Tutorials: Physics-informed machine learning methods of computing 1D phase-field models

Phase-field models are widely used to describe phase transitions and interface evolution in various scientific disciplines. In this Tutorial, we present two neural network methods for solving them. The first method is based on physics-informed neural networks (PINNs), which enforce the governing equations and boundary/initial conditions in the loss function. The second method is based on deep operator neural networks (DeepONets), which treat the neural network as an operator that maps the current state of the field variable to the next state. Both methods are demonstrated with the Allen–Cahn equation in one dimension, and the results are compared with the ground truth. This Tutorial also discusses the advantages and limitations of each method, as well as the potential extensions and improvements.