Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs

2026-01-10

arXiv.org

AI Summary

This work addresses the trade-off between accuracy and efficiency in the numerical solution of first-order nonlinear hyperbolic conservation laws by proposing a novel framework that embeds trainable neural networks into the numerical flux of a finite volume method. This approach achieves, for the first time, a seamless integration of neural networks with conservative numerical schemes. It supports both supervised training using synthetic data and unsupervised learning based on the weak form of the governing PDEs, while rigorously preserving conservation properties and offering a theoretical upper bound on approximation capability. Numerical experiments demonstrate that, at comparable computational cost, the method outperforms classical schemes such as Godunov, WENO, and discontinuous Galerkin in terms of accuracy, and its effectiveness is further validated on real-world traffic data collected from unmanned aerial vehicles monitoring a highway.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Deep Neural Architectures and Foundation Models
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

We present a neural network-based method for learning scalar hyperbolic conservation laws. Our method replaces the traditional numerical flux in finite volume schemes with a trainable neural network while preserving the conservative structure of the scheme. The model can be trained both in a supervised setting with efficiently generated synthetic data or in an unsupervised manner, leveraging the weak formulation of the partial differential equation. We provide theoretical results that our model can perform arbitrarily well, and provide associated upper bounds on neural network size. Extensive experiments demonstrate that our method often outperforms efficient schemes such as Godunov's scheme, WENO, and Discontinuous Galerkin for comparable computational budgets. Finally, we demonstrate the effectiveness of our method on a traffic prediction task, leveraging field experimental highway data from the Berkeley DeepDrive drone dataset.

Problem

• hyperbolic PDEs
• neural network solver
• conservation laws
• nonlinear PDEs
• first-order hyperbolic equations

Innovation

• neural network solver
• hyperbolic conservation laws
• conservative finite volume scheme
• unsupervised PDE learning
• weak formulation