Stability is crucial in applications that require deriving solutions from some input data. Classically, the notion of stability describes robustness under small perturbations of the input and regularization is employed to derive solutions from problems that lack this stable dependence.

In this talk, we visit different problems and methods that lack stability in some, probably less classical, sense. First, we discuss a phase retrieval problem which is not uniformly stable. This non-linear inverse problem cannot be tackled by classical regularization and we highlight possible connections between uniqueness and stability of this problem. Next, we take a look at robustness through the lens of adversarial attacks both for image classification and image reconstruction. While it is known that successful attacks can be designed for data-driven methods, we find that also classical regularization methods can be adversarially attacked.

The last part of the talk is devoted to operator learning and its stability with respect to discretizations. We propose a novel concept of neural operators that by-passes aliasing. These Representation equivalent Neural Operators (ReNOs) establish a unique and stable link between operators on infinite-dimensional spaces and their discrete realizations.