Deep Finite Volume Method for Partial Differential Equations

TL;DR

The Deep Finite Volume Method (DFVM) is a novel deep learning framework for solving high-order PDEs that improves accuracy and reduces computational costs by formulating the problem in a weak form and transforming derivatives.

Contribution

DFVM introduces a weak form formulation and derivative transformation techniques, offering a more accurate and efficient deep learning approach for high-order PDEs compared to existing methods.

Findings

DFVM achieves equal or better accuracy than PINN, DRM, and WAN.

DFVM significantly reduces computational costs.

DFVM outperforms in solving PDEs with nonsmooth solutions, with errors up to two orders of magnitude lower.

Abstract

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}.