  In this work, we propose a set of physics-informed geometric operators (GOs)
to enrich the geometric data provided for training surrogate/discriminative
models, dimension reduction, and generative models, typically employed for
performance prediction, dimension reduction, and creating data-driven
parameterisations, respectively. However, as both the input and output streams
of these models consist of low-level shape representations, they often fail to
capture shape characteristics essential for performance analyses. Therefore,
the proposed GOs exploit the differential and integral properties of
shapes--accessed through Fourier descriptors, curvature integrals, geometric
moments, and their invariants--to infuse high-level intrinsic geometric
information and physics into the feature vector used for training, even when
employing simple model architectures or low-level parametric descriptions. We
showed that for surrogate modelling, along with the inclusion of the notion of
physics, GOs enact regularisation to reduce over-fitting and enhance
generalisation to new, unseen designs. Furthermore, through extensive
experimentation, we demonstrate that for dimension reduction and generative
models, incorporating the proposed GOs enriches the training data with compact
global and local geometric features. This significantly enhances the quality of
the resulting latent space, thereby facilitating the generation of valid and
diverse designs. Lastly, we also show that GOs can enable learning parametric
sensitivities to a great extent. Consequently, these enhancements accelerate
the convergence rate of shape optimisers towards optimal solutions.
