Simulation-based problems involving mixed-variable inputs frequently feature domains that are hierarchical, conditional, heterogeneous, or tree-structured. These characteristics pose challenges for data representation, modeling, and optimization. This paper reviews extensive literature on these structured input spaces and proposes a unified framework that generalizes existing approaches. In this framework, input variables may be continuous, integer, or categorical. A variable is described as meta if its value governs the presence of other decreed variables, enabling the modeling of conditional and hierarchical structures. We further introduce the concept of partially-decreed variables, whose activation depends on contextual conditions. To capture these inter-variable hierarchical relationships, we introduce design space graphs, combining principles from feature modeling and graph theory. This allows the definition of general hierarchical domains suitable for describing complex system architectures. Our framework defines hierarchical distances and kernels to enable surrogate modeling and optimization on hierarchical domains. We demonstrate its effectiveness on complex system design problems, including a neural network and a green-aircraft case study. Our methods are available in the open-source Surrogate Modeling Toolbox (SMT 2.0).

翻译：涉及混合变量输入的仿真问题常呈现层次化、条件性、异构性或树状结构的定义域特征，这些特性对数据表示、建模与优化提出了挑战。本文系统综述了关于此类结构化输入空间的大量文献，并提出一个统一框架以泛化现有方法。在该框架中，输入变量可为连续型、整数型或类别型。若某变量的取值支配其他派生变量的存在性，则将其定义为元变量，从而实现对条件性与层次化结构的建模。我们进一步引入部分派生变量的概念，其激活状态取决于上下文条件。为捕捉变量间的层次关系，我们提出设计空间图，融合了特征建模与图论原理，从而构建适用于描述复杂系统架构的通用层次化定义域。本框架定义了层次化距离与核函数，以支持层次化定义域上的代理建模与优化。我们通过神经网络与绿色飞行器设计等复杂系统案例验证了框架的有效性。相关方法已在开源代理建模工具箱（SMT 2.0）中实现。