Data-driven discovery of phase field dynamics: a physics-informed neural network-based approach to the Allen–Cahn equation

Abstract

This paper aims to analyze the adoption of physics-informed neural networks (PINNs) in solving the Allen–Cahn equation, which represents a fundamental model in phase field dynamics that captures processes such as phase segregation and interface dynamics in materials. To solve the Allen–Cahn equating system under three different initial conditions, PINNs have been employed and shown to achieve very reflective solutions with few numbers of iterations. A comparison with standard numerical solutions verifies the good accuracy of PINN in modeling the nonlinear dynamics of complicated systems. The results indicate that initial conditions play an important role in the rate and nature of phase evolution: lower amplitude initial perturbations reach equilibrium configurations more quickly with minimum interface roughness, whereas higher initial amplitudes represent multi-stage complex interface evolution. It is evident that the dynamics of the Allen–Cahn equation force the phase field toward equilibrium by minimizing the interfacial energy in time. This study further examines the influence of the mobility (L) and interface (ϵ) thickness on phase evolution. Higher mobility accelerates interface migration, thereby enhancing phase separation, although rapidly changing initial conditions present an exception, temporarily increasing interfacial complexity. Similarly, the impact of the interface thickness varies with the initial profile, offering uniform phase separation for smoother configurations, but exhibiting spatially uneven effects when the initial profile contains abrupt variations. These findings highlight PINNs as a highly effective tool for phase field modeling, capable of simulating dynamic systems with accuracy and computational efficiency, thus extending the scope of PINNs in kinetic-controlled applications such as alloy solidification and polymer phase separation.