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<p>This study investigates nonlinear sand-ripple formation in granular media using a hybrid analytical–computational framework. Three nonlinear partial differential equation models are examined, incorporating key physical mechanisms such as surface diffusion, dispersion, and nonlinear steepening. Exact travelling–wave and solitary–wave solutions are derived via an ansatz–based analytical method, providing clear insight into ripple stability and morphology. The analytical solutions are subsequently utilized as training data for an artificial neural network approximation (ANNA), allowing a mesh-free characterization of the solution space. These two ANNA therefore both do well to a level of order&#xa0;<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(10^{-4}\)</EquationSource> </InlineEquation> in mean squared error (<i>MSE</i>), while even the fully generalized form has at best modest loss of accuracy, as expected from additional higher-order non-linear residual interactions. The new hybrid approach presents an inherently powerful and interpretable medium through which to conduct nonlinear wave propagation and pattern formation analyses of granular systems.</p> Graphical Abstract <p></p>

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Analytical solutions and artificial neural network approximations for sand ripples in granular media

  • 期刊论文

  • 发表日期：

    2026年7月31日

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  • Majeed Ahmad Yousif,
  • Zehra Pinar Izgi,
  • Meraa Arab,
  • Hari Mohan Srivastava,
  • Pshtiwan Othman Mohammed

摘要

This study investigates nonlinear sand-ripple formation in granular media using a hybrid analytical–computational framework. Three nonlinear partial differential equation models are examined, incorporating key physical mechanisms such as surface diffusion, dispersion, and nonlinear steepening. Exact travelling–wave and solitary–wave solutions are derived via an ansatz–based analytical method, providing clear insight into ripple stability and morphology. The analytical solutions are subsequently utilized as training data for an artificial neural network approximation (ANNA), allowing a mesh-free characterization of the solution space. These two ANNA therefore both do well to a level of order  \(10^{-4}\) in mean squared error (MSE), while even the fully generalized form has at best modest loss of accuracy, as expected from additional higher-order non-linear residual interactions. The new hybrid approach presents an inherently powerful and interpretable medium through which to conduct nonlinear wave propagation and pattern formation analyses of granular systems.

Graphical Abstract