Many engineering problems require global optimization of expensive-to-evaluate models, in which there is a complex relationship between tunable parameters and the objective. For example, pharmaceutical researchers need to design drugs to fight diseases, and process engineers must synthesize flowsheets to get desired goal of profitability and yield. To achieve this, the practitioners may need to conduct experiments and/ or computer simulations that require heavy use of manpower and/or computational resources. Due to the finite resources and time-constrained market competition, practitioners need to deploy data-efficient algorithms for optimization purposes. Bayesian optimization (BO) is an optimization paradigm that has been effectively used in domains such as drug discovery and material design, aerospace engineering, environmental management, etc. Conventional BO, however, does not account for two important considerations which will be addressed in this work – first, it ignores that there is often some structural knowledge available (e.g., physics or interconnectivity between units); second, in most “real-world” systems there exist uncertainty variables that are not tunable but parameterize the outcome of an experiment or simulation (e.g., particle size variations in raw material). We propose a novel algorithm that addresses the limitations by - i) Maximizing information gain by considering the underlying function network structure; ii) Implementing robust optimization (RO), a two-level optimization problem that optimizes objective while accounting for the adverse effects of the uncertain variables. We demonstrate that the proposed algorithm outperforms existing state-of-the-art methods using synthetic and real-world case studies.