The numerical simulation of strongly non-linear systems at large scale is one of the most persistent challenges in scientific computing. Recent developments in non-linear preconditioning, domain decomposition, and multi-scale methods, as well as in scientific machine learning, allow for the design of new and scalable numerical methods. This minisymposium provides a forum for the presentation and discussion of these new solution methods and their application to complex large scale problems from engineering, medicine, natural sciences, data sciences, or imaging. Our discussion will cover established approaches based on structured matrices, mathematical modeling, and numerical simulation, including variational methods and emerging approaches from the growing field of machine learning, as well as their combination.