Weak neural Galerkin method for nonlinear time-dependent partial differential equations

Abstract

Physics-Informed Neural Networks (PINNs) demonstrate considerable advantages in addressing high-dimensional partial differential equations (PDEs). However, traditional PINNs are constrained in their capacity to represent the temporal evolution of solutions. This study employs the neural Galerkin method in weak form to address nonlinear time-dependent PDEs, providing an accurate depiction of solution dynamics over time. While previous research on the neural Galerkin method is also grounded in the Dirac-Frenkel variational principle, our work distinguishes itself by emphasizing weak formulations, thereby contributing to the advancement of PDEs solution methods. The validity of the proposed method is confirmed through several numerical experiments.