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

Mustafa Kütük, Hamdullah Yücel

TL;DR

The paper develops an energy-dissipation preserving PINN framework for the Allen-Cahn equation with both polynomial and logarithmic energies, including constant and degenerate mobility, and extends to advection and random initial conditions. The key innovation is a weakly enforced energy-dissipation penalty, combined with a Fourier-series analogue for random initial data and adaptive collocation and temporal strategies to capture sharp interfaces efficiently. Across 1D–3D benchmarks, the proposed method achieves consistent energy decay, bounded solutions, and faithful phase separation, surpassing several standard PINN variants. This approach provides a mesh-free, energetically faithful solver for diffuse-interface problems with broad applicability in materials science and related fields.

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.

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

TL;DR

The paper develops an energy-dissipation preserving PINN framework for the Allen-Cahn equation with both polynomial and logarithmic energies, including constant and degenerate mobility, and extends to advection and random initial conditions. The key innovation is a weakly enforced energy-dissipation penalty, combined with a Fourier-series analogue for random initial data and adaptive collocation and temporal strategies to capture sharp interfaces efficiently. Across 1D–3D benchmarks, the proposed method achieves consistent energy decay, bounded solutions, and faithful phase separation, surpassing several standard PINN variants. This approach provides a mesh-free, energetically faithful solver for diffuse-interface problems with broad applicability in materials science and related fields.

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.

Paper Structure

This paper contains 15 sections, 30 equations, 18 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Physics-informed neural network
  3. Penalization of the energy dissipation
  4. Adaptive approaches
  5. Adaptive sampling of collocation points
  6. Adaptivity in the temporal domain
  7. Numerical experiments
  8. 1D Allen--Cahn with polynomial potential
  9. 2D Allen--Cahn with polynomial potential
  10. 3D Allen--Cahn with polynomial potential
  11. 2D Allen--Cahn with logarithmic potential
  12. 2D Allen--Cahn with logarithmic potential and random initials
  13. 2D Allen--Cahn with logarithmic potential and degenerate mobility
  14. 2D Allen--Cahn with advection and polynomial potential
  15. Conclusion

Figures (18)

  • Figure 1: Workflow of the proposed PINN framework.
  • Figure 2: Example \ref{['ex:1D_double']}: Predicted $\widehat{u}(x,t)$ (red) and reference $u_r(x,t)$ (blue) solutions at various time slots.
  • Figure 3: Example \ref{['ex:1D_double']}: Behaviour of absolute error $|\widehat{u} - u_r |$ at various time slots.
  • Figure 4: Example \ref{['ex:1D_double']}: Time evolution of the predicted energy functional $\mathcal{E}(\widehat{u})$.
  • Figure 5: Example \ref{['ex:2D_double']}: Predicted solutions $\widehat{u}(x,t)$ obtained by the proposed methodology at various time slots.
  • ...and 13 more figures