<p>We analyze Jaffrey-Hamel flow of a viscos-elastic fluid with utilizing the Oldroyd-B model within divergent and convergent channels through unsupervised Neural Networks. The governing partial differential equations convert into ordinary differential equation (ODEs) through appropriate transformation. Further, the ODEs transformed into Fourier physics informed neural network (F-PINNs) based error function. To optimize the weights as well as biases of F-PINNs, using heuristic optimization algorithms especially sperm swarm and water cycle optimizers for best accuracy. The results of F-PINNs compared with previous literature for validation. Additionally, the error function optimized over one hundred independent runs for check the efficiency. The F-PINNs results are more accurate as compared to bvpc4 built in function. The error function is optimized using sperm swarm F-PINNs, water cycle F-PINNs, and a hybrid approach combining both sperm swarm and water cycle F-PINN, with optimization values ranging from <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-03}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>03</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-06}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>06</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-07}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>07</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-10}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. Among these, the hybrid method demonstrates superior accuracy compared to the individual approaches. The different three scenarios tested for accuracy with absolute errors ranging from <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq5.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$1.88\times {10}^{-06}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.88</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>06</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$1.13\times {10}^{-08}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.13</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>08</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq7.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$3.46\times {10}^{-05}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3.46</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>05</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_937_Article_IEq8.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$1.66\times {10}^{-06}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.66</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>06</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p> Graphical Abstract <p></p>

Numerical analysis of Oldroyd-B Jaffrey-Hamel flow using fourier Pinns hybridized with sperm swarm and water cycle optimizers

期刊论文
发表日期：

2025年7月4日

查看全文

Muhammad Naeem Aslam,
Nadeem Shaukat,
Muhammad Sarmad Arshad,
Arshad Riaz,
Mohamed Kallel

摘要

We analyze Jaffrey-Hamel flow of a viscos-elastic fluid with utilizing the Oldroyd-B model within divergent and convergent channels through unsupervised Neural Networks. The governing partial differential equations convert into ordinary differential equation (ODEs) through appropriate transformation. Further, the ODEs transformed into Fourier physics informed neural network (F-PINNs) based error function. To optimize the weights as well as biases of F-PINNs, using heuristic optimization algorithms especially sperm swarm and water cycle optimizers for best accuracy. The results of F-PINNs compared with previous literature for validation. Additionally, the error function optimized over one hundred independent runs for check the efficiency. The F-PINNs results are more accurate as compared to bvpc4 built in function. The error function is optimized using sperm swarm F-PINNs, water cycle F-PINNs, and a hybrid approach combining both sperm swarm and water cycle F-PINN, with optimization values ranging from ${10}^{-03}$ $10^{- 03}$ to ${10}^{-06}$ $10^{- 06}$ and ${10}^{-07}$ $10^{- 07}$ to ${10}^{-10}$ $10^{- 10}$ . Among these, the hybrid method demonstrates superior accuracy compared to the individual approaches. The different three scenarios tested for accuracy with absolute errors ranging from $1.88\times {10}^{-06}$ $1.88 \times 10^{- 06}$ to $1.13\times {10}^{-08}$ $1.13 \times 10^{- 08}$ and $3.46\times {10}^{-05}$ $3.46 \times 10^{- 05}$ to $1.66\times {10}^{-06}$ $1.66 \times 10^{- 06}$ .

Graphical Abstract