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

TL;DR

The paper tackles learning solvers for scalar hyperbolic conservation laws by replacing the traditional numerical flux in finite-volume schemes with a neural network while preserving the conservative structure. It introduces Neural Finite Volume Method (NFVM) with a trainable flux $F_{\theta}$ and a stencil-extended variant NFVM$_a^b$, trainable in supervised or unsupervised modes through a weak-form objective, and provides theoretical guarantees on convergence and neural network size. Empirically, NFVM often outperforms baseline first-order schemes (Godunov, LxF, EO), and in several cases rivals higher-order methods (ENO/WENO) or near DG performance on synthetic LWR and Burgers equations; it also demonstrates applicability to real-world traffic data from the Berkeley DeepDrive drone dataset. The work establishes error-propagation and approximation bounds for the learned flux, analyzes sample complexity for supervised and unsupervised training, and shows that a data-driven flux can capture complex traffic dynamics with limited data, offering a shelf-ready, conservation-respecting PDE solver with practical impact on traffic modeling. It opens avenues for higher-dimensional extensions, entropy-enforcing mechanisms, and hybrid data-method approaches to improve robustness and scalability in real-world applications.

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.

Figures (17)

• Figure 1: Lax-Hopf ground truth vs. NFVM$_2^1$ prediction on two different initial conditions. Each flow (column) uses a different trained model. See \ref{['fig:burgers_heatmaps']} for axes and legendt
• Figure 1: LWR flow functions with default parameters from \ref{['app:lwr_flows']}.
• Figure 2: Performance of learned models against baselines in $L_1$ error on evaluation set, with standard deviations reported as error bars. Exact values are reported in \ref{['tab:lwr_metrics']}.
• Figure 3: Performance of learned models against baselines in $L_2$ error on evaluation set, with standard deviations reported as error bars. Exact values are reported in \ref{['tab:lwr_metrics']}.
• Figure 4: Winrates of schemes against each other, computed over the evaluation set of 1000 initial conditions (see \ref{['sec:results']}) for each of the 6 flux functions. A winrate of $N$ for row-scheme $X$ against column-scheme $Y$ means that scheme $X$ has a lower $L_2$ error than scheme $Y$ on $N$% of the initial conditions for that flux.

Theorems & Definitions (37)

• Definition 2.1: Total Variation
• Definition 2.2: Weak solution
• Remark 3.1
• Remark 3.2
• Lemma 4.1
• Proof 1
• Lemma 4.2
• Proof 2
• Remark 4.3: Stability With Respect to Numerical Flux
• Definition 4.4: $\varepsilon$-Proximity