11th CISM–AIMETA Advanced Course on "Machine Learning for Solid Mechanics"

Recent progress in artificial intelligence techniques has delivered tools to solve complex and very high dimensional optimization problems that allow to discover correlations in large data sets, and hence to extract features and patterns from measurements or simulations of complex phenomena. Applications of these machine learning techniques to the field of solid mechanics range from the identification of constitutive equation from full-field time-dependent measurements, to machine learning techniques for the regularization of ill-posed inverse problems in imaging and control, and to the automatic discovery of reduced order models recapitulating the main features of observed patterns in complex multi-physical systems, thanks to the automatic discovery of hidden structures and symmetries in nonlinear dynamical systems. Building upon some introductory lectures on the mathematical foundations of machine learning and on the basic computational tools required, these recent and potentially revolutionary approaches to long standing problems in nonlinear solid mechanics will be discussed in the context of specific case studies from engineering applications. We plan to cover the fundamental mathematical background of machine learning techniques to explain and rationalize the reasons for their success, to illustrate the possibility of automated discovery of dimensionally-reduced hidden structures and symmetries in solid mechanics problems, and to illustrate the potential of machine learning techniques in the context of application to specific classes of material systems ranging from metals, to polymers, to granular materials. The school will be structured according to the following plan: Mathematical background to neural networks and neural operators. Lectures by Carola Schönlieb (Cambridge) and Nikola Kovachki (Nvidia/NYU). Discovery of internal variables and invariant manifolds in history-dependent phenomena. Lectures by Antonio DeSimone (SISSA/Pisa) and Kaushik Bhattacharya (Caltech). Learning from experimental data and connections to numerical methods. Lectures by Dirk Mohr (ETH) and Laurent Stainier (Nantes).