  Partial differential equations play a fundamental role in the mathematical
modelling of many processes and systems in physical, biological and other
sciences. To simulate such processes and systems, the solutions of PDEs often
need to be approximated numerically. The finite element method, for instance,
is a usual standard methodology to do so. The recent success of deep neural
networks at various approximation tasks has motivated their use in the
numerical solution of PDEs. These so-called physics-informed neural networks
and their variants have shown to be able to successfully approximate a large
range of partial differential equations. So far, physics-informed neural
networks and the finite element method have mainly been studied in isolation of
each other. In this work, we compare the methodologies in a systematic
computational study. Indeed, we employ both methods to numerically solve
various linear and nonlinear partial differential equations: Poisson in 1D, 2D,
and 3D, Allen-Cahn in 1D, semilinear Schr\"odinger in 1D and 2D. We then
compare computational costs and approximation accuracies. In terms of solution
time and accuracy, physics-informed neural networks have not been able to
outperform the finite element method in our study. In some experiments, they
were faster at evaluating the solved PDE.
