Cell-Average Based Neural Network Method for Hunter-Saxton Equations (Advances in Applied Mathematics and Mechanics, 2024)
In this paper, we develop a cell-average based neural network (CANN)method for solving the Hunter-Saxton equation with its zero-viscosity and zero-dispersion limits. Motivated from the finite volume schemes, the cell-average basedneural network method is constructed based on the finite volume integrals of the original PDEs. Supervised training is designed to learn the solution average differencebetween two neighboring time steps. The training data set is generated by the cell average based on a single initial value of the given PDE. The training process employsmultiple time levels of cell averages to maintain stability and control temporal accumulation errors. After being well trained based on appropriate meshes, this methodcan be utilized like a regular explicit finite volume method to evolve the solution underlarge time steps. Furthermore, it can be applied to solve different type of initial valueproblems without retraining the neural network. In order to validate the capabilityand robustness of the CANN method, we also utilize it to deal with the corruptedlearning data which is generated from the Gaussian white noise. Several numericalexamples of different types of Hunter-Saxton equations are presented to demonstratethe effectiveness, accuracy, capability, and robustness of the proposed method.