<p>The present study examines the role of copper nanoparticles in addressing the challenges posed by nonlinear magnetohydrodynamics (MHD) Jeffery-Hamel blood flow problem. Employing pertinent transformation techniques, the governing partial differential equations (PDEs) are transformed into nonlinear ordinary differential equations (ODEs). Exploring the utilization of morlet-wavelet based physics informed neural networks (MW-PINNs), the ODEs transformed into error function to handle the blood flow model. A hybridization of the particle swarm optimization (PSO) and neural network algorithm (NNA) is utilized to minimize the error function up to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44379_2025_23_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$(\le {10}^{-10})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>≤</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The primary weights of MW-PINNs are selected randomly later updated using the PSO then using NNA algorithms for enhance accuracy. The results obtained from MW-PINNs -PSO-NNA are compared with 4th order Runge–Kutta method (RK4) and physics informed neural networks (PINNs) as a reference solution for validation. Three cases of the MHD Jeffery-Hamel blood flow with copper nanoparticles were considered for analysis using the MW-PINNs -PSO-NNA approach. The absolute error for three different cases ranging from <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44379_2025_23_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">${5.58245\times 10}^{-07}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>5.58245</mn> <mo>×</mo> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>07</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44379_2025_23_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">${1.33835\times 10}^{-08}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>1.33835</mn> <mo>×</mo> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>08</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44379_2025_23_Article_IEq4.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">${1.80546\times 10}^{-06}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>1.80546</mn> <mo>×</mo> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>06</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44379_2025_23_Article_IEq5.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">${3.03461\times 10}^{-08}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>3.03461</mn> <mo>×</mo> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>08</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44379_2025_23_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">${6.58107\times 10}^{-08}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>6.58107</mn> <mo>×</mo> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>08</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44379_2025_23_Article_IEq7.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">${1.13338\times 10}^{-09}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>1.13338</mn> <mo>×</mo> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>09</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, respectively.</p>

Tuned Morlet-Wavelet based Physics Informed Neural Networks (MW-PINNs) hyperparameters by heuristic algorithm analysis for Jeffery-Hamel blood flow with copper nanoparticles

期刊论文
发表日期：

2025年5月30日

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Faisal Ashraf,
Muhammad Naeem Aslam,
Nadeem Shaukat,
Arshad Riaz,
Muhammad Sarmad Arshad

摘要

The present study examines the role of copper nanoparticles in addressing the challenges posed by nonlinear magnetohydrodynamics (MHD) Jeffery-Hamel blood flow problem. Employing pertinent transformation techniques, the governing partial differential equations (PDEs) are transformed into nonlinear ordinary differential equations (ODEs). Exploring the utilization of morlet-wavelet based physics informed neural networks (MW-PINNs), the ODEs transformed into error function to handle the blood flow model. A hybridization of the particle swarm optimization (PSO) and neural network algorithm (NNA) is utilized to minimize the error function up to $(\le {10}^{-10})$ $(\leq 10^{- 10})$ . The primary weights of MW-PINNs are selected randomly later updated using the PSO then using NNA algorithms for enhance accuracy. The results obtained from MW-PINNs -PSO-NNA are compared with 4th order Runge–Kutta method (RK4) and physics informed neural networks (PINNs) as a reference solution for validation. Three cases of the MHD Jeffery-Hamel blood flow with copper nanoparticles were considered for analysis using the MW-PINNs -PSO-NNA approach. The absolute error for three different cases ranging from ${5.58245\times 10}^{-07}$ ${5.58245 \times 10}^{- 07}$ to ${1.33835\times 10}^{-08}$ ${1.33835 \times 10}^{- 08}$ , ${1.80546\times 10}^{-06}$ ${1.80546 \times 10}^{- 06}$ to ${3.03461\times 10}^{-08}$ ${3.03461 \times 10}^{- 08}$ and ${6.58107\times 10}^{-08}$ ${6.58107 \times 10}^{- 08}$ to ${1.13338\times 10}^{-09}$ ${1.13338 \times 10}^{- 09}$ , respectively.