Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks

Abstract

The $L^2$ gradient flow of the Ginzburg-Landau free energy functional leads to the Allen Cahn equation that is widely used for modeling phase separation. Machine learning methods for solving the Allen-Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives through automatic differentiation, and the large system size required by the space-time approach. To overcome these limitations, we propose a separable neural network-based approximation of the phase field in a minimizing movement scheme to solve the aforementioned gradient flow problem. At each time step, the separable neural network is used to approximate the phase field in space through a low-rank tensor decomposition thereby accelerating the derivative calculations. The minimizing movement scheme naturally allows for the use of Gauss quadrature technique to compute the functional. A `$tanh$' transformation is applied on the neural network-predicted phase field to strictly bounds the solutions within the values of the two phases. For this transformation, a theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that bounding the solution through this transformation is the key to effectively model sharp interfaces through separable neural network. The proposed method outperforms the state-of-the-art machine learning methods for phase separation problems and is an order of magnitude faster than the finite element method.

References in corpus (8)

• A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks
• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications
• Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks
• Separable Physics-Informed Neural Networks
• An Interpretable Machine Learning Model for Deformation of Multi-Walled Carbon Nanotubes
• Deep Ritz Method with Adaptive Quadrature for Linear Elasticity
• Machine Learning for Partial Differential Equations