature for solving many complex engineering problems, including blast-resistant sandwich panels ([PERSON] _et al._, 2016), vehicle-mounted antenna-supporting structures ([PERSON] _et al._, 2016), ultra-high-performance concrete (UHPC) panels, etc. ([PERSON] _et al._, 2018; [PERSON] _et al._, 2018). In the above-mentioned set-based approaches, one of the challenges is to tackle computationally expensive analyses and simulations. Even with the help of parallel computing, increasing computing power, reducing design space, screening significant design variables, among others, it is still difficult to deal with heavy computational costs. In IDEM, an \(n\)-dimensional grid of discrete points is generated at each level of a multilevel design problem; then, the grid design points are evaluated by mapping models across the design hierarchy. However, the exhaustive grid of design points can be burdensome. Meta-models are preferred for replacing computationally expensive analyses; however, they are not well-suited for models with discrete variables, nonlinear responses, or discontinuous responses, which are often found in materials design problems ([PERSON] _et al._, 2018). The prediction accuracy of meta-models is also primarily dependent on the number of experimental sample points; therefore it is difficult to build high-fidelity models with computationally expensive simulations ([PERSON] _et al._, 2016).