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Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws

Taeyoung Kim, Myungjoo Kang

TL;DR

The paper tackles solving hyperbolic conservation laws by replacing traditional numerical fluxes with a neural operator, specifically the Flux Fourier Neural Operator (Flux FNO). It introduces two loss terms—$\

Abstract

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.

Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws

TL;DR

The paper tackles solving hyperbolic conservation laws by replacing traditional numerical fluxes with a neural operator, specifically the Flux Fourier Neural Operator (Flux FNO). It introduces two loss terms—$\

Abstract

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.
Paper Structure (23 sections, 38 equations, 16 figures, 9 tables, 4 algorithms)

This paper contains 23 sections, 38 equations, 16 figures, 9 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Numerical schemes
  3. Physics-inspired machine learning
  4. Various approaches of combining numerical schemes and neural networks
  5. Our contribution
  6. Preliminaries
  7. Hyperbolic conservation laws
  8. Numerical schemes
  9. Fourier neural operator
  10. Algorithms for Flux Fourier Neural Operator
  11. Experiments
  12. Long-term continuous inference
  13. Generalization ability
  14. Combined with 2nd Order Runge-Kutta(RK2) Method
  15. Results on Other Conservation Problems
  16. ...and 8 more sections

Figures (16)

  • Figure 1: Schematic of 1D FNO
  • Figure 2: Schematic of the inference structure for flux Fourier neural operator (Flux FNO)
  • Figure 3: Output of Flux FNO (dashed line with triangle markers) compared with the exact solutions (solid line) for the 1D linear advection problem.
  • Figure 4: Comparison of Flux FNO output with exact solutions and other FNO models at $t=0.4$ (top left), $t=0.8$ (top right), $t=2.5$ (bottom left), and $t=5.0$ (bottom right) for the 1D linear advection problem.
  • Figure 5: Output of Flux FNO (dashed line with triangle markers) compared with the exact solutions (solid line) for the 1D Burgers’ equation problem.
  • ...and 11 more figures