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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. 

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. 

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. 

Abstract Objective. Transcranial magnetic stimulation (TMS) is a non-invasive brain stimulation method that is used to study brain function and conduct neuropsychiatric therapy. Computational methods that are commonly used for electric field (E-field) dosimetry of TMS are limited in accuracy and precision because of possible geometric errors introduced in the generation of head models by segmenting medical images into tissue types. This paper studies E-field prediction fidelity as a function of segmentation accuracy. Approach. The errors in the segmentation of medical images into tissue types are modeled as geometric uncertainty in the shape of the boundary between tissue types. For each tissue boundary realization, we then use an in-house boundary element method to perform a forward propagation analysis and quantify the impact of tissue boundary uncertainties on the induced cortical E-field. Main results. Our results indicate that predictions of E-field induced in the brain are negligibly sensitive to segmentation errors in scalp, skull and white matter (WM), compartments. In contrast, E-field predictions are highly sensitive to possible cerebrospinal fluid (CSF) segmentation errors. Specifically, the segmentation errors on the CSF and gray matter interface lead to higher E-field uncertainties in the gyral crowns, and the segmentation errors on CSF and WM interface lead to higher uncertainties in the sulci. Furthermore, the uncertainty of the average cortical E-fields over a region exhibits lower uncertainty relative to point-wise estimates. Significance. The accuracy of current cortical E-field simulations is limited by the accuracy of CSF segmentation accuracy. Other quantities of interest like the average of the E-field over a cortical region could provide a dose quantity that is robust to possible segmentation errors.