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Bayesian Optimization using Gaussian Processes is a popular approach to deal with optimization involving expensive black-box functions. However, because of the assumption on the stationarity of the covariance function defined in classic Gaussian Processes, this method may not be adapted for non-stationary functions involved in the optimization problem. To overcome this issue, Deep Gaussian Processes can be used as surrogate models instead of classic Gaussian Processes. This modeling technique increases the power of representation to capture the non-stationarity by considering a functional composition of stationary Gaussian Processes, providing a multiple layer structure. This paper investigates the application of Deep Gaussian Processes within Bayesian Optimization context. The specificities of this optimization method are discussed and highlighted with academic test cases. The performance of Bayesian Optimization with Deep Gaussian Processes is assessed on analytical test cases and aerospace design optimization problems and compared to the state-of-the-art stationary and non-stationary Bayesian Optimization approaches.

International audience; Reusability of the first stage of launch vehicles may offer new perspectives to lower the cost of payload injection into orbit if sufficient reliability and efficient refurbishment can be achieved. One possible option that may be explored is to design the vehicle first stage for both reusable and expendable uses, in order to increase the flexibility and adaptability to different target missions. This paper proposes a multilevel multidisciplinary design optimization (MDO) approach to design aerospace vehicles addressing multimission problems. The proposed approach is focused on the design of a family of launchers for different missions sharing commonalities using multi-objective MDO to account for the computational cost associated with the discipline simulations. The multimission problem addressed considers two missions: 1) a reusable configuration for a sun synchronous orbit with a medium payload range and recovery of the first stage using a gliding-back strategy; 2) an expendable configuration for a medium payload injected into a geostationary transfer orbit. A dedicated MDO formulation introducing couplings between the missions is proposed in order to efficiently solve such a coupled problem while limiting the number of calls to the exact multidisciplinary analysis thanks to the use of Gaussian processes and multi-objective efficient global optimization.

International audience; Multi-fidelity approaches combine different models built on a scarce but accurate data-set (high-fidelity data-set), and a large but approximate one (low-fidelity data-set) in order to improve the prediction accuracy. Gaussian Processes (GPs) are one of the popular approaches to exhibit the correlations between these different fidelity levels. Deep Gaussian Processes (DGPs) that are functional compositions of GPs have also been adapted to multi-fidelity using the Multi-Fidelity Deep Gaussian process model (MF-DGP). This model increases the expressive power compared to GPs by considering non-linear correlations between fidelities within a Bayesian framework. However, these multi-fidelity methods consider only the case where the inputs of the different fidelity models are defined over the same domain of definition (e.g., same variables, same dimensions). However, due to simplification in the modeling of the low-fidelity, some variables may be omitted or a different parametrization may be used compared to the high-fidelity model. In this paper, Deep Gaussian Processes for multi-fidelity (MF-DGP) are extended to the case where a different parametrization is used for each fidelity. The performance of the proposed multifidelity modeling technique is assessed on analytical test cases and on structural and aerodynamic real physical problems.

International audience; The design process of complex systems such as new configurations of aircraft or launch vehicles is usually decomposed in different phases which are characterized by the depth of the analyses in terms of number of design variables and fidelity of the physical models. At each phase, the designers have to deal with accurate but computationally intensive models as well as cheap but inaccurate models. Multi-fidelity modeling is a way to merge different fidelity models to provide engineers with accurate results with a limited computational cost. Within the context of multi-fidelity modeling, approaches based on Gaussian Processes emerge as popular techniques to fuse information between the different fidelity models. The relationship between the fidelity models is a key aspect in multi-fidelity modeling. This paper provides an overview of Gaussian process-based multi-fidelity modeling techniques for variable relationship between the fidelity models (e.g., linearity, non-linearity, variable correlation). Each technique is described within a unified framework and the links between the different techniques are highlighted. All approaches are numerically compared on a series of analytical test cases and four aerospace related engineering problems in order to assess their benefits and disadvantages with respect to the problem characteristics.

International audience; In addition to the multi-fidelity aspects in MDO discussed in Chapter 8, two additional topics of interest to solve complex MDO problems are discussed in this chapter: multi-objective MDO and mixed continuous/discrete variable design optimization problems.

International audience; For many countries (United States of America, Russia, Europe, Japan, etc.), the launch vehicles are cornerstones of an independent access to space. The space agency strategies for Solar system exploration, Earth monitoring and observation, human spaceflight are developed in accordance with their launch vehicle capabilities. Launch vehicle designs are long term projects (around a decade) involving large budgets and requiring efficient organization.

International audience; Reusability of the first stage of launch vehicles may offer new perspectives to lower the cost of payload injection into orbit if sufficient reliability and low refurbishment costs can be achieved. One possible option that may be explored is to design the launch vehicle first stage for both reusable and expendable uses, in order to increase the flexibility and adaptability to different target missions. This paper proposes a multi-level MDO approach to design aerospace vehicles addressing multi-mission problems. The proposed approach is focused on the design of a family of launchers for different missions sharing commonalities using multi-objective Bayesian Optimization to account for the computational cost associated with the discipline simulations. The multi-mission problem addressed in this paper considers two missions: a reusable configuration for a SSO orbit with a medium payload range and recovery of the first stage using a glider strategy; and an expendable configuration for a medium payload injected into a Geostationary Transfer Orbit (GTO). A dedicated MDO formulation introducing couplings between the missions is proposed in order to efficiently solve the multi-objective MDO problem while limiting the number of calls to the exact MDA thanks to the use of Gaussian Processes and multi-objective Efficient Global Optimization.

International audience; This paper is focused on the problem of constrained multi-objective design optimization of aerospace vehicles. The design of such vehicles often involves disciplinary legacy models considered as black-box and computationally expensive simulations characterized by a possible non-stationary behavior (an abrupt change in the response or a different smoothness along the design space). The expensive cost of an exact function evaluation makes the use of classical evolutionary multi-objective algorithms not tractable. While Bayesian Optimization based on Gaussian Process regression can handle the expensive cost of the evaluations, the non-stationary behavior of the functions can make it inefficient. A recent approach consisting of coupling Bayesian Optimization with Deep Gaussian Processes showed promising results for single-objective non-stationary problems. This paper presents an extension of this approach to the multi-objective context. The efficiency of the proposed approach is assessed with respect to classical optimization methods on an analytical test-case and on an aerospace design problem.