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

2025-05-29

AI Summary

For hyperbolic conservation laws featuring shocks and discontinuities, conventional finite volume (FV) methods struggle to balance accuracy, generalizability, and computational complexity. This paper introduces Neural FV—a novel framework that for the first time explicitly encodes the conservation structure and entropy stability priors of FV into a neural network architecture, enabling learnable update rules over extended spatiotemporal stencils; it supports joint training via supervised data and unsupervised weak-form residuals. Experiments demonstrate that Neural FV reduces error by an order of magnitude over Godunov’s method on standard conservation law benchmarks, outperforms ENO/WENO schemes, and matches the accuracy of discontinuous Galerkin (DG) methods while incurring significantly lower computational cost. Moreover, in real-world traffic flow modeling, Neural FV exhibits superior fidelity and scalability.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Solving hyperbolic PDEs with shocks and discontinuities
• Improving accuracy and flexibility of finite volume methods
• Learning update rules while preserving conservation structure

Innovation

• Modular neural network generalizes finite volume methods
• Learns update rules preserving conservation structure
• Combines supervised and unsupervised training approaches