用神经网络改进传统有限体积法，更好求解含间断的偏微分方程。
(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs
- 用神经网络学习更广域的更新规则，保持守恒结构。
- 在交通流模拟中误差比经典方法低10倍，性能媲美复杂高阶方法。
- 支持有监督和无监督训练，适合需要高精度与可扩展性的工程场景。
我们提出 (U)NFV，一种模块化神经网络架构，用于求解双曲型守恒律。这类偏微分方程（PDEs）因解中存在激波和间断而难解。传统有限体积（FV）方法虽具收敛到熵解、守恒性等数学优势，但在复杂情形下精度与灵活性不足。神经有限体积通过在扩展的空间-时间模板上学习更新规则，保留守恒结构，克服上述局限。该方法支持基于解数据的有监督训练（NFV）和基于弱形式残差损失的无监督训练（UNFV）。应用于一阶守恒律时，(U)NFV 最低误差比戈杜诺夫方法降低10倍，优于ENO/WENO，性能接近间断伽辽金方法但复杂度远低。在交通建模问题中，无论是基于PDE还是高速公路实测数据，(U)NFV 均能以更高保真度和可扩展性捕捉非线性波动力学。
原文摘要 · Abstract (English)
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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