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In this work, we address operator learning for stochastic homogenization in nonlinear elasticity. A Fourier neural operator is employed to learn the map between the input field describing the material at fine scale and the deformation map. We propose a variationally-consistent loss function that does not involve solution field data. The methodology is tested on materials described either by piecewise constant fields at microscale, or by random fields at mesoscale. High prediction accuracy is obtained for both the solution field and the homogenized response. We show, in particular, that the accuracy achieved with the proposed strategy is comparable to that obtained with the conventional data-driven training method. 

Approximation frameworks for phase-field models of brittle fracture are presented and compared in this work. Such methods aim to address the computational cost associated with conducting full-scale simulations of brittle fracture in heterogeneous materials where material parameters, such as fracture toughness, can vary spatially. They proceed by combining a dimension reduction with learning between function spaces. Two classes of approximations are considered. In the first class, deep learning models are used to perform regression in ad hoc latent spaces. PCA-Net and Fourier neural operators are specifically presented for the sake of comparison. In the second class of techniques, statistical sampling is used to approximate the forward map in latent space, using conditioning. To ensure proper measure concentration, a reduced-order Hamiltonian Monte Carlo technique (namely, probabilistic learning on manifold) is employed. The accuracy of these methods is then investigated on a proxy application where the fracture toughness is modeled as a non-Gaussian random field. It is shown that the probabilistic framework achieves comparable performance in the 𝐿2 sense while enabling the end-user to bypass the art of defining and training deep learning models. 

Traditional linear subspace-based reduced order models (LS-ROMs) can be used to significantly accelerate simulations in which the solution space of the discretized system has a small dimension (with a fast decaying Kolmogorov n-width). However, LS-ROMs struggle to achieve speed-ups in problems whose solution space has a large dimension, such as highly nonlinear problems whose solutions have large gradients. Such an issue can be alleviated by combining nonlinear model reduction with operator learning. Over the past decade, many nonlinear manifold-based reduced order models (NM-ROM) have been proposed. In particular, NM-ROMs based on deep neural networks (DNN) have received increasing interest. This work takes inspiration from adaptive basis methods and specifically focuses on developing an NM-ROM based on Convolutional Neural Network-based autoencoders (CNNAE) with iteration-dependent trainable kernels. Additionally, we investigate DNN-based and quadratic operator inference strategies between latent spaces. A strategy to perform vectorized implicit time integration is also proposed. We demonstrate that the proposed CNN-based NM-ROM, combined with DNN- based operator inference, generally performs better than commonly employed strategies (in terms of prediction accuracy) on a benchmark advection-dominated problem. The method also presents substantial gain in terms of training speed per epoch, with a training time about one order of magnitude smaller than the one associated with a state-of-the-art technique performing with the same level of accuracy. 

Traditional linear subspace-based reduced order models (LS-ROMs) can be used to significantly accelerate simulations in which the solution space of the discretized system has a small dimension (with a fast decaying Kolmogorov 𝑛-width). However, LS-ROMs struggle to achieve speed-ups in problems whose solution space has a large dimension, such as highly nonlinear problems whose solutions have large gradients. Such an issue can be alleviated by combining nonlinear model reduction with operator learning. Over the past decade, many nonlinear manifold-based reduced order models (NM-ROM) have been proposed. In particular, NM-ROMs based on deep neural networks (DNN) have received increasing interest. This work takes inspiration from adaptive basis methods and specifically focuses on developing an NM-ROM based on Convolutional Neural Network-based autoencoders (CNNAE) with iteration-dependent trainable kernels. Additionally, we investigate DNN-based and quadratic operator inference strategies between latent spaces. A strategy to perform vectorized implicit time integration is also proposed. We demonstrate that the proposed CNN-based NM-ROM, combined with DNN- based operator inference, generally performs better than commonly employed strategies (in terms of prediction accuracy) on a benchmark advection-dominated problem. The method also presents substantial gain in terms of training speed per epoch, with a training time about one order of magnitude smaller than the one associated with a state-of-the-art technique performing with the same level of accuracy. 

This work focuses on the representation of model-form uncertainties in phase-field models of brittle fracture. Such uncertainties can arise from the choice of the degradation function for instance, and their consideration has been unaddressed to date. The stochastic modeling framework leverages recent developments related to the analysis of nonlinear dynamical systems and relies on the construction of a stochastic reduced-order model. In the latter, a POD-based reduced-order basis is randomized using Riemannian projection and retraction operators, as well as an information-theoretic formulation enabling proper concentration in the convex hull defined by a set of model proposals. The model thus obtained is mathematically admissible in the almost sure sense and involves a low-dimensional hyperparameter, the calibration of which is facilitated through the formulation of a quadratic programming problem. The relevance of the modeling approach is further assessed on one- and two-dimensional applications. It is shown that model uncertainties can be efficiently captured and propagated to macroscopic quantities of interest. An extension based on localized randomization is also proposed to handle the case where the forward simulation is highly sensitive to sample localization. This work constitutes a methodological development allowing phase-field predictions to be endowed with statistical measures of confidence, accounting for the variability induced by modeling choices. 

The stochastic modeling and calibration of an anisotropic elasto-plastic model for additive manufacturing materials are addressed in this work. We specifically focus on 316L stainless steel, produced by directed energy deposition. Tensile specimens machined from two additive manufactured (AM) box-structures were used to characterize material anisotropy and random spatial variations in elasticity and plasticity material parameters. Tensile specimens were cut parallel (horizontal) and perpendicular (vertical) to the AM deposition plane and were indexed by location. These results show substantial variability in both regimes, with fluctuation levels that differ between specimens loaded in the parallel and perpendicular build directions. Stochastic representations for the stiffness and Hill’s criterion coefficients random fields are presented next. Information-theoretic models are derived within the class of translation random fields, with the aim of promoting identifiability with limited data. The approach allows for the constitutive models to be generated on arbitrary geometries, using the so- called stochastic partial differential approach (to sampling). These representations are then partially calibrated using the aforementioned experimental results, hence enabling subsequent propagation analyses. Sampling is finally exemplified on the considered structure. 

Stochastic mesoscale inhomogeneity of material properties and material symmetries are investigated in a 3D-printed material. The analysis involves a spatially-dependent characterization of the microstructure in 316 L stainless steel, obtained through electron backscatter diffraction imaging. These data are subsequently fed into a Voigt–Reuss–Hill homogenization approxima- tion to produce maps of elasticity tensor coefficients along the path of experimental probing. Information-theoretic stochastic models corresponding to this stiffness random field are then introduced. The case of orthotropic fields is first defined as a high-fidelity model, the realizations of which are consistent with the elasticity maps. To investigate the role of material symmetries, an isotropic approximation is next introduced through ad-hoc projections (using various metrics). Both stochastic representations are identified using the dataset. In particular, the correlation length along the characterization path is identified using a maximum likelihood estimator. Uncertainty propagation is finally performed on a complex geometry, using a Monte Carlo analysis. It is shown that mechanical predictions in the linear elastic regime are mostly sensitive to material symmetry but weakly depend on the spatial correlation length in the considered propagation scenario.