A data-driven approach to solving the Allen–Cahn equation in varying dimensions using physics-informed neural networks (PINNs)

This study investigates a phase-field model governed by the Allen–Cahn equation, incorporating spatially varying forcing that significantly influences system dynamics resulting in pattern formation, controlled interface motion, and novel steady-state solutions. Physics-informed neural networks (PINNs) are employed to solve this system, and the results demonstrate that PINNs can generate highly accurate solutions with relatively few iterations, even under diverse initial conditions. The high accuracy of the PINN in simulating the nonlinear dynamics of complex systems was confirmed in comparison with typical numerical solutions. The findings show that initial conditions have a significant impact on the rate and type of phase evolution: higher initial amplitudes indicate multi-stage complex interface evolution, while lower amplitudes indicate the least amount of interface roughness. The dynamics of the Allen–Cahn equation are shown to minimize the interfacial energy over time, forcing the phase field toward equilibrium. The impact of the mobility and thickness of the interface on the phase evolution was also investigated. Rapidly changing initial circumstances provide an exception, momentarily increasing interfacial complexity, but higher mobility (L) speeds up interface migration, improving phase separation. The effect of the thickness of the interface changes with the starting profile. For smoother configurations, it provides uniform phase separation; however, when the initial profile has abrupt fluctuations, the effects are uneven in space. The scope of PINNs in kinetic-controlled applications such as alloy solidification and polymer phase separation is expanded by these findings, which demonstrate that PINNs are a very useful tool for phase-field modeling and can accurately and computationally simulate dynamic systems.