The complexity of physical processes of interest to contemporary science and engineering continues to grow. As a result, one needs to describe these processes with high-dimensional governing equations, which sometimes are not fully known. For the analysis, prediction, design and control of such processes, there is a broad need for reduced-order models capturing the core of the underlying physical phenomena from equations or data. However, most of these phenomena are characteristically nonlinear, and hence there is no generally applicable mathematical procedure to separate their core behavior from the rest.

This proposed summer course brings together experts in model reduction to introduce participants to the current state of the art in equation-driven and/or data-driven nonlinear model reduction. The emphasis will be on applications of practical relevance, as opposed to simple toy examples. Students will gain insights into at least three major reduced-order modelling approaches: advanced projection-based methods, invariant manifold methods and neural-network based methods. The lectures will touch on both the theoretical basis and the numerical implementation of these approaches. The lecturers include Professors Balakumar Balachandran, Charbel Farhat, Michael Graham, George Haller, Shobhit Jain and Gianluigi Rozza.

Some of the introductory lectures will cover the notion of various low-dimensional invariant manifolds (such as inertial manifolds, attractors and spectral submanifolds) that provide a mathematical foundation for model reduction in high-dimensional nonlinear dynamical systems. A separate tutorial will be offered to the course participants on the use of available open-source codes that can be used to compute reduced-order models from equations (SSMTool) and data (SSMLearn) for various problems in solid mechanics, fluid dynamics, and controls. Several lectures will cover contemporary, neural-network-based modelling techniques for applications in fluids, solids, and controls. As opposed to providing black-box-type approaches, this course will feature systematic uses of neural networks in the discovery and interpolation of low-dimensional invariant sets, such as inertial manifolds or chaotic attractors.
Further lectures will be devoted to the use of neural networks in model-based prediction of critically important societal phenomena, such as the spread of infections in a pandemic or the emergence of highly destructive rogue waves in the ocean. Physics-informed neural networks and their applications to the reduced-order numerical solutions of continuum motion modeled by PDEs will be discussed. Additionally, certified reduced basis methods will be introduced to the students with applications to transport and continuum mechanics. Recent reduction methods aimed at very large systems, such as hyperreduction and moment-matching methods, will also be surveyed.