Approximating Numerical Flux by Fourier Neural Operators for the Hyperbolic Conservation Laws

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.