  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.
