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

Gradient-enhanced physics-informed neural networks based on transfer  learning for inverse problems of the variable coefficient differential  equations