Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

Abstract

In this paper, we develop cell-average based neural network (CANN) method to approximate solutions of nonlinear Cahn-Hilliard equation and Camassa-Holm equation. The CANN method is motivated by the finite volume scheme and evolved from the integral or weak formulation of partial differential equations. The major idea of cell-average based neural network method is to explore a neural network to approximate the solution average difference or evolution between two neighboring time steps. Unlike traditional numerical methods, CANN method is not limited by the CFL restriction and can adapt large time steps for solution evolution, which is a significant advantage that classical numerical methods do not have. Once well trained, this method can be implemented as an fixed explicit finite volume scheme and applied to certain groups of initial value conditions for Cahn-Hilliard equation and Camassa-Holm equation without retraining the neural network. Furthermore, the CANN method also performs very well in handling data with corruption or low quality data generated by Gaussian white noise. Numerical examples are presented to demonstrate effectiveness, accuracy and capability of the proposed method.

The cell-average based neural network for numerical approximation of the nonlinear Schrödinger equation

“…The overall training process and output module are shown in Figure 2. As the Remark 1 in [51], we also use multiple time levels of solution averages in the training process. )…”

“…update Θ * ← Optimizers : SGD, Adam, ect. 10: end while 11: Referring to Remark 2 of [51], it is feasible to employ the proficiently trained CANN model using a single trajectory dataset to address the problem (2.1) under varied initial values u i (x, 0) = u i 0 (x), where i denotes distinct initial conditions. Once the network is trained, it will behave as an explicit one-step finite volume scheme to solve problems with different initial conditions without retraining the network.…”

“…Based on the finite volume method for solving fluid flow problems, we propose a cell average neural network (CANN) method as a novel approach for solving the NLS equation. The idea of CANN method [50,51] is to explore a network to approximate the solution average difference between two neighboring time steps. After well-trained, the CANN method is implemented as a conventional explicit one-step finite volume scheme.…”

Cell-Average Based Neural Network Method for Hunter-Saxton Equations

“…In this paper, we propose a cell-average based neural network (CANN) method [52,53], which is motivated from finite volume method, to solve the Hunter-Saxton equation. The CANN method follows the solution properties and characteristics of the PDEs to build up neural network solvers.…”