In a 1D plasma benchmark, classical conservative finite volume structure achieves rollout MSE of 7.35e-9 and wins 60/64 cases, outperforming neural baselines focused on one-step accuracy.
(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs
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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.
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2026 2representative citing papers
Adding a soft mass-conservation penalty to PINNs for the 1D advection-diffusion equation reduces long-term relative L2 error by 9–67× and mass error by 15–215× compared to vanilla PINNs across Peclet numbers 0.01–20.
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Conservative Discrete Structure Stabilizes Autoregressive Rollouts in a 1D Drift Diffusion Poisson Benchmark
In a 1D plasma benchmark, classical conservative finite volume structure achieves rollout MSE of 7.35e-9 and wins 60/64 cases, outperforming neural baselines focused on one-step accuracy.
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Mass-Conserving Physics-Informed Neural Networks For The One-Dimensional Advection-Diffusion Equation
Adding a soft mass-conservation penalty to PINNs for the 1D advection-diffusion equation reduces long-term relative L2 error by 9–67× and mass error by 15–215× compared to vanilla PINNs across Peclet numbers 0.01–20.