ABSTRACT

In this talk, I will first introduce an Energetic Spectral-Element Time (ESET) marching scheme for solving phase-field equations, derived using an energy variational approach. We prove that the fully implicit ESET method possesses the super-convergence property. In practical computations, treating the nonlinear term with explicit extrapolation yields a computationally efficient semi-implicit ESET scheme. Using this semi-implicit scheme as a Picard iteration solver for the implicit ESET can restore the super-convergence property within a fixed number of iterations. Numerical results demonstrate that this approach outperform the IMEX4 and BDF4 schemes, which were considered most effective fourth-order methods for solving phase-field equations. In the second part of this talk, we extend this spectral element time-marching approach to machine learning methods for solving complex nonlinear parabolic partial differential equations. Combined with a novel adaptive data sampling approach and validation test, this extension results in an efficient deep learning method for solving complex dynamical systems.