Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equations
This addresses numerical solutions for phase-field models in materials science, but it is incremental as it builds on existing PINN methods with specific enhancements.
The paper tackles solving Allen-Cahn equations with various complexities using physics-informed neural networks (PINNs), incorporating energy dissipation as a penalty term and adaptive methods, resulting in consistent energy decrease and phenomena like phase separation.
This paper investigates a numerical solution of Allen-Cahn equation with constant and degenerate mobility, with polynomial and logarithmic energy functionals, with deterministic and random initial functions, and with advective term in one, two, and three spatial dimensions, based on the physics-informed neural network (PINN). To improve the learning capacity of the PINN, we incorporate the energy dissipation property of the Allen-Cahn equation as a penalty term into the loss function of the network. To facilitate the learning process of random initials, we employ a continuous analogue of the initial random condition by utilizing the Fourier series expansion. Adaptive methods from traditional numerical analysis are also integrated to enhance the effectiveness of the proposed PINN. Numerical results indicate a consistent decrease in the discrete energy, while also revealing phenomena such as phase separation and metastability.