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Auxetic structures, known for their negative Poisson's ratio, exhibit effective elastic properties heavily influenced by their underlying geometry and base material properties. While periodic homogenization of auxetic unit cells can be used to investigate these properties, it is computationally expensive and limits design space exploration and inverse analysis. In this paper, the fast Fourier transform (FFT)-based homogenization approach is adopted to efficiently generate data for developing surrogate models, bypassing concerns about periodic mesh generation and boundary conditions typically associated with the finite element method (FEM). Surrogate models are developed for the real-time prediction of the effective elastic properties of auxetic unit cells with orthogonal voids of different shapes. The generated surrogate models accept geometric parameters and base material properties as inputs to predict the effective elastic constants in real-time. This rapid evaluation enables a practical inverse analysis framework for obtaining the optimal design parameters that yield the desired effective response. The performance of the generated surrogate models is rigorously examined through a train/test split methodology, a parametric study, and an inverse problem. Finally, a graphical user interface (GUI) is developed, offering real-time prediction of the effective tangent stiffness and performing inverse analysis to determine optimal geometric parameters.

Abstract Trajectory planning aims at computing an optimal trajectory through the minimization of a cost function. This paper considers four different scenarios: (i) the first concerns a given trajectory on which a cost function is minimized by a acting on the velocity along it; (ii) the second considers trajectories expressed parametrically, from which an optimal path and the velocity along it are computed; (iii), the case in which only the departure and arrival points of the trajectory are known, and the optimal path must be determined; and finally, (iv) the case involving uncertainty in the environment in which the trajectory operates. When the considered cost functions are expressed analytically, the application of Euler–Lagrange equations constitutes an appealing option. However, in many applications, complex cost functions are learned by using black-box machine learning techniques, for instance deep neural networks. In such cases, a neural approach of the trajectory planning becomes an appealing alternative. Different numerical experiments will serve to illustrate the potential of the proposed methodologies on some selected use cases.

Abstract The simulation of magnetic bearings involves highly non-linear physics, with high dependency on the input variation. Moreover, such a simulation is time consuming and can’t run, within realistic computation time for control purposes, when using classical computation methods. On the other hand, classical model reduction techniques fail to achieve the required precision within the allowed computation window. To address this complexity, this work proposes a combination of physics-based computing methods, model reduction techniques and machine learning algorithms, to tackle the requirements. The physical model used to represent the magnetic bearing is the classical Cauer Ladder Network method, while the model reduction technique is applied on the error of the physical model’s solution. Later on, in the latent space a machine learning algorithm is used to predict the evolution of the correction in the latent space. The results show an improvement of the solution without scarifying the computation time. The solution is computed in almost real-time (few milliseconds), and compared to the finite element reference solution.

In the emerging field of mechanical metamaterials, using periodic lattice structures as a primary ingredient is relatively frequent. However, the choice of aperiodic lattices in these structures presents unique advantages regarding failure, e.g., buckling or fracture, because avoiding repeated patterns prevents global failures, with local failures occurring in turn that can beneficially delay structural collapse. Therefore, it is expedient to develop models for computing efficiently the effective mechanical properties in lattices from different general features while addressing the challenge of presenting topologies (or graphs) of different sizes. In this paper, we develop a deep learning model to predict energetically-equivalent mechanical properties of linear elastic lattices effectively. Considering the lattice as a graph and defining material and geometrical features on such, we show that Graph Neural Networks provide more accurate predictions than a dense, fully connected strategy, thanks to the geometrically induced bias through graph representation, closer to the underlying equilibrium laws from mechanics solved in the direct problem. Leveraging the efficient forward-evaluation of a vast number of lattices using this surrogate enables the inverse problem, i.e., to obtain a structure having prescribed specific behavior, which is ultimately suitable for multiscale structural optimization problems.

The efficient solution (fast and accurate) of parametric partial differential equations (pPDE) is of major interest in many domains of science and engineering, enabling evaluations of the quantities of interest, optimization, control, and uncertainty propagation—all them under stringent real-time constraints. Different methodologies have been proposed in the past within the model order reduction (MOR) community, based on the use of reduced bases (RB) or the separated representation at the heart of the so-called proper generalized decompositions (PGD). In PGD, an alternate-direction strategy is employed to circumvent the integration issues of operating in multi-dimensional domains. Recently, physics informed neural networks (PINNs), a particular collocation schema where the unknown field is approximated by a neural network (NN), have emerged in the domain of scientific machine learning. PNNs combine the versatility of NN-based approximation with the ease of collocating pPDE. The present paper proposes a combination of both procedures to find an efficient solution for pPDE, that can either be viewed as an efficient collocation procedure for PINN, or as a monolithic PGD that bypasses the use of the fixed-point alternated directions.

Monitoring a deep geological repository for radioactive waste during the operational phases relies on a combination of fit-for-purpose numerical simulations and online sensor measurements, both producing complementary massive data, which can then be compared to predict reliable and integrated information (e.g., in a digital twin) reflecting the actual physical evolution of the installation over the long term (i.e., a century), the ultimate objective being to assess that the repository components/processes are effectively following the expected trajectory towards the closure phase. Data prediction involves using historical data and statistical methods to forecast future outcomes, but it faces challenges such as data quality issues, the complexity of real-world data, and the difficulty in balancing model complexity. Feature selection, overfitting, and the interpretability of complex models further contribute to the complexity. Data reconciliation involves aligning model with in situ data, but a major challenge is to create models capturing all the complexity of the real world, encompassing dynamic variables, as well as the residual and complex near-field effects on measurements (e.g., sensors coupling). This difficulty can result in residual discrepancies between simulated and real data, highlighting the challenge of accurately estimating real-world intricacies within predictive models during the reconciliation process. The paper delves into these challenges for complex and instrumented systems (multi-scale, multi-physics, and multi-media), discussing practical applications of machine and deep learning methods in the case study of thermal loading monitoring of a high-level waste (HLW) cell demonstrator (called ALC1605) implemented at Andra’s underground research laboratory.

Learning linear and nonlinear dynamical systems from available data is a timely topic in scientific machine learning. Learning must be performed while enforcing the numerical stability of the learned model, the existing knowledge within an informed or augmented setting, or by taking into account the multiscale dynamics—for both linear and nonlinear dynamics. However, when the final objective of such a learned dynamical system is to be used for control purposes, learning transformed dynamics can be advantageous. Therefore, many alternatives exists, and the present paper focuses on two of them: the first based on the discovery and use of the so-called flat control and the second one based on the use of the Koopman theory. The main contributions when addressing the first is the discovery of the flat output transformation by using an original neural framework. Moreover, when using the Koopman theory, this paper proposes an original procedure for learning parametric dynamics in the latent space, which is of particular interest in control-based engineering applications.

There was an error in the original publication, section data availability, stating “Data is available upon request” [...]

A deep geological repository for radioactive waste, such as Andra’s Cigéo project, requires long-term (persistent) monitoring. To achieve this goal, data from a network of sensors are acquired. This network is subject to deterioration over time due to environmental effects (radioactivity, mechanical deterioration of the cell, etc.), and it is paramount to assess each sensor’s integrity and ensure data consistency to enable the precise monitoring of the facilities. Graph neural networks (GNNs) are suitable for detecting faulty sensors in complex networks because they accurately depict physical phenomena that occur in a system and take the sensor network’s local structure into consideration in the predictions. In this work, we leveraged the availability of the experimental data acquired in Andra’s Underground Research Laboratory (URL) to train a graph neural network for the assessment of data integrity. The experiment considered in this work emulated the thermal loading of a high-level waste (HLW) demonstrator cell (i.e., the heating of the containment cell by nuclear waste). Using real experiment data acquired in Andra’s URL in a deep geological layer was one of the novelties of this work. The used model was a GNN that inputted the temperature field from the sensors (at the current and past steps) and returned the state of each individual sensor, i.e., faulty or not. The other novelty of this work lay in the application of the GraphSAGE model which was modified with elements of the Graph Net framework to detect faulty sensors, with up to half of the sensors in the network being faulty at once. This proportion of faulty sensors was explained by the use of distributed sensors (optic fiber) and the environmental effects on the cell. The GNNs trained on the experimental data were ultimately compared against other standard classification methods (thresholding, artificial neural networks, etc.), which demonstrated their effectiveness in the assessment of data integrity.

When training a parametric surrogate to represent a real-world complex system in real time, there is a common assumption that the values of the parameters defining the system are known with absolute confidence. Consequently, during the training process, our focus is directed exclusively towards optimizing the accuracy of the surrogate’s output. However, real physics is characterized by increased complexity and unpredictability. Notably, a certain degree of uncertainty may exist in determining the system’s parameters. Therefore, in this paper, we account for the propagation of these uncertainties through the surrogate using a standard Monte Carlo methodology. Subsequently, we propose a novel regression technique based on optimal transport to infer the impact of the uncertainty of the surrogate’s input on its output precision in real time. The OT-based regression allows for the inference of fields emulating physical reality more accurately than classical regression techniques, including advanced ones.

Machine learning approaches are currently used to understand or model complex physical systems. In general, a substantial number of samples must be collected to create a model with reliable results. However, collecting numerous data is often relatively time-consuming or expensive. Moreover, the problems of industrial interest tend to be more and more complex, and depend on a high number of parameters. High-dimensional problems intrinsically involve the need for large amounts of data through the curse of dimensionality. That is why new approaches based on smart sampling techniques have been investigated to minimize the number of samples to be given to train the model, such as active learning methods. Here, we propose a technique based on a combination of the Fisher information matrix and sparse proper generalized decomposition that enables the definition of a new active learning informativeness criterion in high dimensions. We provide examples proving the performances of this technique on a theoretical 5D polynomial function and on an industrial crash simulation application. The results prove that the proposed strategy outperforms the usual ones.