Recently, there has been a growing interest for mixed-categorical meta-models based on Gaussian process (GP) surrogates. In this setting, several existing approaches use different strategies either by using continuous kernels (e.g., continuous relaxation and Gower distance based GP) or by using a direct estimation of the correlation matrix. In this paper, we present a kernel-based approach that extends continuous exponential kernels to handle mixed-categorical variables. The proposed kernel leads to a new GP surrogate that generalizes both the continuous relaxation and the Gower distance based GP models. We demonstrate, on both analytical and engineering problems, that our proposed GP model gives a higher likelihood and a smaller residual error than the other kernel-based state-of-the-art models. Our method is available in the open-source software SMT.
Giovanni Catalani, Siddhant Agarwal, Xavier Bertrand et al.
This paper presents a methodology to learn surrogate models of steady state fluid dynamics simulations on meshed domains, based on Implicit Neural Representations (INRs). The proposed models can be applied directly to unstructured domains for different flow conditions, handle non-parametric 3D geometric variations, and generalize to unseen shapes at test time. The coordinate-based formulation naturally leads to robustness with respect to discretization, allowing an excellent trade-off between computational cost (memory footprint and training time) and accuracy. The method is demonstrated on two industrially relevant applications: a RANS dataset of the two-dimensional compressible flow over a transonic airfoil and a dataset of the surface pressure distribution over 3D wings, including shape, inflow condition, and control surface deflection variations. On the considered test cases, our approach achieves a more than three times lower test error and significantly improves generalization error on unseen geometries compared to state-of-the-art Graph Neural Network architectures. Remarkably, the method can perform inference five order of magnitude faster than the high fidelity solver on the RANS transonic airfoil dataset. Code is available at https://gitlab.isae-supaero.fr/gi.catalani/aero-nepf
Paul Saves, Youssef Diouane, Nathalie Bartoli et al.
Recently, there has been a growing interest in mixed-categorical metamodels based on Gaussian Process (GP) for Bayesian optimization. In this context, different approaches can be used to build the mixed-categorical GP. Many of these approaches involve a high number of hyperparameters; in fact, the more general and precise the strategy used to build the GP, the greater the number of hyperparameters to estimate. This paper introduces an innovative dimension reduction algorithm that relies on partial least squares regression to reduce the number of hyperparameters used to build a mixed-variable GP. Our goal is to generalize classical dimension reduction techniques commonly used within GP (for continuous inputs) to handle mixed-categorical inputs. The good potential of the proposed method is demonstrated in both structural and multidisciplinary application contexts. The targeted applications include the analysis of a cantilever beam as well as the optimization of a green aircraft, resulting in a significant 439-kilogram reduction in fuel consumption during a single mission.
Mohammad Daffa Robani, Paul Saves, Pramudita Satria Palar et al.
Surrogate models are of high interest for many engineering applications, serving as cheap-to-evaluate time-efficient approximations of black-box functions to help engineers and practitioners make decisions and understand complex systems. As such, the need for explainability methods is rising and many studies have been performed to facilitate knowledge discovery from surrogate models. To respond to these enquiries, this paper introduces SMT-EX, an enhancement of the open-source Python Surrogate Modeling Toolbox (SMT) that integrates explainability techniques into a state-of-the-art surrogate modelling framework. More precisely, SMT-EX includes three key explainability methods: Shapley Additive Explanations, Partial Dependence Plot, and Individual Conditional Expectations. A peculiar explainability dependency of SMT has been developed for such purpose that can be easily activated once the surrogate model is built, offering a user-friendly and efficient tool for swift insight extraction. The effectiveness of SMT-EX is showcased through two test cases. The first case is a 10-variable wing weight problem with purely continuous variables and the second one is a 3-variable mixed-categorical cantilever beam bending problem. Relying on SMT-EX analyses for these problems, we demonstrate its versatility in addressing a diverse range of problem characteristics. SMT-Explainability is freely available on Github: https://github.com/SMTorg/smt-explainability .
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.
The Surrogate Modeling Toolbox (SMT) is an open-source Python package that offers a collection of surrogate modeling methods, sampling techniques, and a set of sample problems. This paper presents SMT 2.0, a major new release of SMT that introduces significant upgrades and new features to the toolbox. This release adds the capability to handle mixed-variable surrogate models and hierarchical variables. These types of variables are becoming increasingly important in several surrogate modeling applications. SMT 2.0 also improves SMT by extending sampling methods, adding new surrogate models, and computing variance and kernel derivatives for Kriging. This release also includes new functions to handle noisy and use multifidelity data. To the best of our knowledge, SMT 2.0 is the first open-source surrogate library to propose surrogate models for hierarchical and mixed inputs. This open-source software is distributed under the New BSD license.
Paul Saves, Nathalie Bartoli, Youssef Diouane et al.
Multidisciplinary design optimization methods aim at adapting numerical optimization techniques to the design of engineering systems involving multiple disciplines. In this context, a large number of mixed continuous, integer and categorical variables might arise during the optimization process and practical applications involve a large number of design variables. Recently, there has been a growing interest in mixed variables constrained Bayesian optimization but most existing approaches severely increase the number of the hyperparameters related to the surrogate model. In this paper, we address this issue by constructing surrogate models using less hyperparameters. The reduction process is based on the partial least squares method. An adaptive procedure for choosing the number of hyperparameters is proposed. The performance of the proposed approach is confirmed on analytical tests as well as two real applications related to aircraft design. A significant improvement is obtained compared to genetic algorithms.
Robin Grapin, Youssef Diouane, Joseph Morlier et al.
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.
Nathalie Bartoli, Thierry Lefebvre, Rémi Lafage et al.
This work aims at developing new methodologies to optimize computational costly complex systems (e.g., aeronautical engineering systems). The proposed surrogate-based method (often called Bayesian optimization) uses adaptive sampling to promote a trade-off between exploration and exploitation. Our in-house implementation, called SEGOMOE, handles a high number of design variables (continuous, discrete or categorical) and nonlinearities by combining mixtures of experts for the objective and/or the constraints. Additionally, the method handles multi-objective optimization settings, as it allows the construction of accurate Pareto fronts with a minimal number of function evaluations. Different infill criteria have been implemented to handle multiple objectives with or without constraints. The effectiveness of the proposed method was tested on practical aeronautical applications within the context of the European Project AGILE 4.0 and demonstrated favorable results. A first example concerns a retrofitting problem where a comparison between two optimizers have been made. A second example introduces hierarchical variables to deal with architecture system in order to design an aircraft family. The third example increases drastically the number of categorical variables as it combines aircraft design, supply chain and manufacturing process. In this article, we show, on three different realistic problems, various aspects of our optimization codes thanks to the diversity of the treated aircraft problems.
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, we propose a new approach that leverages Bayesian Optimization (BO) to optimize the hyper-parameters 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 less 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.
Paul Saves, Pramudita Satria Palar, Muhammad Daffa Robani et al.
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.
For developing innovative systems architectures, modeling and optimization techniques have been central to frame the architecting process and define the optimization and modeling problems. In this context, for system-of-systems the use of efficient dedicated approaches (often physics-based simulations) is highly recommended to reduce the computational complexity of the targeted applications. However, exploring novel architectures using such dedicated approaches might pose challenges for optimization algorithms, including increased evaluation costs and potential failures. To address these challenges, surrogate-based optimization algorithms, such as Bayesian optimization utilizing Gaussian process models have emerged.
Mouadh Yagoubi, David Danan, Milad Leyli-Abadi et al.
The integration of machine learning (ML) into the physical sciences is reshaping computational paradigms, offering the potential to accelerate demanding simulations such as computational fluid dynamics (CFD). Yet, persistent challenges in accuracy, generalization, and physical consistency hinder the practical deployment of ML models in scientific domains. To address these limitations and systematically benchmark progress, we organized the ML4CFD competition, centered on surrogate modeling for aerodynamic simulations over two-dimensional airfoils. The competition attracted over 240 teams, who were provided with a curated dataset generated via OpenFOAM and evaluated through a multi-criteria framework encompassing predictive accuracy, physical fidelity, computational efficiency, and out-of-distribution generalization. This retrospective analysis reviews the competition outcomes, highlighting several approaches that outperformed baselines under our global evaluation score. Notably, the top entry exceeded the performance of the original OpenFOAM solver on aggregate metrics, illustrating the promise of ML-based surrogates to outperform traditional solvers under tailored criteria. Drawing from these results, we analyze the key design principles of top submissions, assess the robustness of our evaluation framework, and offer guidance for future scientific ML challenges.
Giovanni Catalani, Jean Fesquet, Xavier Bertrand et al.
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 discretization-invariant 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 in 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.