VC-PINN: Variable Coefficient Physical Information Neural Network For Forward And Inverse PDE Problems with Variable Coefficient

TL;DR

VC-PINN is a novel neural network approach that effectively solves forward and inverse variable coefficient PDEs, handling high-dimensional problems and complex coefficients with improved stability and unified framework.

Contribution

The paper introduces VC-PINN, integrating ResNet structures to address vanishing gradients and unify linear and nonlinear variable coefficients in PDEs, advancing neural PDE solving techniques.

Findings

Successfully applied to multiple variable coefficient PDEs.

Handles high-dimensional problems with complex coefficients.

Demonstrates robustness against noise and unifies forward/inverse problems.

Abstract

The paper proposes a deep learning method specifically dealing with the forward and inverse problem of variable coefficient partial differential equations -- Variable Coefficient Physical Information Neural Network (VC-PINN). The shortcut connections (ResNet structure) introduced into the network alleviates the "Vanishing gradient" and unifies the 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.