Gradient-enhanced PINN with residual unit for studying forward-inverse problems of variable coefficient equations

Physics-informed neural network (PINN) is a powerful emerging method for studying forward-inverse problems of partial differential equations (PDEs), even from limited sample data. Variable coefficient PDEs, which model real-world phenomena, are of considerable physical significance and research value. This study proposes a gradient-enhanced PINN with residual unit (R-gPINN) method to solve the data-driven solution and function discovery for variable coefficient PDEs. On the one hand, the proposed method incorporates residual units into the neural networks to mitigate gradient vanishing and network degradation, unify linear and nonlinear coefficient problem. We present two types of residual unit structures in this work to offer more flexible solutions in problem-solving. On the other hand, by including gradient terms of variable coefficients, the method penalizes collocation points that fail to satisfy physical properties. This enhancement improves the network's adherence to physical constraints and aligns the prediction function more closely with the objective function. Numerical experiments including solve the forward-inverse problems of variable coefficient Burgers equation, variable coefficient KdV equation, variable coefficient Sine–Gordon equation, and high-dimensional variable coefficient Kadomtsev–Petviashvili equation. The results show that using R-gPINN method can greatly improve the accuracy of predict solution and predict variable coefficient in solving variable coefficient equations.