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Traditionally, classical numerical schemes have been employed to solvepartial differential equations (PDEs) using computational methods. Recently,neural network-based methods have emerged. Despite these advancements, neuralnetwork-based methods, such as physics-informed neural networks (PINNs) andneural operators, exhibit deficiencies in robustness and generalization. Toaddress these issues, numerous studies have integrated classical numericalframeworks with machine learning techniques, incorporating neural networks intoparts of traditional numerical methods. In this study, we focus on hyperbolicconservation laws by replacing traditional numerical fluxes with neuraloperators. To this end, we developed loss functions inspired by establishednumerical schemes related to conservation laws and approximated numericalfluxes using Fourier neural operators (FNOs). Our experiments demonstrated thatour approach combines the strengths of both traditional numerical schemes andFNOs, outperforming standard FNO methods in several respects. For instance, wedemonstrate that our method is robust, has resolution invariance, and isfeasible as a data-driven method. In particular, our method can make continuouspredictions over time and exhibits superior generalization capabilities without-of-distribution (OOD) samples, which are challenges that existing neuraloperator methods encounter.

Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws