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Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks

Revanth Mattey, Susanta Ghosh

TL;DR

This work addresses the computational challenges of solving the $L^2$ gradient flow given by the Allen–Cahn equation for phase-field evolution. It introduces SDMM, which combines separable neural networks for spatial phase-field representation with a minimizing movement time-stepping scheme, and employs a $\tanh$ transformation to bound the solution within $[-1,1]$ while preserving unconditional energy stability. The method leverages Gauss quadrature for accurate energy evaluations and forward automatic differentiation for efficient derivatives, achieving order-of-magnitude speedups over finite element methods and outperforming SPINN in sharp-interface regimes. The results demonstrate high accuracy across star-shaped, coarsening, and random initial conditions and indicate broad potential for applying SDMM to other gradient-flow driven PDEs.

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.

Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks

TL;DR

This work addresses the computational challenges of solving the gradient flow given by the Allen–Cahn equation for phase-field evolution. It introduces SDMM, which combines separable neural networks for spatial phase-field representation with a minimizing movement time-stepping scheme, and employs a transformation to bound the solution within while preserving unconditional energy stability. The method leverages Gauss quadrature for accurate energy evaluations and forward automatic differentiation for efficient derivatives, achieving order-of-magnitude speedups over finite element methods and outperforming SPINN in sharp-interface regimes. The results demonstrate high accuracy across star-shaped, coarsening, and random initial conditions and indicate broad potential for applying SDMM to other gradient-flow driven PDEs.

Abstract

The 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 `' 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.
Paper Structure (29 sections, 1 theorem, 36 equations, 20 figures, 2 tables)

This paper contains 29 sections, 1 theorem, 36 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Gradient Flow for Phase Field Problems
  4. $L^2$ Gradient Flow
  5. Allen Cahn Equation
  6. Phase Field Modeling using Separable Neural Networks
  7. Separable Neural Networks
  8. Separable Physics Informed Neural Networks (SPINN)
  9. Challenges of SPINN for Sharp Interface Phase Field Evolution
  10. Proposed Method: Separable Deep Minimizing Movement (SDMM) Scheme
  11. Minimizing Movement Scheme
  12. Unconditional Stability of Minimizing Movement Scheme
  13. Proposed Method: Separable Neural Networks Based Minimizing Movement Scheme for Phase Field Problems
  14. Loss Function
  15. Numerical Integration
  16. ...and 14 more sections

Key Result

Theorem 1

Let a phase field, $\Tilde{\phi}$, is transformed to $\Tilde{\phi} \mapsto \tanh{\Tilde{\phi}} \,=\,\phi$, then the minimizing movement scheme for the Ginzburg-Landau functional on the transformed phase field $\phi$ is unconditionally stable, i.e., satisfies the equation (eq:UnconditionalStability).

Figures (20)

  • Figure 1: (Top, Middle) Space-time phase filed contour by the SPINN method and the reference solution. (Bottom) Solution at various time obtained by the SPINN method () and the reference solution () obtained via Chebfun driscoll2014chebfun
  • Figure 2: (a) Errors ($\mathcal{E}_{\text{SDMM}}$) in the SDMM method with respect to the reference solution, (b) Simulation (wall-clock) times for different mesh sizes. We found that the SDMM method provides erroneous solutions for mesh size coarser than $128^2$.
  • Figure 3: Bar graph comparing the computational wall time for the proposed SDMM method using GPU and the numerical method employing 24 CPU cores. Details of the CPU and GPU systems used for the simulations are given in \ref{['sec:comp_resources']}.
  • Figure 4: Cross section of the solution predicted at time, t = 0.02 secs with number of elements (a) $2048^2$ (b) $128^2$.
  • Figure 5: Solution evolution of the star-shaped interface at various time snapshots using the proposed SDMM approach.
  • ...and 15 more figures

Theorems & Definitions (1)

  • Theorem 1