Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws (Communications in Computational Physics, 2025)
Traditionally, classical numerical schemes have been employed to solvepartial differential equations (PDEs) using computational methods. Recently, neuralnetwork-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 machinelearning techniques, incorporating neural networks into parts of traditional numericalmethods. 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 experimentsdemonstrated 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 without-of-distribution (OOD) samples, which are challenges that existing neural operatormethods encounter.