Deep Finite Volume Method for Partial Differential Equations

TL;DR

The paper tackles efficient, accurate solving of high-order PDEs with deep learning by introducing the Deep Finite Volume Method (DFVM), a weak-form, conservation-based loss built from control-volume quadratures. By transforming higher-order derivatives into first-order terms via flux evaluations and boundary integrals, DFVM achieves high accuracy, especially for non-smooth solutions, while reducing computational cost relative to strong-form approaches like PINN. Across Poisson, biharmonic, Cahn–Hilliard, and Black–Scholes problems, DFVM often outperforms PINN, DRM, and WAN, and scales better to high dimensions, with an adaptive variant (ATS-DFVM) further improving sampling efficiency. These results suggest DFVM as a robust, scalable framework that blends finite-volume intuition with neural networks for a wide range of PDEs, with open-source code available on GitHub for broader adoption.

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}.

Figures (5)

• Figure 3.1: The exact solution and approximate solution by different learning methods.
• Figure 3.2: Training curves of three methods for \ref{['eq:Poisson']}.
• Figure 3.3: ATS-DFVM training points in the $(x_1,x_2)$ plane. For display purpose, images only show the slices of $x_3 = \cdots = x_{10} = 0$.
• Figure 3.4: Results of C-H equation.
• Figure 3.5: Results of Black-Scholes Equation.