  We propose gradient-enhanced PINNs based on transfer learning (TL-gPINNs) for
inverse problems of the function coefficient discovery in order to overcome
deficiency of the discrete characterization of the PDE loss in neural networks
and improve accuracy of function feature description, which offers a new angle
of view for gPINNs. The TL-gPINN algorithm is applied to infer the unknown
variable coefficients of various forms (the polynomial, trigonometric function,
hyperbolic function and fractional polynomial) and multiple variable
coefficients simultaneously with abundant soliton solutions for the well-known
variable coefficient nonlinear Schr\"{o}odinger equation. Compared with the
PINN and gPINN, TL-gPINN yields considerable improvement in accuracy. Moreover,
our method leverages the advantage of the transfer learning technique, which
can help to mitigate the problem of inefficiency caused by extra loss terms of
the gradient. Numerical results fully demonstrate the effectiveness of the
TL-gPINN method in significant accuracy enhancement, and it also outperforms
gPINN in efficiency even when the training data was corrupted with different
levels of noise or hyper-parameters of neural networks are arbitrarily changed.
