Many results on micromorphic media can be found in the literature, where the equations of motion and the energetic foundations of the Eringen micromorphic continuum have been well established. The present paper has been devoted to the continuum modeling of a finite number of interacting, separated continua (microdomains) at the microscale, in which the strain energy has been formulated through a generalization of the Cauchy-Green deformation tensor, resulting in a degenerate metric at the considered scale, and a (6 & times; 6) Green-Lagrange strain tensor. The equilibrium equations have been obtained by systematically applying the method of virtual power. For one of the first time, boundary layer conditions appear in micromorphic mechanics. The paper concludes with a discussion on the number of constitutive parameters, shown to coincide, in number, with those of classical (Cauchy) elasticity, together with the recovery of micropolar continua as a special case and wide spectrum of applications of the proposed framework. Further details concerning the algebraic expressions of the tensors involved are provided in the Appendices.

We present a nonlinear micropolar continuum model that includes a diffusion-reaction equation for the remodeling of bone with a trabecular lattice microstructure. This micropolar formulation enables the modeling of bone's adaptive response to mechanical stimuli. The remodeling process is described as a time evolution regulated by a feedback mechanism that adjusts both the orientation of the trabecular lattice microarchitecture, thanks to the micropolar formulation, and the mechanical properties, related to key morphological features such us bone density. The junctions between trabeculae are modeled as nodal points within the micropolar framework, incorporating the following energy densities: mechanical deformation, mechano-biologic contributions, and Rayleigh-type dissipation terms that control the time evolution of remodeling. We present the weak form of the energetic formulation, which includes the diffusion-reaction equation. The cases of isotropic and orthotropic material symmetries class are deduced from the general formulation. This weak formulation is adapted to the development of a computational model based on the use of the finite element method. At the end of this paper, we discuss aspects related to validation, model complexity, and possible clinical applicability.

Numerical simulations are crucial for modeling complex systems, but calibrating them becomes challenging when data are noisy or incomplete and likelihood evaluations are computationally expensive. Bayesian calibration offers an interesting way to handle uncertainty, yet computing the posterior distribution remains a major challenge under such conditions. To address this, we propose a sequential surrogate-based approach that incrementally improves the approximation of the log-likelihood using Gaussian Process Regression. Starting from limited evaluations, the surrogate and its gradient are refined step by step. At each iteration, new evaluations of the expensive likelihood are added only at informative locations, that is to say where the surrogate is most uncertain and where the potential impact on the posterior is greatest. The surrogate is then coupled with the Metropolis-Adjusted Langevin Algorithm, which uses gradient information to efficiently explore the posterior. This approach accelerates convergence, handles relatively high-dimensional settings, and keeps computational costs low. We demonstrate its effectiveness on both a synthetic benchmark and an industrial application involving the calibration of high-speed train parameters from incomplete sensor data.

This paper introduces a new probabilistic framework for supervised learning in neural systems. It is designed to model complex, uncertain systems whose random outputs are strongly non-Gaussian given deterministic inputs. The architecture itself is a random object stochastically generated by a latent anisotropic Gaussian random field defined on a compact, boundaryless, multiply-connected manifold. The goal is to establish a novel conceptual and mathematical framework in which neural architectures are realizations of a geometry-aware, field-driven generative process. Both the neural topology and synaptic weights emerge jointly from a latent random field. A reduced-order parameterization governs the spatial intensity of an inhomogeneous Poisson process on the manifold, from which neuron locations are sampled. Input and output neurons are identified via extremal evaluations of the latent field, while connectivity is established through geodesic proximity and local field affinity. Synaptic weights are conditionally sampled from the field realization, inducing stochastic output responses even for deterministic inputs. To ensure scalability, the architecture is sparsified via percentile-based diffusion masking, yielding geometry-aware sparse connectivity without ad hoc structural assumptions. Supervised learning is formulated as inference on the generative hyperparameters of the latent field, using a negative log-likelihood loss estimated through Monte Carlo sampling from single-observation-per-input datasets. The paper initiates a mathematical analysis of the model, establishing foundational properties such as well-posedness, measurability, and a preliminary analysis of the expressive variability of the induced stochastic mappings, which support its internal coherence and lay the groundwork for a broader theory of geometry-driven stochastic learning.

We present a quantum computing formulation to address a challenging problem in the development of probabilistic learning on manifolds (PLoM). It involves solving the spectral problem of the high-dimensional Fokker-Planck (FKP) operator, which remains beyond the reach of classical computing. Our ultimate goal is to develop an efficient approach for practical computations on quantum computers. For now, we focus on an adapted formulation tailored to quantum computing. The methodological aspects covered in this work include the construction of the FKP equation, where the invariant probability measure is derived from a training dataset, and the formulation of the eigenvalue problem for the FKP operator. The eigen equation is transformed into a Schr & ouml;dinger equation with a potential V, a non-algebraic function that is neither simple nor a polynomial representation. To address this, we propose a methodology for constructing a multivariate polynomial approximation of V, leveraging polynomial chaos expansion within the Gaussian Sobolev space. This approach preserves the algebraic properties of the potential and adapts it for quantum algorithms. The quantum computing formulation employs a finite basis representation, incorporating second quantization with creation and annihilation operators. Explicit formulas for the Laplacian and potential are derived and mapped onto qubits using Pauli matrix expressions. Additionally, we outline the design of quantum circuits and the implementation of measurements to construct and observe specific quantum states. Information is extracted through quantum measurements, with eigenstates constructed and overlap measurements evaluated using universal quantum gates.

In this paper, the problem of helium gas transport in graphene nanochannels is investigated by a multilevel approach coupling surrogate modeling and Monte Carlo simulations. Under the rarefaction condition, the Knudsen diffusion is the dominant gas flow regime, characterized by gas-wall collisions. Furthermore, at low temperatures, helium gas atoms can be adsorbed on the surface and move like Brownian particles. To address the problem, the collision data between gas atoms and wall surfaces in equilibrium state are first generated by molecular dynamics simulations. A general statistical surrogate model based on polynomial chaos expansion accounting for adsorption and surface diffusion is constructed. The model exploits the Langevin dynamics analogy to satisfy physical constraints on the velocity distribution, the decaying time-dependent velocity correlation, and time-dependent displacement behavior. The trained model is then used in Monte Carlo simulations to compute various transport quantities in long nanochannels. Numerical results from the Monte Carlo study show that the approach can capture the size effect on the diffusion coefficient and transmission probability. Without being limited to helium and graphene, the developed method can be applied to any gas and solid couple and the problem of construction of kinetic boundary conditions.

This paper presents an innovative, computationally tractable approach for modeling and quantifying model-form uncertainty (MFU) in viscous computational fluid dynamics (CFD) models. It distinguishes between two sources of uncertainty: those related to turbulence modeling and other sources such as wall and far-field boundary conditions. The proposed approach comprises two complementary and coupled methods for uncertainty quantification (UQ): one targeting uncertainties in Reynolds stress modeling; and the other addressing all remaining model-form and parametric uncertainties. The first method decomposes the Reynolds stress tensor into a trace-vanishing deviatoric component and a spherical part. It then constructs a hyperparameterized probability model for the eigenvalues of the deviatoric component, based on its spectral algebraic properties. Further probabilistic modeling yields a complete hyperparameterized model for the Reynolds stress tensor, with each realization corresponding to an admissible turbulence model within a specific family. The second method adapts a recently developed nonparametric probabilistic approach for modeling and quantifying MFU to the context of this study. It relies on a probabilistic, projection-based model order reduction (PMOR) technique that is also hyperparameterized, ensuring computational tractability for UQ. The hyperparameters for both methods are simultaneously determined by formulating and minimizing an appropriate data-driven probabilistic loss function. Additionally, the methodology accounts for the uncertainties associated with PMOR, which is introduced to achieve efficient Monte Carlo simulations. The efficacy of the overall approach proposed for UQ in large-scale CFD computations is demonstrated through the Reynolds-averaged Navier-Stokes-based aerodynamic analysis of a rigid NASA Common Research Model configuration in the transonic flow regime, for which wind tunnel data is available.

This paper discusses wave propagation in unbounded particle-based materials described by a second-gradient continuum model, recently introduced by the authors, to provide an identification technique. The term particle-based materials denotes materials modeled as assemblies of particles, disregarding typical granular material properties such as contact topology, granulometry, grain sizes, and shapes. This work introduces a center-symmetric second-gradient continuum resulting from pairwise interactions. The corresponding Euler-Lagrange equations (equilibrium equations) are derived using the least action principle. This approach unveils non-classical interactions within subdomains. A novel, symmetric, and positive-definite acoustic tensor is constructed, allowing for an exploration of wave propagation through perturbation techniques. The properties of this acoustic tensor enable the extension of an identification procedure from Cauchy (classical) elasticity to the proposed second-gradient continuum model. Potential applications concern polymers, composite materials, and liquid crystals.

The main novelty of this paper consists of presenting a statistical artificial neural network (ANN)-based model for a robust prediction of the frequency-dependent aeroacoustic liner impedance using an aeroacoustic computational model (ACM) dataset of small size. The model, focusing on percentage of open area (POA) and sound pressure level (SPL) at a zero Mach number, takes into account uncertainties using a probabilistic formulation. The main difficulty in training an ANN-based model is the small size of the ACM dataset. The probabilistic learning carried out using the probabilistic learning on manifolds (PLoM) algorithm addresses this difficulty as it allows constructing a very large training dataset from learning the probabilistic model from a small dataset. A prior conditional probability model is presented for the PCA-based statistical reduced representation of the frequency-sampled vector of the log-resistance and reactance. It induces some statistical constraints that are not straightforwardly taken into account when training such an ANN-based model by classical optimizations methods under constraints. A second novelty of this paper consists of presenting an alternate solution that involves using conditional statistics estimated with learned realizations from PLoM. A numerical example is presented.

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.

This paper proposes, for particle-based materials, a higher-order nonlocal elasticity continuum model that includes the Piola peridynamics and the Eringen nonlocal elasticity. When referring to particle-based materials, we denote systems that can be modeled as assemblies of material points (or particles). Note that this paper is not devoted to granular materials, then factors such as the topology of contacts, granulometry, grain sizes, shapes, and geometric structure are not considered. Additionally, when referring to Piola peridynamics, we specifically denote the particular peridynamic model developed by Piola, which differs from the commonly adopted approach to peridynamics. The proposed higher-order nonlocal elasticity continuum model offers several advantages. First, it can describe interactions between material points over longer ranges than those considered by Eringen nonlocal elasticity. Second, it exhibits similar characteristics to gradient-type theories and Piola peridynamics, enabling the consideration of more complex external and contact actions, including N th order forces and stresses. Furthermore, the proposed deterministic model is developed to lay the foundation for a stochastic formulation applicable to uncertain particle-based materials. We want to emphasize that the aim of this paper is not to unify Eringen nonlocal elasticity with the various existing peridynamic models.

A formulation and an algorithm are presented to construct a truncated polynomial chaos representation of a vector-valued random output. This representation depends on a vector-valued random input with a known probability measure and a vector-valued random latent variable with an unknown probability measure. The construction of this PCE representation relies solely on a training set comprising a small number of independent realizations of the non-Gaussian dependent random output and input vectors. The training set consists of heterogeneous data, which poses challenges in accurately estimating the chaos coefficients. Despite the heterogeneity of the data, the proposed formulation and algorithm allow for the construction of a highly accurate global surrogate model. Additionally, we propose an alternative approach by constructing a surrogate model based on prior separation of the heterogeneous dataset into subsets, each containing “quasi-homogeneous” data. The separation method is designed to account for a partial overlap of the probability measure supports associated with the subsets. The identification of the PCE is performed offline. By utilizing the PCE, a fast online surrogate model is obtained, enabling analysis of large dynamical systems beyond the computational capabilities currently available. An application to atomic collisions of Helium on a graphite substrate is presented, where the training set was generated by Molecular Dynamics simulations done in a previous paper. The obtained results demonstrate accuracy of the proposed approach.

This article is the second part of a previous article devoted to the deterministic aspects. Here, we present a comprehensive study on the development and application of a novel stochastic second-gradient continuum model for particle-based materials. An application is presented concerning colloidal crystals. Since we are dealing with particle-based materials, factors such as the topology of contacts, particle sizes, shapes, and geometric structure are not considered. The mechanical properties of the introduced second-gradient continuum are modeled as random fields to account for uncertainties. The stochastic computational model is based on a mixed finite element (FE), and the Monte Carlo (MC) numerical simulation method is used as a stochastic solver. Finally, the resulting stochastic second-gradient model is applied to analyze colloidal crystals, which have wide-ranging applications. The simulations show the effects of second-order gradient on the mechanical response of a colloidal crystal under axial load, for which there could be significant fluctuations in the displacements.