Digital twins of additive manufacturing processes aim to provide process engineers with the input they need to control the quality of the build process, and to support certification of the produced part. This often involves the collection and analysis of huge amounts of data (data-driven modelling), and sometimes also fine-grained simulations (physics-based modelling). Both of these modelling approaches are typically resource intensive, both in terms of memory usage and compute power. Nevertheless, since the process is of limited duration, such resource requirements can often be met without difficulty. However, it is also of interest to create a digital twin of the manufactured part itself that can be used in downstream applications, and to document the status of the part throughout its lifetime. Such digital twins are intended to live together with the part and to be exchangeable between systems. In that context, it is desirable that the models are represented as compactly as possible. In this talk we will discuss the transition from process digital twins to product digital twins, including what types of information should be brought forward from the build process. We will have a particular focus on mathematical methods for compactly representing the geometry of the as-built model, including implicit representations based on splines, and neural representations.