  In this work, we assess the ability of physics-informed neural networks
(PINNs) to solve increasingly-complex coupled ordinary differential equations
(ODEs). We focus on a pair of benchmarks: discretized partial differential
equations and harmonic oscillators, each of which has a tunable parameter that
controls its complexity. Even by varying network architecture and applying a
state-of-the-art training method that accounts for "difficult" training
regions, we show that PINNs eventually fail to produce correct solutions to
these benchmarks as their complexity -- the number of equations and the size of
time domain -- increases. We identify several reasons why this may be the case,
including insufficient network capacity, poor conditioning of the ODEs, and
high local curvature, as measured by the Laplacian of the PINN loss.
