Deep Finite Volume Method for High-Dimensional Partial Differential Equations

In this paper, we propose the deep finite volume method (DFVM), a novel deep learning method for solving %high-order (order $\geq 2$) partial differential equations (PDEs). The key idea is to design a new loss function based on the local conservation property over the so-called {\it control volumes}, derived from the original PDE. Since the DFVM is designed according to a {\it weak instead of strong} form of the PDE, it may achieve better accuracy than the strong-form-based deep learning method such as the well-known PINN, when the to-be-solved PDE has an insufficiently smooth solution. Moreover, since the calculation of second-order derivatives of neural networks has been transformed to that of first-order derivatives which can be implemented directly by the Automatic Differentiation mechanism(AD), the DFVM usually has a computational cost much lower than that of the methods which need to compute second-order derivatives by the AD. Our numerical experiments show that compared to some deep learning methods in the literature such as the PINN, DRM, and WAN, the DFVM obtains the same or higher accurate approximate solutions by consuming significantly lower computational cost. Moreover, for some PDE with a nonsmooth solution, the relative error of approximate solutions by DFVM is two orders of magnitude less than that by the PINN.