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Mass-preserving Spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearity and singularity

Huang Qiumei, Ma Jiaxuan, Xu Zhen

TL;DR

The paper addresses the challenge of accurately solving Cahn-Hilliard equations with high-order derivatives, strong nonlinearity, and singular potentials while preserving mass. It introduces a mass-preserving spatio-temporal adaptive PINN that decomposes the time domain into subintervals, trains independent networks per interval, uses residual-based adaptive sampling in space, and imposes a soft mass constraint in the loss. The method transforms the fourth-order CH equation into a second-order system to reduce computational cost and adapts time-interval lengths based on the energy decrease rate, achieving improved accuracy and mass conservation across GL and FH potentials, including 3D and coupled CH systems. The numerical results demonstrate substantial improvements over baseline PINN with uniform time division, particularly in handling singularities and maintaining mass, suggesting practical applicability to complex phase-field models.

Abstract

As one kind important phase field equations, Cahn-Hilliard equations contain spatial high order derivatives, strong nonlinearities, and even singularities. When using the physics informed neural network (PINN) to simulate the long time evolution, it is necessary to decompose the time domain to capture the transition of solutions in different time. Moreover, the baseline PINN can't maintain the mass conservation property for the equations. We propose a mass-preserving spatio-temporal adaptive PINN. This method adaptively dividing the time domain according to the rate of energy decrease, and solves the Cahn-Hilliard equation in each time step using an independent neural network. To improve the prediction accuracy, spatial adaptive sampling is employed in the subdomain to select points with large residual value and add them to the training samples. Additionally, a mass constraint is added to the loss function to compensate the mass degradation problem of the PINN method in solving the Cahn-Hilliard equations. The mass-preserving spatio-temporal adaptive PINN is employed to solve a series of numerical examples. These include the Cahn-Hilliard equations with different bulk potentials, the three dimensional Cahn-Hilliard equation with singularities, and the set of Cahn-Hilliard equations. The numerical results demonstrate the effectiveness of the proposed algorithm.

Mass-preserving Spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearity and singularity

TL;DR

The paper addresses the challenge of accurately solving Cahn-Hilliard equations with high-order derivatives, strong nonlinearity, and singular potentials while preserving mass. It introduces a mass-preserving spatio-temporal adaptive PINN that decomposes the time domain into subintervals, trains independent networks per interval, uses residual-based adaptive sampling in space, and imposes a soft mass constraint in the loss. The method transforms the fourth-order CH equation into a second-order system to reduce computational cost and adapts time-interval lengths based on the energy decrease rate, achieving improved accuracy and mass conservation across GL and FH potentials, including 3D and coupled CH systems. The numerical results demonstrate substantial improvements over baseline PINN with uniform time division, particularly in handling singularities and maintaining mass, suggesting practical applicability to complex phase-field models.

Abstract

As one kind important phase field equations, Cahn-Hilliard equations contain spatial high order derivatives, strong nonlinearities, and even singularities. When using the physics informed neural network (PINN) to simulate the long time evolution, it is necessary to decompose the time domain to capture the transition of solutions in different time. Moreover, the baseline PINN can't maintain the mass conservation property for the equations. We propose a mass-preserving spatio-temporal adaptive PINN. This method adaptively dividing the time domain according to the rate of energy decrease, and solves the Cahn-Hilliard equation in each time step using an independent neural network. To improve the prediction accuracy, spatial adaptive sampling is employed in the subdomain to select points with large residual value and add them to the training samples. Additionally, a mass constraint is added to the loss function to compensate the mass degradation problem of the PINN method in solving the Cahn-Hilliard equations. The mass-preserving spatio-temporal adaptive PINN is employed to solve a series of numerical examples. These include the Cahn-Hilliard equations with different bulk potentials, the three dimensional Cahn-Hilliard equation with singularities, and the set of Cahn-Hilliard equations. The numerical results demonstrate the effectiveness of the proposed algorithm.
Paper Structure (14 sections, 35 equations, 21 figures, 2 tables, 3 algorithms)

This paper contains 14 sections, 35 equations, 21 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Review of physics informed neural networks
  4. Improved methods for PINN
  5. Adaptive Sampling in Space and Time
  6. Mass Constraint
  7. Mass-preserving spatio-temporal adaptive PINN
  8. Numerical results
  9. Cahn-Hilliard equation with Ginzburg-Landau potential in two dimensions(2D)
  10. Cahn-Hilliard equation with Flory-Huggins potential in two dimensions(2D)
  11. Simulations for the high-dimensional Cahn-Hilliard equation and the system of Cahn-Hilliard equations
  12. Cahn-Hilliard equation with Flory-Huggins potential in three dimensions(3D)
  13. System of Cahn-Hilliard equations in two dimensions
  14. Conclusion

Figures (21)

  • Figure 1: Fully connected neural network
  • Figure 2: Adaptive sampling in space
  • Figure 3: Adaptive sampling in time
  • Figure 4: Predicted solutions and realitive errors of 2D Cahn-Hilliard equation with Ginzburg-Landau potential given by spatio-temporal adaptive PINN with time step $\Delta t=0.2$.
  • Figure 5: The numerical energy(E) and mass(M) of the predicted solutions $\hat{u}_\theta(\bm{x},t)$
  • ...and 16 more figures