Traditionally, classical numerical schemes have been employed to solve partial differential equations (PDEs) using computational methods. Recently, neural network-based methods have emerged. Despite these advancements, neural network-based methods, such as physics-informed neural networks (PINNs) and neural operators, exhibit deficiencies in robustness and generalization. To address these issues, numerous studies have integrated classical numerical frameworks with machine learning techniques, incorporating neural networks into parts of traditional numerical methods. In this study, we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators. To this end, we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators (FNOs). Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs, outperforming standard FNO methods in several respects. For instance, we demonstrate that our method is robust, has resolution invariance, and is feasible as a data-driven method. In particular, our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution (OOD) samples, which are challenges that existing neural operator methods encounter.

翻译：传统上，经典数值格式已被用于通过计算方法求解偏微分方程。近年来，基于神经网络的方法逐渐兴起。尽管取得了这些进展，但基于神经网络的方法（如物理信息神经网络和神经算子）在鲁棒性和泛化能力方面仍存在不足。为解决这些问题，大量研究将经典数值框架与机器学习技术相结合，将神经网络融入传统数值方法的组成部分。在本研究中，我们聚焦于双曲守恒律，用神经算子替代传统数值通量。为此，我们借鉴与守恒律相关的成熟数值格式，开发了损失函数，并利用傅里叶神经算子逼近数值通量。实验表明，我们的方法结合了传统数值格式与傅里叶神经算子的优势，在多个方面优于标准傅里叶神经算子方法。例如，我们证明该方法具有鲁棒性、分辨率不变性，并可作为数据驱动方法使用。特别地，该方法能够随时间进行连续预测，并且在处理分布外样本时展现出卓越的泛化能力，而这是现有神经算子方法面临的挑战。