  In this paper we present a systematic, data-driven approach to discovering
"bespoke" coarse variables based on manifold learning algorithms. We illustrate
this methodology with the classic Kuramoto phase oscillator model, and
demonstrate how our manifold learning technique can successfully identify a
coarse variable that is one-to-one with the established Kuramoto order
parameter. We then introduce an extension of our coarse-graining methodology
which enables us to learn evolution equations for the discovered coarse
variables via an artificial neural network architecture templated on numerical
time integrators (initial value solvers). This approach allows us to learn
accurate approximations of time derivatives of state variables from sparse flow
data, and hence discover useful approximate differential equation descriptions
of their dynamic behavior. We demonstrate this capability by learning ODEs that
agree with the known analytical expression for the Kuramoto order parameter
dynamics at the continuum limit. We then show how this approach can also be
used to learn the dynamics of coarse variables discovered through our manifold
learning methodology. In both of these examples, we compare the results of our
neural network based method to typical finite differences complemented with
geometric harmonics. Finally, we present a series of computational examples
illustrating how a variation of our manifold learning methodology can be used
to discover sets of "effective" parameters, reduced parameter combinations, for
multi-parameter models with complex coupling. We conclude with a discussion of
possible extensions of this approach, including the possibility of obtaining
data-driven effective partial differential equations for coarse-grained
neuronal network behavior.
