(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs
This addresses accuracy and flexibility limitations in computational fluid dynamics and traffic modeling, offering a scalable solution for engineers and scientists, though it is incremental as it builds on existing finite volume methods.
The paper tackles solving hyperbolic PDEs with shocks and discontinuities by introducing (U)NFV, a neural network architecture that generalizes finite volume methods, achieving up to 10x lower error than Godunov's method and rivaling discontinuous Galerkin solvers with less complexity.
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.