  A physics-informed neural network (PINN), which has been recently proposed by
Raissi et al [J. Comp. Phys. 378, pp. 686-707 (2019)], is applied to the
partial differential equation (PDE) of liquid film flows. The PDE considered is
the time evolution of the thickness distribution $h(x,t)$ owing to the Laplace
pressure, which involves 4th-order spatial derivative and 4th-order nonlinear
term. Even for such a PDE, it is confirmed that the PINN can predict the
solutions with sufficient accuracy. Nevertheless, some improvements are needed
in training convergence and accuracy of the solutions. The precision of
floating-point numbers is a critical issue for the present PDE. When the
calculation is executed with a single precision floating-point number, the
optimization is terminated due to the loss of significant digits. Calculation
of the automatic differentiation (AD) dominates the computational time required
for training, and becomes exponentially longer with increasing order of
derivatives. By splitting the original 4th-order one-variable PDE into
2nd-order two-variable PDEs, the computational time for each training iteration
is greatly reduced. The sampling density of training data also significantly
affects training convergence. For the problem considered in this study, mproved
convergence was obtained by allowing the sampling density of training data to
be greater in earlier time ranges, where the rapid flattening of the thickness
occurs.
