  Singularly-perturbed ordinary differential equations often exhibit Stokes'
phenomenon, which describes the appearance and disappearance of oscillating
exponentially small terms across curves in the complex plane known as Stokes
curves. These curves originate at singular points in the leading-order solution
to the differential equation. In many important problems, it is impossible to
obtain a closed-form expression for these leading-order solutions, and it is
therefore challenging to locate these singular points. We present evidence that
the analytic leading-order solution of a linear differential equation can be
replaced with a rational approximation based on a numerical leading-order
solution using the adaptive Antoulas-Anderson (AAA) method. We show that the
subsequent exponential asymptotic analysis accurately predicts the
exponentially small behaviour present in the solution. We explore the
limitations of this approach, and show that for sufficiently small values of
the asymptotic parameter, this approach breaks down; however, the range of
validity may be extended by increasing the number of poles in the rational
approximation. We finish by presenting a related nonlinear problem and
discussing the challenges that arise when attempting to apply this method to
nonlinear problems.
