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<p>This paper investigates the convective–diffusive transport equation, known as the equilibrium dispersive (ED) model in liquid chromatography. The analytical solution of the model exists in literature and is used here as a benchmark. A data-driven approach is employed using an Artificial Neural Network (ANN) combined with Bayesian Intelligent Regularization (IBR) to approximate the solution and analyze the transport behavior for different key parameters: interstitial mobile-phase velocity (<i>v</i>), injected mass (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(c_0\)</EquationSource> </InlineEquation>), and dispersion coefficient (<i>D</i>). These parameters control the peak profiles, where variations in <i>v</i> alter retention time, <i>D</i> affects peak broadening, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(c_0\)</EquationSource> </InlineEquation> modifies peak magnitude. The ANN-IBR method is trained, validated, and tested on high-quality data generated from the analytical solution using MATLAB’s bvp4c solver with the Lobatto-IIIA scheme. Performance is evaluated using Mean Squared Error (MSE), regression analysis (RA), and error histograms (EH). We have done a comprehensive comparative analysis of the applied scheme with Levenberg–Marquardt (LM) algorithm and Stochastic Gradient Descent (SGD) method. The results demonstrate that the ANN-IBR framework accurately captures the dynamics of chromatographic transport and provides a robust computational tool for modeling systems.</p>

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Investigation of convective-diffusive model in liquid chromatography using bayesian neural network approach

  • 期刊论文

  • 发表日期：

    2026年4月15日

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  • Farman U Khan,
  • Rafay Mustafa,
  • Naveed Ahmed,
  • Dania Saleem

摘要

This paper investigates the convective–diffusive transport equation, known as the equilibrium dispersive (ED) model in liquid chromatography. The analytical solution of the model exists in literature and is used here as a benchmark. A data-driven approach is employed using an Artificial Neural Network (ANN) combined with Bayesian Intelligent Regularization (IBR) to approximate the solution and analyze the transport behavior for different key parameters: interstitial mobile-phase velocity (v), injected mass ( \(c_0\) ), and dispersion coefficient (D). These parameters control the peak profiles, where variations in v alter retention time, D affects peak broadening, and \(c_0\) modifies peak magnitude. The ANN-IBR method is trained, validated, and tested on high-quality data generated from the analytical solution using MATLAB’s bvp4c solver with the Lobatto-IIIA scheme. Performance is evaluated using Mean Squared Error (MSE), regression analysis (RA), and error histograms (EH). We have done a comprehensive comparative analysis of the applied scheme with Levenberg–Marquardt (LM) algorithm and Stochastic Gradient Descent (SGD) method. The results demonstrate that the ANN-IBR framework accurately captures the dynamics of chromatographic transport and provides a robust computational tool for modeling systems.