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PFWNN: A deep learning method for solving forward and inverse problems of phase-field models

Gang Bao, Chang Ma, Yuxuan Gong

TL;DR

The paper tackles the numerical difficulty of forward and inverse problems for phase-field models, notably Allen–Cahn and Cahn–Hilliard equations, by introducing Phase-Field Weak-form Neural Networks (PFWNN). PFWNN parameterizes weak solutions with a periodic layer to enforce boundary conditions and employs locally supported test functions (CSRBFs) for efficient, region-focused learning, along with a time-causal residual training strategy. For inverse problems, a secondary network models the energy derivative $f(\phi)$, enabling reconstruction of the energy functional from spatio-temporal data, with optional auxiliary networks (e.g., $\mu^{NN}$) for CH. Across 1D and 2D AC and CH benchmarks, PFWNN delivers high-accuracy forward solutions and reliable inverse recovery of $f(\phi)$, often outperforming Ada-PINN, and demonstrates the potential to efficiently solve high-order, time-dependent PDEs with sharp interfaces.

Abstract

Phase-field models have been widely used to investigate the phase transformation phenomena. However, it is difficult to solve the problems numerically due to their strong nonlinearities and higher-order terms. This work is devoted to solving forward and inverse problems of the phase-field models by a novel deep learning framework named Phase-Field Weak-form Neural Networks (PFWNN), which is based on the weak forms of the phase-field equations. In this framework, the weak solutions are parameterized as deep neural networks with a periodic layer, while the test function space is constructed by functions compactly supported in small regions. The PFWNN can efficiently solve the phase-field equations characterizing the sharp transitions and identify the important parameters by employing the weak forms. It also allows local training in small regions, which significantly reduce the computational cost. Moreover, it can guarantee the residual descending along the time marching direction, enhancing the convergence of the method. Numerical examples are presented for several benchmark problems. The results validate the efficiency and accuracy of the PFWNN. This work also sheds light on solving the forward and inverse problems of general high-order time-dependent partial differential equations.

PFWNN: A deep learning method for solving forward and inverse problems of phase-field models

TL;DR

The paper tackles the numerical difficulty of forward and inverse problems for phase-field models, notably Allen–Cahn and Cahn–Hilliard equations, by introducing Phase-Field Weak-form Neural Networks (PFWNN). PFWNN parameterizes weak solutions with a periodic layer to enforce boundary conditions and employs locally supported test functions (CSRBFs) for efficient, region-focused learning, along with a time-causal residual training strategy. For inverse problems, a secondary network models the energy derivative , enabling reconstruction of the energy functional from spatio-temporal data, with optional auxiliary networks (e.g., ) for CH. Across 1D and 2D AC and CH benchmarks, PFWNN delivers high-accuracy forward solutions and reliable inverse recovery of , often outperforming Ada-PINN, and demonstrates the potential to efficiently solve high-order, time-dependent PDEs with sharp interfaces.

Abstract

Phase-field models have been widely used to investigate the phase transformation phenomena. However, it is difficult to solve the problems numerically due to their strong nonlinearities and higher-order terms. This work is devoted to solving forward and inverse problems of the phase-field models by a novel deep learning framework named Phase-Field Weak-form Neural Networks (PFWNN), which is based on the weak forms of the phase-field equations. In this framework, the weak solutions are parameterized as deep neural networks with a periodic layer, while the test function space is constructed by functions compactly supported in small regions. The PFWNN can efficiently solve the phase-field equations characterizing the sharp transitions and identify the important parameters by employing the weak forms. It also allows local training in small regions, which significantly reduce the computational cost. Moreover, it can guarantee the residual descending along the time marching direction, enhancing the convergence of the method. Numerical examples are presented for several benchmark problems. The results validate the efficiency and accuracy of the PFWNN. This work also sheds light on solving the forward and inverse problems of general high-order time-dependent partial differential equations.
Paper Structure (16 sections, 1 theorem, 34 equations, 14 figures, 1 algorithm)

This paper contains 16 sections, 1 theorem, 34 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Weak Neural Networks for phase-field Models
  3. The Framework of the PFWNN
  4. Neural Networks Representing Weak Solutions
  5. Calculation of the Loss Functions
  6. Inverse Problems
  7. Numerical Experiments
  8. Numerical Results for the Allen-Cahn Equations
  9. The 1D case
  10. The 2D case
  11. Numerical Results for the Cahn-Hilliard Equations
  12. The 1D case
  13. The 2D case
  14. The Measurement Data for the Inverse Problems
  15. Conclusions
  16. ...and 1 more sections

Key Result

Lemma 2.1

($C^{\infty}$ periodic conditions dong2021method) Let $p(x)$ be a given smooth periodic function with period $L$ on the real axis, i.e. $p(x +L) = p(x)$ for all $x \in (-\infty, \infty)$, and $f(x)$ denote an arbitrary smooth function. Define $\phi(x) =f(p) =f(p(x))$. Then $\phi^{(i)} (x) = \phi^{(i

Figures (14)

  • Figure 1: The sketch of the PFWNN. The dotted box should be added when solving the inverse problems.
  • Figure 2: The plots of the test functions in 1D (left) and 2D (right).
  • Figure 3: Results for 1D Allen-Cahn system. Spatio-temporal solutions: (a) The reference solution $\phi$, (b) The predicted solution $\phi^{NN}$, (c) The point-wise error of the PFWNN. (d) The point-wise error of Ada-PINN. (e) Loss of the PFWNN and Ada-PINN vs. computation times for the last sub-interval. (f) Relative errors of solution $\phi^{NN}$ for the PFWNN and Ada-PINN vs. computation times for the last sub-interval. (g) The three plots are the predicted solutions of the PFWNN and Ada-PINN vs. reference solutions at different timestamps.
  • Figure 4: Results for 1D Allen-Cahn system. Spatio-temporal solutions: (a) The reference solution $\phi$, (b) The predicted solution $\phi^{NN}$, (c) The point-wise error of the PFWNN. (d) The three plots are the predicted solutions vs. reference solutions at different timestamps.
  • Figure 5: Results for the inverse problem of 1D Allen-Cahn system. (a) Spatio-temporal energy functional: Top: The reference energy functional $f(\phi)$, Middle: The predicted solution $f^{NN}(\phi^{NN})$, Bottom: The point-wise error. (b) The predicted energy functional $f^{NN}$ vs. reference solutions $f$. (c) Relative errors for solution $f^{NN}$ vs. computation times.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4