Article citationsMore>>

Raissi, M., Perdikaris, P. and Karniadakis, G.E. (2019) Physics-informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics, 378, 686-707. https://doi.org/10.1016/j.jcp.2018.10.045

has been cited by the following article:

• TITLE: Crank-Nicolson Finite Difference Scheme for the 3D Allen-Cahn Equation with PINN Comparison

  AUTHORS: Bo Liu, Lijuan Wang

  KEYWORDS: Allen-Cahn Equation, Crank-Nicolson Scheme, Maximum Principle, Physics-Informed Neural Network (PINN)

  JOURNAL NAME: Applied Mathematics, Vol.16 No.10, October 31, 2025

  ABSTRACT: In this paper, we derive a second-order Crank-Nicolson finite-difference scheme for the three-dimensional Allen-Cahn equation and present an estimate of the scheme’s truncation error. We prove the existence of the numerical solution and establish stepwise uniqueness, from which the global uniqueness follows given the initial value u 0. We also establish conditional convergence in the L∞ -norm, and demonstrate a discrete maximum principle for the scheme. Numerical Examples 1 and 2 validate the discrete maximum principle, while Numerical Example 3 employs a physics-informed neural network (PINN), using the Crank-Nicolson solution as the reference, to compute the errors between the two approaches in the L2 and L∞ norms. Owing to the high dimensionality of the problem, visualization is performed via isosurface plots on identical time slices; these plots show strong agreement between the Crank-Nicolson and PINN solutions. The numerical results collectively confirm the accuracy and effectiveness of the proposed scheme.