The evaluation of real-world engineering systems is expensive. With physical models and inputs, computer simulations can accurately deliver information about systems. Although cheaper than experimental or industrial data, simulations can be considerably slow when the systems involve multi-scale, multiphase phenomena and dynamic behaviors. Data-driven surrogates can represent the original physical models by building a relationship between inputs and outputs (responses). For some complex systems, there are even no physical models available, and thus data-driven surrogates together with the design of experiments (DoE) seem to be the only choice. Surrogates are cheap-to-evaluate and can directly be employed for optimization, control and design problems in many engineering fields, e.g. chemical engineering, 1,2,9,10 pharmaceutical manufacturing, supply chain management, and aerospace engineering. The traditional formulation of surrogates is polynomials, moving least-squares and radial basis functions, etc., but the low accuracy is often criticized. In machine learning, artificial neural networks (ANNs) and Gaussian Processes (GPs) have been identified as promising formulations for surrogate generation. ANNs are regarded as universal approximators and allow to fit multiple outputs simultaneously. The flexible structure and various activation functions enable ANNs to accurately fit any linear or nonlinear relationship of inputs/outputs. GPs belongs to a non-parametric model type, which has excellent fitting performance and can predict uncertainty. However, GP is generally implemented for the single output, whereas the formulation for fitting multiple outputs is complex. An alternative solution is to use independent GPs for each output, while the correlations of outputs are missing, which can cause worse predictions for highly-coupled outputs. Data is one of the prerequisites to train a surrogate. When the available data is limited, data generation is necessary, but it can be extremely expensive for real-world engineering systems. To demonstrate it, we consider an example of a chemical process -pressure swing adsorption (PSA). Details about PSA can be found in Section 4, where we use PSA as one of the case studies. Data is randomly sampled, while the surrogate is trained iteratively. As shown in Figure , the time for data generation has a significantly higher order of magnitude than that for surrogate training, which is one of the common problems in chemical process systems.