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

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.