Adaptive Deep Learning Approximation for Allen-Cahn Equation.

Solving general non-linear partial differential equations (PDE) precisely and efficiently has been a long-lasting challenge in the field of scientific computing. Based on the deep learning framework for solving non-linear PDEs physics-informed neural networks (PINN), we introduce an adaptive collocation strategy into the PINN method to improve the effectiveness and robustness of this algorithm when selecting the initial data to be trained. Instead of merely training the neural network once, multi-step discrete time models are considered when predicting the long time behaviour of solutions of the Allen-Cahn equation. Numerical results concerning solutions of the Allen-Cahn equation are presented, which demonstrate that this approach can improve the robustness of original neural networks approximation.