  One of the requirements of the population-based approach to Structural Health
Monitoring (SHM) proposed in the earlier papers in this sequence, is that
structures be represented by points in an abstract space. Furthermore, these
spaces should be metric spaces in a loose sense; i.e. there should be some
measure of distance applicable to pairs of points; similar structures should
then be close in the metric. However, this geometrical construction is not
enough for the framing of problems in data-based SHM, as it leaves undefined
the notion of feature spaces. Interpreting the feature values on a
structure-by-structure basis as a type of field over the space of structures,
it seems sensible to borrow an idea from modern theoretical physics, and define
feature assignments as sections in a vector bundle over the structure space.
With this idea in place, one can interpret the effect of environmental and
operational variations as gauge degrees of freedom, as in modern gauge field
theories. This paper will discuss the various geometrical structures required
for an abstract theory of feature spaces in SHM, and will draw analogies with
how these structures have shown their power in modern physics. In the second
part of the paper, the problem of determining the normal condition cross
section of a feature bundle is addressed. The solution is provided by the
application of Graph Neural Networks (GNN), a versatile non-Euclidean machine
learning algorithm which is not restricted to inputs and outputs from vector
spaces. In particular, the algorithm is well suited to operating directly on
the sort of graph structures which are an important part of the proposed
framework for PBSHM. The solution of the normal section problem is demonstrated
for a heterogeneous population of truss structures for which the feature of
interest is the first natural frequency.
