VC-PINN: Variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient

The paper proposes a deep learning method specifically dealing with the forward and inverse problem of variable coefficient partial differential equations – variable coefficient physics-informed neural network (VC-PINN). The shortcut connections (ResNet structure) introduced into the network alleviate the “vanishing gradient” problem and unify linear and nonlinear coefficients. The developed method was applied to four equations including the variable coefficient Sine–Gordon (vSG), the generalized variable coefficient Kadomtsev–Petviashvili equation (gvKP), the variable coefficient Korteweg–de Vries equation (vKdV), the variable coefficient Sawada–Kotera equation (vSK). Numerical results show that VC-PINN is successful in the case of high dimensionality, various variable coefficients (polynomials, trigonometric functions, fractions, oscillation attenuation coefficients), and the coexistence of multiple variable coefficients. We also conducted an in-depth analysis of VC-PINN in a combination of theory and numerical experiments, including four aspects: the necessity of ResNet; the relationship between the convexity of variable coefficients and learning; anti-noise analysis; the unity of forward and inverse problems/relationship with standard PINN.