(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs

TL;DR

The paper introduces (U)NFV, a neural network-based extension of finite volume methods that improves accuracy and flexibility in solving hyperbolic PDEs, especially with shocks and discontinuities.

Contribution

It presents a novel neural finite volume framework that learns update rules and supports both supervised and unsupervised training, outperforming classical methods.

Findings

Achieves up to 10x lower error than Godunov's method.

Outperforms ENO/WENO schemes and rivals discontinuous Galerkin methods.

Effectively models traffic wave dynamics with higher fidelity.

Abstract

We introduce (U)NFV, a modular neural network architecture that generalizes classical finite volume (FV) methods for solving hyperbolic conservation laws. Hyperbolic partial differential equations (PDEs) are challenging to solve, particularly conservation laws whose physically relevant solutions contain shocks and discontinuities. FV methods are widely used for their mathematical properties: convergence to entropy solutions, flow conservation, or total variation diminishing, but often lack accuracy and flexibility in complex settings. Neural Finite Volume addresses these limitations by learning update rules over extended spatial and temporal stencils while preserving conservation structure. It supports both supervised training on solution data (NFV) and unsupervised training via weak-form residual loss (UNFV). Applied to first-order conservation laws, (U)NFV achieves up to 10x lower error than Godunov's method, outperforms ENO/WENO, and rivals discontinuous Galerkin solvers with far less complexity. On traffic modeling problems, both from PDEs and from experimental highway data, (U)NFV captures nonlinear wave dynamics with significantly higher fidelity and scalability than traditional FV approaches.