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.

翻译：相场模型已被广泛应用于研究相变现象。然而，由于其强非线性和高阶项的存在，数值求解这些问题十分困难。本研究致力于通过一种新颖的深度学习框架——相场弱形式神经网络（PFWNN）——来求解相场模型的正问题和反问题，该框架基于相场方程的弱形式。在此框架中，弱解被参数化为包含周期性层的深度神经网络，而测试函数空间则由紧支于小区域内的函数构成。PFWNN能够利用弱形式高效求解表征急剧相变的相场方程，并识别重要参数。它还允许在小区域内进行局部训练，从而显著降低计算成本。此外，该方法能保证残差沿时间推进方向递减，从而增强了算法的收敛性。文中给出了多个基准问题的数值算例。结果验证了PFWNN的高效性与准确性。本工作也为求解一般高阶时间依赖偏微分方程的正反问题提供了启示。