Table 2.

Comparison of PINNs using different strategies for robustness to solve the 1D nonlinear Schrödinger 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). However, if the SGP does not have a sufficient number of IPs the error increases as seen when 10 IPs are used. Multiple domain cPINNs have worse performance. 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)	0.0105
PINN (σ = 0.1)	0.0289
PINN (σ = 0.1, L 1 regularization with $\lambda ={10}^{-4}$)	0.1613
PINN (σ = 0.1, L 2 regularization with $\lambda ={10}^{-4}$)	0.2681
cPINN-2 (no error)	0.2745
cPINN-2 (σ = 0.1, no smoothing)	0.4782
cPINN-3 (no error)	0.0258
cPINN-3 (σ = 0.1, no smoothing)	0.4178
GP-smoothed PINN (σ = 0.1, 50 IPs for u and v)	0.0125
SGP-smoothed PINN (σ = 0.1, 10 IPs for u and v)	0.0231
SGP-smoothed PINN (σ = 0.1, 29 and 20 IPs for u and v)	0.0123