## 4 Discussion  
In the case where the PDE was driven by initial and boundary conditions without a source term, it can be seen that PINNs accurately account for both insulated and convective boundary condition types. For the convective boundary condition case, both the hybrid and all-Tanh functions performed near-equally well, with the slight advantage to the hybrid. PINN models incorporating a convective boundary condition underperformed in cases with low boundary sampling but became more accurate as the sampling in that region was increased. This is seen as evidence that more complex functions require more randomly sampled data to better represent the function at the boundaries to the PINN. An alternative could be to sample the boundary values at varying time steps to better represent the function as a whole, rather than randomly sampling values from a tabulated function. While both the hybrid and all-Tanh combinations performed well, the hybrid approach performed noticeably better in both insulated and convective boundary condition cases. The three-layer skin experiment yielded the most accurate results from the models, where the all-Tanh model, utilizing the S&T method, outperformed the others. This is especially evident when comparing true vs. predicted thermal profiles. In Figure 9, the all-Tanh model closely fit the shape and magnitude of the curves around the dermis layer (layer 2) where other models had difficulty predicting such as the hybrid combination also found in Figure 9.  
While the problem of the three-layer skin experiment with a source term should be more difficult, the models for this experiment performed exceptionally better than those of the non-source. It is suspected that, because of the smarter sampling of the collocation points, being more concentrated in the first two layers of the domain which had the most drastic and varied changes in temperature, it was able to predict more accurately overall. In the non-source experiments, there was no such smart sampling and the entire domain was sampled more evenly across the domain for the collocation points. This is evidence in support of smart sampling of collocation points, where areas of interest receive more concentrated sampling to yield better results on a per-application basis. Run failure was low for all experiments, being roughly 5% for the non-source experiments, and 0% for all but one case in the source experiment. The hybrid model for the source experiment saw about 33% failure rate and it is suspected that the dominant use of Tanh in the architecture still affected it despite the use of the ELU activation function before the output layer. The evidence for this lies in the 0% failure rate of both the all-Tanh model using the S&T method and the all-ELU model. From this observation, it appears that the S&T method was superior to the use of ELU as the last activation function. Convergence rates for the non-source experiments were similar for the all-Tanh and hybrid models, but substantially different for the all-ELU model. With the optimizer used in these experiments, the all-ELU model did not converge efficiently enough to prevent early stopping. Similarly, with the source experiments, the all-ELU model had earlier training stoppage than the all-Tanh and hybrid models. The difference for the source experiment, though, is that there was a slightly more exaggerated difference between the average number of iterations between the all-Tanh and the hybrid, showing that the all-Tanh trained most efficiently, for more iterations, than the other two models. Hyperparameter values \(a\)1 and \(a\)2 were not optimized for any of the experiments, and were arbitrarily adjusted until coherent results were consistently observed. It is not known whether more refined adjustments in these parameters will yield non-negligible improvements in predictions, though current performances in the experiments are already considered sufficient.