Physics-informed neural networks incorporating energy dissipation for the phase-field model of ferroelectric microstructure evolution

Abstract

Physics-informed neural networks (PINNs) are an emerging technique to solve partial differential equations (PDEs). In this work, we propose a simple but effective PINN approach for the phase-field model of ferroelectric microstructure evolution. This model is a time-dependent, nonlinear, and high-order PDE system of multi-physics, challenging to be solved using a baseline PINN. Considering that the acquisition of steady microstructures is one of the primary focuses in simulations of ferroelectric microstructure evolution, we simplify the time-dependent PDE system to be a static problem. This static problem, however, is ill-posed. To overcome this issue, a term originated from the law of energy dissipation is embedded into the loss function as an extra constraint for the PINN. With this modification, the PINN successfully predicts the steady ferroelectric microstructure without tracking the evolution process. In addition, although the proposed PINN approach cannot tackle the dynamic problem in a straightforward fashion, it is of benefit to the PINN prediction of the evolution process by providing labeled data. These data are crucial because they help the PINN avoid the propagation failure, a common failure mode of PINNs when predicting dynamic behaviors. The above mentioned advantages of the proposed PINN approach are demonstrated through a number of examples.