Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equations

TL;DR

This paper develops an energy dissipation preserving physics-informed neural network (PINN) for solving the Allen-Cahn equation, incorporating energy properties into the training process, and demonstrates its effectiveness across various conditions and dimensions.

Contribution

The paper introduces a novel PINN framework that enforces energy dissipation, improving accuracy and stability in solving Allen-Cahn equations with complex features.

Findings

Consistent decrease in discrete energy observed in simulations

Effective handling of random initial conditions via Fourier series

Captures phenomena like phase separation and metastability

Abstract

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.