Table 3.

Comparison of PINNs using different strategies for robustness to solve the 1D Burgers’ equation. The introduction of error in the initial condition causes a significant increase in MSE for the standard PINN. GP-smoothing reduces the MSE to nearly as low as the PINN with no error. SGP-smoothing is also effective in reducing error and uses fewer inducing points (IPs). Results quoted for L ₁ and L ₂ regularizations are taken from the best performance observed over choices of $\lambda \in \{{10}^{-n}{\}}_{n=1}^5$.

Model	MSE
PINN (no error)	0.0116
PINN (σ = 0.5)	0.1982
PINN (σ = 0.1, L 1 regularization with $\lambda ={10}^{-4}$)	0.0392
PINN (σ = 0.1, L 2 regularization with $\lambda ={10}^{-4}$)	0.0293
PINN (σ = 0.5, Cole-Hopf regularizer)	0.1125
cPINN-2 (no error)	0.0161
cPINN-2 (σ = 0.5, no smoothing)	0.0834
cPINN-2 (σ = 0.5, Cole-Hopf regularizer)	0.0891
cPINN-3 (no error)	2.782$\times {10}^{-5}$
cPINN-3 (σ = 0.5, no smoothing)	0.0854
cPINN-3 (σ = 0.5, Cole-Hopf regularizer)	0.0329
UQ-PINN [20] ($\sigma =0.5)$	0.1248
GP-smoothed PINN (σ = 0.5, 50 IPs)	0.0384
SGP-smoothed PINN (σ = 0.5, 41 IPs)	0.0080