用可训练神经网络替代传统数值通量，高效求解非线性双曲型偏微分方程。
Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs
- 以神经网络替代有限体积法中的数值通量，保持守恒结构。
- 在同等计算开销下，性能优于Godunov、WENO和DG等经典方法。
- 支持有监督与无监督训练，适合交通流等实际物理系统建模。
我们提出一种基于神经网络的方法来学习标量双曲守恒律。该方法将传统有限体积格式中的数值通量替换为可训练的神经网络，同时保持格式的守恒结构。模型可在有监督设置下使用高效生成的合成数据训练，也可通过偏微分方程的弱形式进行无监督训练。我们提供了理论结果，表明模型可任意逼近最优解，并给出了神经网络规模的上界。大量实验表明，该方法在相同计算预算下，通常优于Godunov、WENO和不连续伽辽金（Discontinuous Galerkin）等高效算法。最后，我们在交通预测任务中验证了该方法的有效性，使用伯克利深度驾驶无人机数据集中的实测高速公路数据。
原文摘要 · Abstract (English)
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.
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