Figure 6
From: Investigating molecular transport in the human brain from MRI with physics-informed neural networks
Adaptive refinement yields PINN solutions that are consistent with a diffusion model. (a) Upper row: Output \(c(x,t=22\, \mathrm {h}, \theta _{\mathrm {opt}})\) of PINNs models trained with \(p=1\) and \(p=1\) & RAR and FEM solution for \((\alpha , \beta , \gamma ) = (10 ^{-6}, 0.1, 0.01)\). Lower row: Zoom into a sagittal slice of data at \(24\,\)h compared PINN and FEM solutions. The PINN prediction after training without RAR overfits the data. Compare also to Fig. 5. (b) Green: PINN estimates for the diffusion coefficient with RAR or RAE and different initial learning rates (\(p=1\) in all cases). Blue: \(\ell ^1\)-norm of the residual after training. It can be seen that lower learning rate leads to a lower residual norm and an estimate for the diffusion coefficient closer to the FEM approach.
