  We present an approach to reduced-order modelling that builds off recent
graph-theoretic work for representation, exploration, and analysis of computed
states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351,
501-530, 2019). We extend a non-local calculus on finite weighted graphs to
build such models by exploiting polynomial expansions and Taylor series. In the
general framework for non-local calculus on graphs, the graph edge weights are
intricately linked to the embedding of the graph, and consequently to the
definition of the derivatives. In a previous communication (Duschenes and
Garikipati, arXiv:2105.01740), we have shown that radially symmetric,
continuous edge weights derived from, for example Gaussian functions, yield
inconsistent results in the resulting non-local derivatives when compared
against the corresponding local, differential derivative definitions. Taking
inspiration from finite difference methods, we algorithmically compute edge
weights, considering the embedding of the local neighborhood of each graph
vertex. Given this procedure, we ensure the consistency of the non-local
derivatives in this setting, a crucial requirement for numerical applications.
We show that we can achieve any desired orders of accuracy of derivatives, in a
chosen number of dimensions without symmetry assumptions in the underlying
data. Finally, we present two example applications of extracting reduced-order
models using this non-local calculus, in the form of ordinary differential
equations from parabolic partial differential equations of progressively
greater complexity.
