  In this paper, we introduce the Deep Finite Volume Method (DFVM), an
innovative deep learning framework tailored for solving high-order (order
\(\geq 2\)) partial differential equations (PDEs). Our approach centers on a
novel loss function crafted from local conservation laws derived from the
original PDE, distinguishing DFVM from traditional deep learning methods. By
formulating DFVM in the weak form of the PDE rather than the strong form, we
enhance accuracy, particularly beneficial for PDEs with less smooth solutions
compared to strong-form-based methods like Physics-Informed Neural Networks
(PINNs). A key technique of DFVM lies in its transformation of all second-order
or higher derivatives of neural networks into first-order derivatives which can
be comupted directly using Automatic Differentiation (AD). This adaptation
significantly reduces computational overhead, particularly advantageous for
solving high-dimensional PDEs. Numerical experiments demonstrate that DFVM
achieves equal or superior solution accuracy compared to existing deep learning
methods such as PINN, Deep Ritz Method (DRM), and Weak Adversarial Networks
(WAN), while drastically reducing computational costs. Notably, for PDEs with
nonsmooth solutions, DFVM yields approximate solutions with relative errors up
to two orders of magnitude lower than those obtained by PINN. The
implementation of DFVM is available on GitHub at
\href{https://github.com/Sysuzqs/DFVM}{https://github.com/Sysuzqs/DFVM}.
