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

TL;DR

This work introduces Neural Finite Volume ((U)NFV), a conservation-aware neural framework that generalizes classical finite volume methods by learning flux updates over extended spatiotemporal stencils. It provides two training paradigms: NFV (supervised) and UNFV (unsupervised via a weak-form residual), enabling accurate solutions to hyperbolic conservation laws even when traditional data are scarce or noisy. Across Burgers' and LWR traffic models, NFV variants outperform standard FV schemes and rival higher-order methods, with NFV$_5^5$ approaching discontinuous Galerkin performance at lower complexity, and NFV$_{11}^{11}$ delivering best accuracy with scalable parameter growth. The methodology further demonstrates practical impact by modeling real-world highway data (I-24 MOTION), where NFV yields superior predictive fidelity and stability, suggesting broad applicability to data-rich and data-scarce physical systems alike.

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.

Figures (14)

• Figure 1: Prediction of entropy solutions of hyperbolic PDEs.Top: NFV$^5_5$ prediction vs. the Godunov scheme for Burgers' equation at a fixed time. Mid: Entropic solution $u(t,x)$ for Burgers' equation over domain $(t,x) \in [0,1]^2$, and its corresponding initial condition. Bottom: Entropic solution $u(t,x)$ for the LWR equation with different fluxes sharing the same initial condition.
• Figure 2: Example stencil for $\text{FV}_5^2$, which takes in a rectangular stencil of 2 time steps times 5 space cells to compute the next cell average (in red) using \ref{['eq:update_rule']} (specifically, both the in-flow and out-flow are computed using the illustrated 2x4 sub-stencils).
• Figure 3: Comparison of numerical schemes across flow functions. Each cell shows the proportion of the evaluation set on which the row scheme outperforms the column scheme. DG, the only FEM tested, is rarely beaten. NFV$_3^1$ and UNFV$_3^1$ outperform other first-order schemes and rival higher-order ones, making them strong choices depending on the equation.
• Figure 4: Comparison of the final density of the Burgers' equation (left) and LWR triangular equation (right) for NFV$_5^5$ and the Godunov Scheme. The proposed method displays an excellent approximation of the exact solution, capturing sharp features such as discontinuities and points of non-differentiability. It contains some minor oscillations in the solution, which are not present in the Godunov scheme. The latter, however, fails to capture the discontinuities and points of non-differentiability, offering a very smoothed solution.
• Figure 5: Convergence plots on Greenshields' flux. The $L_2$ error is computed against the exact solution on the evaluation set for different mesh discretizations. We report both error average and standard deviation, on a log-log scale. The dashed vertical line illustrates the discretization on which NFV$_3^1$ and UNFV$_3^1$ were trained; the models generalize to smaller discretizations. The ratio $\Delta t / \Delta x = 0.1$ remains constant as the mesh is refined.