PFWNN: A deep learning method for solving forward and inverse problems of phase-field models

Abstract: Phase-field models have been widely used to investigate the phase transformation phenomena. However, it is difficult to solve the problems numerically due to their strong nonlinearities and higher-order terms. This work is devoted to solving forward and inverse problems of the phase-field models by a novel deep learning framework named Phase-Field Weak-form Neural Networks (PFWNN), which is based on the weak forms of the phase-field equations. In this framework, the weak solutions are parameterized as deep neural networks with a periodic layer, while the test function space is constructed by functions compactly supported in small regions. The PFWNN can efficiently solve the phase-field equations characterizing the sharp transitions and identify the important parameters by employing the weak forms. It also allows local training in small regions, which significantly reduce the computational cost. Moreover, it can guarantee the residual descending along the time marching direction, enhancing the convergence of the method. Numerical examples are presented for several benchmark problems. The results validate the efficiency and accuracy of the PFWNN. This work also sheds light on solving the forward and inverse problems of general high-order time-dependent partial differential equations.