Physics-informed neural network applied to surface-tension-driven liquid film flows

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 diffusion of the thickness occurs.