  In this paper, we introduce a shallow (one-hidden-layer) physics-informed
neural network for solving partial differential equations on static and
evolving surfaces. For the static surface case, with the aid of level set
function, the surface normal and mean curvature used in the surface
differential expressions can be computed easily. So instead of imposing the
normal extension constraints used in literature, we write the surface
differential operators in the form of traditional Cartesian differential
operators and use them in the loss function directly. We perform a series of
performance study for the present methodology by solving Laplace-Beltrami
equation and surface diffusion equation on complex static surfaces. With just a
moderate number of neurons used in the hidden layer, we are able to attain
satisfactory prediction results. Then we extend the present methodology to
solve the advection-diffusion equation on an evolving surface with given
velocity. To track the surface, we additionally introduce a prescribed hidden
layer to enforce the topological structure of the surface and use the network
to learn the homeomorphism between the surface and the prescribed topology. The
proposed network structure is designed to track the surface and solve the
equation simultaneously. Again, the numerical results show comparable accuracy
as the static cases. As an application, we simulate the surfactant transport on
the droplet surface under shear flow and obtain some physically plausible
results.
