Explainable machine learning techniques have gained increasing attention in engineering applications, especially in aerospace design and analysis, where understanding how input variables influence predictive models is essential. Partial Dependence Plots (PDPs) are widely used for interpreting black-box models by showing the average effect of an input variable on the prediction. However, their global sensitivity metric can be misleading when strong interactions are present, as averaging tends to obscure interaction effects. To address this limitation, we propose a global sensitivity metric based on Individual Conditional Expectation (ICE) curves. The method computes the expected feature importance across ICE curves, along with their standard deviation, to more effectively capture the influence of interactions. The proposed metrics are model-agnostic and can be applied to any predictive model, including but not limited to surrogate models. Furthermore, we provide a mathematical proof demonstrating that the PDP-based sensitivity is a lower bound of the proposed ICE-based metric under additive and multiplicative separability. In addition, we introduce an ICE-based correlation value to quantify how interactions modify the relationship between inputs and the output. Comparative evaluations were performed on three cases: a 5-variable analytical function, a 5-variable wind-turbine fatigue problem, and a 9-variable airfoil aerodynamics case, where ICE-based sensitivity was benchmarked against PDP, SHapley Additive exPlanations (SHAP), and Sobol’ indices. The results show that ICE-based feature importance provides richer insights than the traditional PDP-based approach, while visual interpretations from PDP, ICE, and SHAP complement one another by offering multiple perspectives.

Aircraft Maneuver Load Alleviation (MLA) is a multidisciplinary problem involving the coupled physics of aerodynamics, flight dynamics, and structural analysis. This work presents an analytical model-based investigation of MLA through movable control surfaces for various wing platforms, aiming to understand the underlying physics, identify key influencing parameters, and inform the formulation of associated multidisciplinary optimization problems. The investigation starts with a rigid wing and a single-point aileron, and incrementally increases modeling complexity by introducing finite aileron span, aircraft trim conditions, alternative lift distributions (e.g., Prandtl’s Bell-Shaped) and wing flexibility to further improve the fidelity. Key findings reveal that optimal aileron placement for load alleviation is highly sensitive to trim effects, shifting from approximately 71% span for a rigid wing to 82% span when trim is considered, and further to 64.33% span for a trimmed wing with a Bell-Shaped lift distribution. Trim effects are identified as the dominant factor due to compensatory pitch angle changes, while wing planform (lift distribution) also has a significant influence. In contrast, the aileron span and wing flexibility (for straight wings) exhibit comparatively minor impacts on optimal placement. Furthermore, the analysis demonstrates that while flaps can also achieve load alleviation with optimal placement near the wing root, the optimal aileron placement is independent of flap usage. These analytical insights provide foundational understanding, crucial for guiding preliminary aircraft design and streamlining subsequent computationally expensive high-fidelity Multidisciplinary Design Optimization (MDO) efforts.

Modern design optimization frameworks aim first and foremost for models with the most accurate predictions without balancing computational overhead. It remains a reason why scaled architecture and multidisciplinary design optimization problems are difficult to address, even with sample-efficient Bayesian optimizers. In this paper, a metric quantifying the computational energy footprint is introduced within a Bayesian optimization framework to guide the parameter setting of a model towards configurations that balance both performance and frugality. The computer experiments highlighted existing tradeoffs between optimum convergence and the underlying energy footprint, and sometimes resulted in both a better-found optimum and lower energy consumption.

Graph Neural Networks have been applied to learn the flight and structural dynamics of a High-Altitude Long-Endurance aircraft in response to discrete gusts. The graph network methodology enables the development of a model for structural displacements, loads and aircraft flight dynamics leveraging on the inductive bias provided by the physical connections. Neural Ordinary Differential Equations have been integrated with Graph Neural Network in a novel architecture using exogenous inputs. The results demonstrate promising capabilities in model approximation improving on traditional graph networks, in particular for long term predictions in time by reducing integration drift errors. Even without targeted software optimization, the surrogate model provides an approximately 200-fold increase in computational speed compared to the original simulation environment.

This study introduces SEGOMOE, a Bayesian optimization tool for optimizing complex, computationally expensive systems, especially in aeronautics. It efficiently handles mixed design variables (continuous, discrete, categorical, hierarchical) using adaptive Gaussian process models. SEGOMOE combines expert models to address nonlinearities in objectives and constraints, leveraging the open-source Surrogate Modeling Toolbox (SMT). The tool supports multi-fidelity data and solves both single- and multi-objective problems, including hidden constraints and high-dimensional decomposition. Validated through benchmarks and real-world aeronautical applications, SEGOMOE proves to be robust and versatile for tackling multidisciplinary challenges.

This work evaluates Deep Gaussian Processes (DGPs) as probabilistic surrogate models for predicting thrust and torque of small Unmanned Aerial Vehicle (UAV) propellers. Unlike deterministic neural-network models, DGPs provide predictive uncertainty estimates in addition to mean predictions. Compared with standard Gaussian Processes, their hierarchical composition of multiple GP layers gives greater flexibility for representing complex nonlinear relationships between propeller geometry, operating conditions, and aerodynamic response. The models are trained and evaluated using experimental wind-tunnel data from 19 propellers with diameters between 7 and 20 inches. DGPs are compared with Multi-Layer Perceptrons and sparse Gaussian Processes using diameter-based train-test splits, where complete propeller diameters are withheld from training to assess generalization to unseen geometries. The results show that DGPs achieve competitive or improved predictive accuracy while providing more reliable uncertainty-aware predictions than the sparse Gaussian Process baseline. These findings indicate that DGPs are promising surrogate models for early-stage UAV propeller design, particularly when experimental data are limited and predictive uncertainty is important.

Ensuring high accuracy and efficiency of predictive models is paramount in the aerospace industry, particularly in the context of multidisciplinary design and optimization processes. These processes often require numerous evaluations of complex objective functions, which can be computationally expensive and time-consuming. To build efficient and accurate predictive models with limited high-fidelity data, we propose a new approach that leverages Bayesian Optimization (BO) to optimize the hyperparameters of a lightweight and accurate Neural Network (NN) for aerodynamic performance prediction. To clearly describe the interplay between design variables, hierarchical and categorical kernels are used in the BO formulation. We demonstrate the efficiency of our approach through two comprehensive case studies, where the optimized NN significantly outperforms baseline models and other publicly available NNs in terms of accuracy and parameter efficiency. For the drag coefficient prediction task, the Mean Absolute Percentage Error (MAPE) of our optimized model drops from 0.1433% to 0.0163%, which is nearly an order of magnitude improvement over the baseline model. Additionally, our model achieves a MAPE of 0.82% on a benchmark aircraft self-noise prediction problem, significantly outperforming existing models (where their MAPE values are around 2 to 3%) while requiring fewer computational resources. The results highlight the potential of our framework to enhance the scalability and performance of NNs in large-scale MDO problems, offering a promising solution for the aerospace industry.

This paper introduces a novel surrogate modeling framework for aerodynamic applications based on Neural Fields. The proposed approach, MARIO (Modulated Aerodynamic Resolution Invariant Operator), addresses non parametric geometric variability through an efficient shape encoding mechanism and exploits the discretizationinvariant nature of Neural Fields. It enables training on significantly downsampled meshes, while maintaining consistent accuracy during full-resolution inference. These properties allow for efficient modeling of diverse flow conditions, while reducing computational cost and memory requirements compared to traditional CFD solvers and existing surrogate methods. The framework is validated on two complementary datasets that reflect industrial constraints. First, the AirfRANS dataset consists of a two-dimensional airfoil benchmark with non-parametric shape variations. Performance evaluation of MARIO on this case demonstrates an order of magnitude improvement in prediction accuracy over existing methods across velocity, pressure, and turbulent viscosity fields, while accurately capturing boundary layer phenomena and aerodynamic coefficients. Second, the NASA Common Research Model features three-dimensional pressure distributions on a full aircraft surface mesh, with parametric control surface deflections. This configuration confirms MARIO's accuracy and scalability. Benchmarking against state-of-the-art methods demonstrates that Neural Field surrogates can provide rapid and accurate aerodynamic predictions under the computational and data limitations characteristic of industrial applications.

The simulation of complex systems increasingly relies on sophisticated but fundamentally opaque computational black-box simulators. Surrogate models play a central role in reducing the computational cost of complex systems simulations across a wide range of scientific and engineering domains. Notwithstanding, they inevitably inherit and often exacerbate this black-box nature, obscuring how input variables drive physical responses. Conversely, Explainable Artificial Intelligence (XAI) offers powerful tools to unpack these models. Yet, XAI methods struggle with engineering-specific constraints, such as highly correlated inputs, dynamical systems, and rigorous reliability requirements. Consequently, surrogate modeling and XAI have largely evolved as distinct fields of research, despite their strong complementarity. To reconnect these approaches, this state-of-the-art survey provides a structured perspective that maps existing XAI techniques onto the various stages of surrogate modeling workflows for design and exploration. To ground this synthesis, we draw upon illustrative applications across both equation-based simulations and agent-based modeling. We survey a broad spectrum of techniques, highlighting their strengths for revealing interactions and supporting human comprehension. Finally, we identify pressing open challenges, including the explainability of dynamical systems and the handling of mixed-variable systems, and propose a research agenda to make explainability a core, embedded element of simulation-driven workflows from model construction through decision-making. By transforming opaque emulators into explainable tools, this agenda empowers practitioners to move beyond accelerating simulations to extracting actionable insights from complex system behaviors.

Modern engineering design optimization increasingly relies on surrogate models to reduce the computational cost associated with repeated high-fidelity simulations. While standard Gaussian Processes are widely used, their assumption of stationarity can limit predictive accuracy in problems involving nonlinear material behavior, structural dynamics, or responses that vary across regions of the design space, particularly when only limited training data are available. This study evaluates the potential of deep probabilistic surrogate models, specifically Deep Gaussian Processes (DGPs) and Deep Kernel Learning (DKL), for constrained structural optimization. The work systematically benchmarks DGP and DKL side by side, clarifying their respective strengths and limitations. The approach is first validated on the classical 10-bar truss benchmark and subsequently applied to the optimization of a post-tensioned concrete bridge girder aimed at minimizing environmental impact. The results indicate that both DGP and DKL identify high-quality feasible designs with fewer high-fidelity evaluations, achieve faster convergence than standard GP-based approaches, and exhibit improved consistency across repeated optimization runs. These findings highlight the potential of deep probabilistic surrogates to accelerate structural design by reducing the number of computationally expensive simulations required in practice.

Complex systems are increasingly explored through simulation-driven engineering workflows that combine physics-based and empirical models with optimization and analytics. Despite their power, these workflows face two central obstacles: (1) high computational cost, since accurate exploration requires many expensive simulator runs; and (2) limited transparency and reliability when decisions rely on opaque blackbox components. We propose a workflow that addresses both challenges by training lightweight emulators on compact designs of experiments that (i) provide fast, low-latency approximations of expensive simulators, (ii) enable rigorous uncertainty quantification, and (iii) are adapted for global and local Explainable Artificial Intelligence (XAI) analyses. This workflow unifies every simulation-based complex-system analysis tool, ranging from engineering design to agent-based models for socio-environmental understanding. In this paper, we proposea comparative methodology and practical recommendations for using surrogate-based explainability tools within the proposed workflow. The methodology supports continuous and categorical inputs, combines global-effect and uncertainty analyses with local attribution, and evaluates the consistency of explanations across surrogate models, thereby diagnosing surrogate adequacy and guiding further data collection or model refinement. We demonstrate the approach on two contrasting case studies: a multidisciplinary design analysis of a hybrid-electric aircraft and an agent-based model of urban segregation. Results show that the surrogate model and XAI coupling enables large-scale exploration in seconds, uncovers nonlinear interactions and emergent behaviors, identifies key design and policy levers, and signals regions where surrogates require more data or alternative architectures.

This paper presents a framework and one pedagogical application of motion tracking algorithms applied to structural dynamics. The aim of this work is to show the ability of high speed camera to study the dynamic characteristics of simple mechanical systems using a marker less and simultaneous Single Input Multiple Output (SIMO) broadband analysis. KLT (Kanade-Lucas-Tomasi) trackers are used as virtual sensors on mechanical systems video. First we introduce the paradigm of virtual sensors in the field of modal analysis using video processing. Then we present a pedagogical example of flexible beam (Fishing rod) video. From KLT tracking we extracted displacements data (virtual sensors) which are then enhanced using filtering and smoothing and then we can identify natural frequency and damping ratio from classical modal analysis. The experimental results (mode shapes) are compared to an analytical flexible beam model showing high correlation but also showing the limitation of linear analysis. The main interest of this paper is that displacements are simply measured using only video at FPS (Frame Per Second) that respects the Nyquist frequency. There is no target needed on the structure only few critical pixels that are good features to track and which become virtual sensors.

Gaussian process regression (GPR) is a widely used regression model, but it has poor scalability. Sparse approximation methods improve scalability by using inducing points to approximate the GPR, but determining the optimal number and placement of these points is challenging. Increasing the number of inducing points generally improves the predictive accuracy, but it comes at a computational cost. This article presents a method to estimate the necessary number of inducing points for accurate predictions of finite element method (FEM) analyses using approximate GPR. The approach leverages the proper orthogonal decomposition (POD) technique, using its modes to determine the inducing points. Results demonstrate that the proposed method identifies a sufficient number of inducing points for approximate GPR to achieve predictive accuracy comparable to full GPR, but with half the training time. This approach ensures computational efficiency without significant loss in accuracy, making it a valuable tool for scalable regression in engineering applications. POD has previously been combined with GPR to provide computationally efficient predictions for the full solution field across unseen variable combinations, treating spatial components separately via reduced basis functions. However, this work treats the spatial component as a variable within the GPR approximation, allowing continuous spatial predictions. This ensures that the covariance in the spatial dimension is captured by a single GPR. The method is applied to simulations of a three-span, post-tensioned concrete girder bridge.

This paper introduces a comprehensive open-source framework for developing correlation kernels, with a particular focus on user-defined and composition of kernels for surrogate modeling. By advancing kernel-based modeling techniques, we incorporate frequency-aware elements that effectively capture complex mechanical behaviors and timefrequency dynamics intrinsic to aircraft systems. Traditional kernel functions, often limited to exponential-based methods, are extended to include a wider range of kernels such as exponential squared sine and rational quadratic kernels, along with their respective firstand second-order derivatives. The proposed methodologies are first validated on a sinus cardinal test case and then applied to forecasting Mauna-Loa Carbon Dioxide (CO 2 ) concentrations and airline passenger traffic. All these advancements are integrated into the open-source Surrogate Modeling Toolbox (SMT 2.0), providing a versatile platform for both standard and customizable kernel configurations. Furthermore, the framework enables the combination of various kernels to leverage their unique strengths into composite models tailored to specific problems. The resulting framework offers a flexible toolset for engineers and researchers, paving the way for numerous future applications in metamodeling for complex, frequency-sensitive domains.

Bayesian optimization is an advanced tool to perform ecient global optimization It consists on enriching iteratively surrogate Kriging models of the objective and the constraints both supposed to be computationally expensive of the targeted optimization problem Nowadays efficient extensions of Bayesian optimization to solve expensive multiobjective problems are of high interest The proposed method in this paper extends the super efficient global optimization with mixture of experts SEGOMOE to solve constrained multiobjective problems To cope with the illposedness of the multiobjective inll criteria different enrichment procedures using regularization techniques are proposed The merit of the proposed approaches are shown on known multiobjective benchmark problems with and without constraints The proposed methods are then used to solve a biobjective application related to conceptual aircraft design with ve unknown design variables and three nonlinear inequality constraints The preliminary results show a reduction of the total cost in terms of function evaluations by a factor of 20 compared to the evolutionary algorithm NSGA-II.

The main objective of this study is to develop an alternative criterion for modeshape classification, as the currently available one, MAC (Modal Assurance Criteria), is only a vector correlation representing modeshape similarities. This new method is developed to provide a set of features (Fourier Descriptors) for comparing modeshapes with “local” similarities of higher interest than “global” similarities using nodal lines. These lines are able to characterize modeshapes very easily. So when damage occurs, we are able to track the few descriptors changes to localise the damage. We validated our method on a CFCF plate demonstrating the quality of the damage localisation on a severe case (close to boundary conditions).

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