On the Construction of Admissible Representations for Scientific Machine Learning and Uncertainty Quantification, with Various Applications in Computational Mechanics

Abstract

The representation of information stands at the core of scientific machine learning and uncertainty quantification in computational mechanics. The main purpose is to construct "optimal" representations that on the one hand satisfy constraints related to well-posedness and, on the other hand, exhibit desired properties related to expressivity, identifiability, and so on. The aim of this dissertation is to develop such representations for two classes of problems involving nonlinear hyperelastic materials and operators producing nonsmooth solution fields, respectively.We first consider the construction of rectified deep learning models for constitutive modeling in nonlinear elasticity. The method combines unconstrained neural networks with integral transformations to enforce convexity. This construction, together with the use of a mechanistic-informed parameterization (to satisfy objectivity, for instance), ensures the admissibility of the model. Applications to real datasets on isotropic and anisotropic soft tissues are presented to assess the relevance of the approach in terms of prediction accuracy and training cost.We then address the stochastic modeling and identification of anisotropic strain energy density functions on patient-specific geometries (more specifically, human arterial walls). An information-theoretic formulation is proposed, integrating constraints related to well-posedness (e.g., growth conditions) and linearization. The stochastic partial differential equation approach is used to define the latent Gaussian field on the complex domain, using a new parameterization of the diffusion field. The modeling capabilities are demonstrated by considering experimental results on artery layers. Finally, uncertainty propagation is performed to quantify the impact on the mechanical response (in a static regime).We next focus on concurrent multiscale approaches (without separation of scales) and develop a statistical surrogate to approximate the apparent constitutive model (mapping the right Cauchy-Green tensor to the homogenized second Piola-Kirchhoff stress tensor). We formulate the problem through conditional statistics, and use probabilistic learning on manifolds as generative model. The FE^2 method is employed to build datasets relevant to forward prediction and inverse problems. We show that the proposed methodology enables accurate estimations (in the sense of probability laws) for both scenarios, despite the high levels of nonlinearity and stochasticity exhibited by the system and the small amount of data used for training.Finally, we develop a nonlinear manifold reduced-order model based on convolutional neural network-based autoencoders. The encoder specifically involves iteration-dependent trainable kernels inspired by adaptive basis methods, with the aim of promoting interpretability and reducing the number of parameters to be trained. We investigate operator inference strategies between latent spaces, and propose a strategy to perform vectorized implicit time integration. We demonstrate that the proposed architecture and algorithms generally perform better than commonly employed strategies on a benchmark advection-dominated problem, in terms of both prediction accuracy and training speed.