Cell-average based neural network method for hyperbolic and parabolic partial differential equations
This is an incremental improvement for computational physics, offering a mesh-dependent method that avoids CFL restrictions and adapts to any time step size.
The authors tackled the challenge of solving hyperbolic and parabolic partial differential equations by proposing a cell-average based neural network method, which achieves first-order convergence for the Heat equation and captures discontinuities with minimal numerical diffusion.
Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. A simple feed forward network is forced to learn the solution average evolution between two neighboring time steps. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme, thus is mesh dependent. Different to traditional numerical methods, our method can be relieved from the explicit scheme CFL restriction and can adapt to any time step size for solution evolution. For Heat equation, first order of convergence is observed and the errors are related to the spatial mesh size but are observed independent of the mesh size in time. The cell-average based neural network method can sharply evolve contact discontinuity with almost zero numerical diffusion introduced. Shock and rarefaction waves are well captured for nonlinear hyperbolic conservation laws.