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<p>Traditional linear schemes, which set appropriate weights to achieve the desired convergence order, are reliable for resolving smooth wave phenomena but lose accuracy and spectral fidelity on coarser meshes. To address this, recent work has turned to machine learning-enhanced numerical methods for improved precision and efficiency. In this paper, we introduce a data-driven finite-difference approach for solving hyperbolic equations with smooth solutions on coarse grids. Our neural network learns high-fidelity solutions and adapts its weights to local flow characteristics, yielding superior accuracy. Remarkably, although we do not explicitly optimize spectral behavior during training, our learned schemes exhibit enhanced spectral properties compared to classical linear methods. A suite of numerical experiments confirms the method’s accuracy and computational gains. Notably, for the 3D inviscid Taylor–Green vortex, the WLNN method achieves comparable accuracy on a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(32^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>32</mn> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> grid as the UP5 scheme does on a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(48^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>48</mn> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> grid, while requiring only about one-third of the computational cost.</p>

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Learning high-accuracy numerical schemes for hyperbolic equations on coarse meshes

  • 期刊论文

  • 发表日期：

    2026年3月18日

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  • Jinrui Zhou,
  • Yiqi Gu,
  • Hua Shen,
  • Liwei Xu,
  • Juan Zhang,
  • Guanyu Zhou

摘要

Traditional linear schemes, which set appropriate weights to achieve the desired convergence order, are reliable for resolving smooth wave phenomena but lose accuracy and spectral fidelity on coarser meshes. To address this, recent work has turned to machine learning-enhanced numerical methods for improved precision and efficiency. In this paper, we introduce a data-driven finite-difference approach for solving hyperbolic equations with smooth solutions on coarse grids. Our neural network learns high-fidelity solutions and adapts its weights to local flow characteristics, yielding superior accuracy. Remarkably, although we do not explicitly optimize spectral behavior during training, our learned schemes exhibit enhanced spectral properties compared to classical linear methods. A suite of numerical experiments confirms the method’s accuracy and computational gains. Notably, for the 3D inviscid Taylor–Green vortex, the WLNN method achieves comparable accuracy on a \(32^3\) 32 3 grid as the UP5 scheme does on a \(48^3\) 48 3 grid, while requiring only about one-third of the computational cost.