  The numerical modeling of thin shell structures is a challenge, which has
been met by a variety of finite element (FE) and other formulations -- many of
which give rise to new challenges, from complex implementations to artificial
locking. As a potential alternative, we use machine learning and present a
Physics-Informed Neural Network (PINN) to predict the small-strain response of
arbitrarily curved shells. To this end, the shell midsurface is described by a
chart, from which the mechanical fields are derived in a curvilinear coordinate
frame by adopting Naghdi's shell theory. Unlike in typical PINN applications,
the corresponding strong or weak form must therefore be solved in a
non-Euclidean domain. We investigate the performance of the proposed PINN in
three distinct scenarios, including the well-known Scordelis-Lo roof setting
widely used to test FE shell elements against locking. Results show that the
PINN can accurately identify the solution field in all three benchmarks if the
equations are presented in their weak form, while it may fail to do so when
using the strong form. In the thin-thickness limit, where classical methods are
susceptible to locking, training time notably increases as the differences in
scaling of the membrane, shear, and bending energies lead to adverse numerical
stiffness in the gradient flow dynamics. Nevertheless, the PINN can accurately
match the ground truth and performs well in the Scordelis-Lo roof benchmark,
highlighting its potential for a drastically simplified alternative to
designing locking-free shell FE formulations.
