Roger Ghanem

This work aims to enhance the small amount of data obtained from simulations using statistical methods, in order to improve the analysis of the underlying physical properties of plasma behavior, particularly electron density and electron temperature in the boundary region of the tokamak. Electron density and electron temperature are important observables for characterizing plasma behavior in the boundary region of the tokamak, as they influence transport properties and confinement regimes. While the simulations used in this work are fully deterministic, we aim to extract statistical information by interpreting numerical simulations as samples from underlying random fields. This approach allows for the construction of marginal and joint probability density functions (PDFs) that provide physical insight beyond standard deterministic interpretation and capture key structural features of the plasma behavior in the boundary region of the tokamak, including the role of triangularity, the impact of the diffusion coefficient, and the implications for plasma behavior. Numerical simulations are performed with the Gkeyll gyrokinetic code. Despite their resolution, the computational cost limits the number of available simulation data, suggesting a role for advanced analysis techniques capable of extracting meaningful physical insights from a small dataset. To this end, we explore a statistical framework based on probabilistic learning on manifolds (PLoM), a nonparametric method designed for small-sample inference.

This work presents a novel framework for physically consistent model error characterization and operator learning for reduced-order models of non-equilibrium chemical kinetics. By leveraging the Bayesian framework, we identify and infer sources of model and parametric uncertainty within the Coarse-Graining Methodology across a range of initial conditions. The model error is embedded into the chemical kinetics model to ensure that its propagation to quantities of interest remains physically consistent. For operator learning, we develop a methodology that separates time dynamics from other input parameters. Karhunen-Loeve Expansion (KLE) is employed to capture time dynamics, yielding temporal modes, while Polynomial Chaos Expansion (PCE) is subsequently used to map model error and input parameters to KLE coefficients. The proposed model offers three significant advantages: i) Separating time dynamics from other inputs ensures stability of chemistry surrogate when coupled with fluid solvers; ii) The framework fully accounts for model and parametric uncertainty, enabling robust probabilistic predictions; iii) The surrogate model is highly interpretable, with visualizable time modes and a PCE component that facilitates analytical calculation of sensitivity indices. We apply this framework to O2-O chemistry system under hypersonic flight conditions, validating it in both a 0D adiabatic reactor and coupled simulations with a fluid solver in a 1D shock case. Results demonstrate that the surrogate is stable during time integration, delivers physically consistent probabilistic predictions accounting for model and parametric uncertainty, and achieves maximum relative error below 10%. This work represents a significant step forward in enabling probabilistic predictions of non-equilibrium chemistry with coupled fluid solvers, offering a physically accurate approach for hypersonic flow predictions.

After years of operation in reactors at nuclear power plants, nuclear fuel assemblies (FA) become high-level radioactive waste known as spent nuclear fuel (SNF). In the absence of a permanent disposal solution, SNF is stored in sealed stainless-steel dry storage canisters. Damage to the FA can occur during handling, storage, or transportation, and detecting such damage is critical prior to a long-term disposal. However, direct visual inspection is costly and sometimes not feasible because the canisters are sealed, necessitating non-destructive evaluation techniques for their internal condition assessment. In this study, a 2/3-scale physical canister mock-up with mock-up FA was employed to simulate multiple FA damage modes. Experimental modal analysis was performed to obtain frequency response functions (FRF) from the exterior surface of the canister bottom plate. A variational autoencoder (VAE) was trained exclusively on FRF data from the undamaged canister to learn the response of the healthy structure. FRF from the canister with FA damage were then processed through the trained VAE to generate reconstruction error signals. These signals were analyzed using unsupervised machine learning models, including isolation forest and local outlier factor, to detect anomalies. Additionally, a probabilistic approach was developed by fitting a Gaussian mixture model (GMM) to the latent space of the trained VAE. The resulting anomaly scores showed strong separation between healthy and damaged samples, with higher scores corresponding to greater damage severity. The GMM-based method achieved an F1 score of 0.998 on the testing dataset. These results demonstrate that the VAE-based framework can effectively detect FA damage within sealed canisters using unlabeled FRF data. This offers a practical solution for detecting anomalous behavior in real-world SNF canister inspections.

Abstract Nuclear fuel assemblies (FA) become high-level radioactive waste known as spent nuclear fuel (SNF) after several years of operation in nuclear reactors. Currently, a considerable portion of SNF is temporarily stored in sealed stainless-steel dry storage canisters. Handling, storage or transportation events (normal operations or accidents) can cause potential damage to the FAs inside the canisters. Damage to FAs inside the canisters needs to be identified for safety purposes during storage or before and after transportation. Due to the difficulty of a visual inspection of the sealed canisters, non-destructive evaluation (NDE) is critical to identify the potential internal damage. In this study, a high-fidelity finite element (FE) model was used to simulate different levels of FA damage. Wasserstein generative adversarial networks (WGAN) were developed to learn and generate noise using data from previously conducted experiments, which was then added to the numerically obtained frequency response functions (FRF) to bridge the gap between experimental and numerical domains. The noisy computational data were analyzed by a multi-task extreme gradient boosting (XGBoost) model to identify the damage level and location. The XGBoost achieved macro-F1 scores of 0.998 and 0.900 for damage detection and localization tasks in the FE dataset and perfect scores of 1.0 for the same in the experimental dataset. The results demonstrate that the machine learning (ML)-aided NDE method was successful in identifying various damage modes within SNF canisters even in the presence of noise levels observed in actual large-scale experiments.

We present a generative learning framework for probabilistic sampling based on an extension of the Probabilistic Learning on Manifolds (PLoM) approach, which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a limited (yet representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to uncover the intrinsic low-dimensional manifold structure, and (c) a reduced-order Ito Stochastic Differential Equation (ISDE) to sample from the learned distribution. A key challenge arises, however, when the number of available data points N is small and the dimensionality of the diffusion-map basis approaches N, resulting in overfitting and loss of generalization. To overcome this limitation, we propose an enabling extension that implements a synthesis of Double Diffusion Maps -- a technique capable of capturing multiscale geometric features of the data -- with Geometric Harmonics (GH), a nonparametric reconstruction method that allows smooth nonlinear interpolation in high-dimensional ambient spaces. This approach enables us to solve a full-order ISDE directly in the latent space, preserving the full dynamical complexity of the system, while leveraging its reduced geometric representation. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an Itô stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps, a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

This research presents a novel methodology for multi-fidelity modelling in coarse-graining models (CGM) to investigate thermo-chemical non-equilibrium effects in hypersonic flows. The focus is on combining abundant solutions from inexpensive yet inaccurate low-fidelity model with sparse set of accurate but expensive mid and high-fidelity model samples. Principal Component Analysis (PCA) is first used to reduce the dimensionality of quantities of interest followed by which a Polynomial Chaos Expansion (PCE) is constructed to map the random input parameters to PCA coefficients. Furthermore, to account for the error introduced by low-fidelity model and insufficient number of samples from mid and high-fidelity models, the PCE coefficients are assumed to be probabilistic whose posterior distributions are obtained by Bayesian inference. The surrogate modelling methodology developed in this work is well suited for high dimensional quantities of interest and strikes a balance between computational cost and accuracy. The multi-fidelity surrogate model is subsequently used for forward propagation of uncertainty in free-stream conditions to obtain probabilistic predictions of the quantities of interest including species mole fractions and translational temperature around a spherical reentry vehicle.

International audience

This work presents a novel framework for physically consistent model error characterization and operator learning for reduced-order models of non-equilibrium chemical kinetics. By leveraging the Bayesian framework, we identify and infer sources of model and parametric uncertainty within the Coarse-Graining Methodology across a range of initial conditions. The model error is embedded into the chemical kinetics model to ensure that its propagation to quantities of interest remains physically consistent. For operator learning, we develop a methodology that separates time dynamics from other input parameters. Karhunen-Loeve Expansion (KLE) is employed to capture time dynamics, yielding temporal modes, while Polynomial Chaos Expansion (PCE) is subsequently used to map model error and input parameters to KLE coefficients. The proposed model offers three significant advantages: i) Separating time dynamics from other inputs ensures stability of chemistry surrogate when coupled with fluid solvers; ii) The framework fully accounts for model and parametric uncertainty, enabling robust probabilistic predictions; iii) The surrogate model is highly interpretable, with visualizable time modes and a PCE component that facilitates analytical calculation of sensitivity indices. We apply this framework to O2-O chemistry system under hypersonic flight conditions, validating it in both a 0D adiabatic reactor and coupled simulations with a fluid solver in a 1D shock case. Results demonstrate that the surrogate is stable during time integration, delivers physically consistent probabilistic predictions accounting for model and parametric uncertainty, and achieves maximum relative error below 10%. This work represents a significant step forward in enabling probabilistic predictions of non-equilibrium chemistry with coupled fluid solvers, offering a physically accurate approach for hypersonic flow predictions.

Spent nuclear fuel (SNF) assemblies (FAs), composed of bundled radioactive fuel rods, are stored in stainless-steel canisters as an interim dry storage option until permanent storage solutions are available. Accidental damage to these canisters may occur during handling or transportation events. If such events occur, it is necessary to assess the integrity of FAs before long-term storage. Since the canisters are sealed shut and can only be opened in special handling facilities, non-destructive evaluation (NDE) is critical for detecting the FAs from the canister's exterior surface. In this study, two multi-task machine learning (ML) classifiers, a k -nearest neighbors ( k -NN) and a convolutional neural network (CNN), are developed to simultaneously detect and localize internal FA damage. The classifiers were trained and tested on a dataset collected via experimental modal analysis on a 2/3-scale mock-up canister. The canister was excited from the bottom plate while accelerometers were also attached on the bottom plate at arbitrary locations to record the structural response. The differences in frequency response functions (FRFs) between an intact fully loaded canister basket (FLCB) system and the canister with simulated internal damage were calculated and used as input to the ML models. Results showed that both classifiers achieved high accuracy on the testing set. The k -NN classifier produced macro-F1 scores of 1.00 for the damage detection task and 0.996 for the localization task. The macro-F1 scores of the CNN were 0.991 for damage detection and 0.964 for damage localization. Additionally, dropout layers were added to the fully connected layers of the CNN to introduce model uncertainty. By testing the model 1,000 times, probability density functions (PDFs) were generated, and it was confirmed that the CNN produced confident predictions in both the damage detection and localization tasks. This multi-task ML method contributes to advancing NDE of SNF canisters and holds potential for field applications in inspecting actual SNF canisters.

PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an Itô equation from a stochastic dissipative Hamiltonian dynamical system, acting as the MCMC generator, for which the KDE-estimated probability measure with the training dataset is the invariant measure. PLoM performs a projection on a reduced-order vector basis related to the training dataset, using the diffusion maps (DMAPS) basis constructed with a time-independent isotropic kernel. In this paper, we propose a new ISDE projection vector basis built from a transient anisotropic kernel, providing an alternative to the DMAPS basis to improve statistical surrogates for stochastic manifolds with heterogeneous data. The construction ensures that for times near the initial time, the DMAPS basis coincides with the transient basis. For larger times, the differences between the two bases are characterized by the angle of their spanned vector subspaces. The optimal instant yielding the optimal transient basis is determined using an estimation of mutual information from Information Theory, which is normalized by the entropy estimation to account for the effects of the number of realizations used in the estimations. Consequently, this new vector basis better represents statistical dependencies in the learned probability measure for any dimension. Three applications with varying levels of statistical complexity and data heterogeneity validate the proposed theory, showing that the transient anisotropic kernel improves the learned probability measure.

In the field of computational mechanics, mesh refinement is essential for achieving high-fidelity solutions in finite element method (FEM) simulations. However, detailed modeling of composite materials within large complex systems can be computationally expensive or even intractable. This paper introduces a novel framework for global mesh refinement using statistical learning. By simultaneously observing low-fidelity (coarse-mesh) and high-fidelity (fine-mesh) solutions of FEM simulations, we learn the joint probability distribution of the observed quantities. This distribution encodes the correction from coarse to fine-mesh solutions, allowing us to predict high-fidelity solutions from coarse-mesh observations using statistical conditioning. We utilize data-driven representative volume elements (RVEs) to collect and combine snapshots of solutions across different mesh resolutions. Our framework is validated through multiple case studies, including elastic and bilinear material models, and various levels of refinement. The results demonstrate satisfactory prediction accuracy, even with significant mesh refinement. Additionally, we address irregular discretizations by incorporating an intermediate interpolation step to regular grids. Our approach significantly reduces the number of required function evaluations while maintaining high accuracy, thus enhancing the efficiency of FEM simulations. By discovering and leveraging statistical dependencies between the characteristics of stochastic solutions at different mesh resolutions, the proposed approach provides a milestone in alleviating the computational burden in stochastic finite elements. This work also highlights the potential of probabilistic learning methods in multiscale modeling and offers a promising direction for future research in computational mechanics and materials science.

This work introduces a novel framework for the prediction of design allowables of composite laminates with reduced experimental cost. Building on high-fidelity simulations, polynomial chaos expansions (PCE) are first used to build probabilistic models for the material properties (input parameters). The coefficients of the models are themselves randomized by perturbing them with new stochastic degrees of freedom, thus accounting for uncertainty on the uncertainty and generating a family of distributions for these parameters. High-fidelity simulations are used to generate samples of the quantity of interest (QoI) which is then represented by a new PCE in terms of the previously described stochastic degrees of freedom. With this construction, the distribution of the QoI can be cheaply estimated and updated with experimental observations. The framework is applied to a hybrid carbon-glass fiber composite laminate under 3-point bending. We demonstrate how the updated distribution of the QoI can be used to predict the A-basis design allowable.

Cross scale simulation aims to develop computational models that can capture the behavior of a system at multiple scales, from microscopic to macroscopic level. To explore the constitutive relationship of materials, a widely used conceptual and numerical tool called Representative Volume Element (RVE) is introduced to solicit information from their subscales in finite element analysis. It characterizes the behavior at one quadrature point of a physical device by homogenizing the local behavior over finer scales. By replicating the RVE throughout the analysis domain, a homogeneous finite element model can be synthesized, simplifying the analysis of a heterogeneous material in the presence of uncertainty, such as variations in loading, material properties of micro-constituents, and modeling physical behavior. Moreover, the generation of many similar data at subscale introduced by homogenization in RVE is usually tackled by statistical methods. In this work, we explore the mechanical behavior of a hybrid composite material, consisting of glass and carbon fibers embedded in an epoxy resin for use as a battery enclosure, using a novel energy density informed neural network architecture. We model the material properties of fiber tows and resin as random variables and use input strain tensors from a macro-scale three-point bend test simulation. History of stress tensors and strain energy density are extracted as outputs for supervise learning. In the neural network architecture, Physical constraints are implemented by introducing a novel energy density layer for the prediction of stress along the direction of tows. A long-short-term-memory (LSTM) neural network training on the same dataset is also conducted for comparison. Elimination of spurious discontinuities is obtained by PCA (Principal Components Analysis) in the stress-strain curve when predicting stresses in directions transverse to the tows, YY and ZZ directions. We develop a user-defined subroutine in LS-Dyna and conduct a mesoscale simulation to verify its accuracy.

Multifidelity uncertainty quantification (MF UQ) sampling approaches have been shown to significantly reduce the variance of statistical estimators while preserving the bias of the highest-fidelity model, provided that the low-fidelity models are well correlated. However, maintaining a high level of correlation can be challenging, especially when models depend on different input uncertain parameters, which drastically reduces the correlation. Existing MF UQ approaches do not adequately address this issue. In this work, we propose a new sampling strategy that exploits a shared space to improve the correlation among models with dissimilar parametrization. We achieve this by transforming the original coordinates onto an auxiliary manifold using the adaptive basis (AB) method~\cite{Tipireddy2014}. The AB method has two main benefits: (1) it provides an effective tool to identify the low-dimensional manifold on which each model can be represented, and (2) it enables easy transformation of polynomial chaos representations from high- to low-dimensional spaces. This latter feature is used to identify a shared manifold among models without requiring additional evaluations. We present two algorithmic flavors of the new estimator to cover different analysis scenarios, including those with legacy and non-legacy high-fidelity data. We provide numerical results for analytical examples, a direct field acoustic test, and a finite element model of a nuclear fuel assembly. For all examples, we compare the proposed strategy against both single-fidelity and MF estimators based on the original model parametrization.

View Video Presentation: https://doi.org/10.2514/6.2023-3331.vid This study investigates uncertainty propagation and sensitivity analysis of state-specific dissociation and excitation rate coefficients in the context of macroscopic quantities of interest such as species mole fraction evolution and quasi-steady-state (QSS) rate coefficient. To accomplish this, an isothermal isochoric zero-dimensional chemical reactor is solved for various bath conditions. To handle the computational complexity of the master equations, three different coarse-graining methods are utilized: a 200-bin energy-based lumping model, a 3-bin energy-based lumping model, and a 10-bin spectral clustering-based model. The results show that while the uncertainty propagation is sensitive to the type of coarse-graining, the spectral clustering method produces the least model error when compared to the other coarse-grained models employed. Moreover, when an uncertainty factor of 5 is applied to the state-specific dissociation rate coefficients, it leads to an approximate ± 10% uncertainty range around the nominal values of the QSS rate coefficient. The sensitivity analysis conducted using the 200-bin model reveals that the most influential factor affecting the QSS rate coefficient and dissociation time is the mono-quantum vibrational excitation from low-lying levels. Additionally, at low temperatures, the high-lying dissociation rate coefficients contribute significantly to the uncertainty of the studied quantities, while at high temperatures, the dissociation from low to moderate-lying vibrational energy states plays a crucial role. These findings underscore the critical role played by vibrational excitation in determining the behavior of reactive systems at different temperature regimes.

Presented are manufacturing control systems for composite-material structures, methods for assembling/operating such systems, and transfer molding techniques for predicting and ameliorating void conditions in fiber-reinforced polymer panels. A method for forming a composite-material construction includes receiving a start signal indicating a fiber-based preform is inside a mold cavity, and transmitting a command signal to inject pressurized resin into the mold to induce resin flow within the mold cavity and impregnate the fiber-based preform. An electronic controller receives, from a distributed array of sensors attached to the mold, signals indicative of pressure and/or temperature at discrete locations on an interior face of the mold cavity. The controller determines a measurement deviation between a calibrated baseline value and the pressure and/or temperature values for each of the discrete locations. If any one of the respective measurement deviations exceeds a calibrated threshold, a void signal is generated to flag a detected void condition.

We present a novel stochastic optimal control framework that accounts for various types of uncertainties, with application to reentry trajectory planning. The formulation of the optimal trajectory control problem is presented in the context of an indirect method where a functional objective associated with the terminal vehicle speed is to be minimized. Uncertain input parameters in the optimal trajectory control model, including aerodynamic parameters and initial and terminal conditions, are modeled as aleatory random variables, while the statistical parameters of these aleatory distributions are themselves random variables. The parametric and model uncertainties are simultaneously propagated through an extended polynomial chaos expansion (EPCE) formalism. Several metrics are described to evaluate response statistics and presented as insightful tools for robust decision making. Specifically, the response probability density function (PDF) reflecting influence of both epistemic and aleatory uncertainties is obtained. By sampling over the random variables representing model error, an ensemble of response PDFs is generated and the associated failure probability is estimated as a random variable with its own polynomial chaos expansion. Besides, the sensitivity index functions of response PDF with respect to the statistical parameters are evaluated. Coupling parametric and model uncertainties within the EPCE framework leads to a robust and efficient paradigm for multilevel uncertainty propagation and PDF characterization in general optimal control problems.

Multifidelity uncertainty quantification (MF UQ) sampling approaches have been shown to significantly reduce the variance of statistical estimators while preserving the bias of the highest-fidelity model, provided that the low-fidelity models are well correlated. However, maintaining a high level of correlation can be challenging, especially when models depend on different input uncertain parameters, which drastically reduces the correlation. Existing MF UQ approaches do not adequately address this issue. In this work, we propose a new sampling strategy that exploits a shared space to improve the correlation among models with dissimilar parametrization. We achieve this by transforming the original coordinates onto an auxiliary manifold using the adaptive basis (AB) method~\cite{Tipireddy2014}. The AB method has two main benefits: (1) it provides an effective tool to identify the low-dimensional manifold on which each model can be represented, and (2) it enables easy transformation of polynomial chaos representations from high- to low-dimensional spaces. This latter feature is used to identify a shared manifold among models without requiring additional evaluations. We present two algorithmic flavors of the new estimator to cover different analysis scenarios, including those with legacy and non-legacy high-fidelity data. We provide numerical results for analytical examples, a direct field acoustic test, and a finite element model of a nuclear fuel assembly. For all examples, we compare the proposed strategy against both single-fidelity and MF estimators based on the original model parametrization.