1. A brief introduction to supervised learning and its application to control of PDEs via PINNs,
2. DeepXDE: a Python library for scientific machine learning and physics-informed learning,
3. What does curse of dimensionality mean in Machine Learning?,
4. DeepONets: a promising tool to break the curse of dimensionality in control and parametric problems for PDEs, and
5. Some challenging open problems in the field.

In this presentation, I will survey several research directions aimed at addressing these challenges, with a particular emphasis on how machine learning techniques can be integrated into classical solution strategies. I will discuss the difficulties inherent to multi-query settings, outline emerging opportunities created by combining model-based and data-driven approaches, and highlight open questions that motivate further exploration.

In the context of above-mentioned problems, I will begin with a brief overview of the mathematical damage modeling. I will specifically focus on the models in which material damages can be described by a scalar damage field $ζ = ζ(t, x)$, which acts as an internal variable measuring the fractional decrease in the stress-strain response. In this case, the corresponding elasticity system may exhibit a degeneration. This means that, for certain damage field $ζ(t, x)$, this problem may lack the uniqueness of weak solutions due to the Lavrentieff phenomenon. So, one of the principal point at this stage is to study the solvability issues for the proposed modes and highlight some questions which seem to be open for nowadays.

The next point, I would like to touch on, is the formulation of possible and plausible statements for sustainable optimal control problems within the framework of `life-cycle optimization`. Because of the degeneration in the principle part of differential operators and the $L^1$-properties of the right-hand sides of PDEs involved in the proposed models, issues of solvability and the existence of approximate solutions of the corresponding optimal control problems remain under-researched. At the end of my presentation I would like to dwell on the development of machine learning approach to the study and practical implementation of the aforementioned OCPs. In this context, I am going to discuss some issues closely related to the deep neural network application to our business and highlight some key questions that need to be answered.