  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.
