# Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws

arXiv: https://arxiv.org/abs/2401.01783 · HTML: https://mlsanity.com/paper/2401.01783/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 42 references that are in the map.
- First posted on arXiv 3 Jan 2024.

## Builds on (most cited first, 20 of 20)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [An Introduction to Convolutional Neural Networks](https://mlsanity.com/paper/1511.08458/) (2015)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Generalization Error in Deep Learning](https://mlsanity.com/paper/1808.01174/) (2018)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)
- [Bounding the Rademacher Complexity of Fourier neural operators](https://mlsanity.com/paper/2209.05150/) (2022)
- [Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning](https://mlsanity.com/paper/1905.11079/) (2019)
- [Critical Investigation of Failure Modes in Physics-informed Neural Networks](https://mlsanity.com/paper/2206.09961/) (2022)
- [Constraint-Aware Neural Networks for Riemann Problems](https://mlsanity.com/paper/1904.12794/) (2019)
- [Scale-invariant Learning by Physics Inversion](https://mlsanity.com/paper/2109.15048/) (2021)

## Built on by (most cited first, 11 of 11)

- [Recurrent Neural Operators: Stable Long-Term PDE Prediction](https://mlsanity.com/paper/2505.20721/) (2025)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Diagnosing Failure Modes of Neural Operators Across Diverse PDE Families](https://mlsanity.com/paper/2601.11428/) (2026)
- [Neural Operators Learn the Local Physics of Magnetohydrodynamics](https://mlsanity.com/paper/2404.16015/) (2024)
- [Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge](https://mlsanity.com/paper/2602.15184/) (2026)
- [Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs](https://mlsanity.com/paper/2601.06388/) (2026)
- [FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates](https://mlsanity.com/paper/2602.01941/) (2026)
- [From Data to Laws: Neural Discovery of Conservation Laws Without False Positives](https://mlsanity.com/paper/2603.20474/) (2026)
- [A Robust In-Context Model for Conservation Laws: Injecting Context into Flux Neural Operators via Recurrent Vision Transformers](https://mlsanity.com/paper/2605.05488/) (2026)
- [HypNO: A Graph-Based Neural Operator with Physics-Informed Message Passing for Hyperbolic Conservation Laws](https://mlsanity.com/paper/2607.20541/) (2026)
- [Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws](https://mlsanity.com/paper/2609.32887/) (2026)

## Abstract (arXiv)

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.