Classical numerical schemes exist for solving PDEs numerically, and recently, neural network-based methods have been developed. However, methodologies using neural networks, such as PINN and neural operators, lack robustness and generalization power. To compensate for such drawbacks, there are many types of research combining classical numerical schemes and machine learning methods by replacing a small portion of the numerical schemes with neural networks. In this work, we focus on hyperbolic conservation laws and replace numerical fluxes in the numerical schemes by neural operator. For this, we construct losses that are motivated by numerical schemes for conservation laws and approximate numerical flux by FNO. Through experiments, we show that our methodology has advantages of both numerical schemes and FNO by comparing with original methods. For instance, we demonstrate our method gains robustness, resolution invariance property, and feasibility of a data-driven method. Our method especially has the ability to predict continuously in time and generalization power on the out-of-distribution samples, which are challenges to be tackled for existing neural operator methods.

翻译：经典数值格式已广泛用于偏微分方程的数值求解，近年来基于神经网络的方法也得到了发展。然而，诸如PINN和神经算子等神经网络方法缺乏鲁棒性和泛化能力。为弥补这一缺陷，大量研究将经典数值格式与机器学习方法相结合，通过用神经网络替换数值格式中的小部分组件来实现。本文聚焦双曲守恒律，用神经算子替代数值格式中的数值通量。为此，我们基于守恒律数值格式的启发构造损失函数，并通过FNO逼近数值通量。通过实验，我们证明该方法兼具数值格式与FNO的优势。例如，我们的方法在鲁棒性、分辨率不变性及数据驱动方法的可行性上均有提升。尤其值得关注的是，该方法具备连续时间预测能力及对分布外样本的泛化能力，这些正是现有神经算子方法亟待解决的挑战。