Figure 13.

The SGP-smoothed PINN-evaluated $|h(x,t)|$ at different timeslices, $t=0,0.39,0.79,1.37$, from left to right. In this case the training data on the initial timeslice is subject to measurement errors, modeled by a Gaussian random variable with zero mean and a standard deviation of 0.1. SGP based smoothing was used on initial timeslice before training the PINN using 10 inducing points for each $u(x,0)$ and $v(x,0)$ for the top row and 29 inducing points for $u(x,0)$ and 20 for $v(x,0)$ for the bottom row. The points marked with the blue cross (x) pointer in the leftmost set of plots indicate the samples on the initial timeslice used to train the PINN. The grey band in the subsequent plots represents the uncertainty associated with the PINN-evaluated approximation of $|h(x,t)|$.