Multioutput FOSLS Deep Neural Network for Solving Allen–Cahn Equation

This paper utilizes feed-forward neural networks to approximate solutions and their scaled gradients of the Allen–Cahn equation. A first-order system least-squares (FOSLS) formulation is used to split the problem into a first-order system and then the converted minimization problem is approximated by using a deep learning approach. These equations are strongly nonlinear differential equations with a very small positive parameter \(\boldsymbol{\epsilon } \) (interfacial width), which creates steep interfaces as it approaches zero, making neural network difficult to approximate solution and its gradients accurately. To overcome this difficulty, instead of approximating the problem in the entire time interval, we divide it into a finite number of time segments and approximate both the solution and its gradients over each time segment. The solution found in the previous time segment is used as the initial condition for the current time segment. A novel transfer learning is incorporated by training the network into successive time segments to improve the accuracy and efficiency of the neural network. The present approach not only provides a direct approximation of the solution but its gradient as well. Numerical results for one, two, and three space dimensions are presented, and the energy law, an essential property of the Allen‒Cahn equations, is also verified.