<p>This paper presents the development of integrated intelligent computing using physics-informed neural networks to solve the mathematical normal-tumor immune-unhealthy diet model. The model considers vitamin intervention as a moderating factor within one day. A nonlinear activation function called sigmoid was used for the model across three different scenarios to define the fitness or error function. For computing the optimized biases and weights of physics-informed neural networks, hybridization of heuristic algorithms, particularly particle swarm optimization and neural networks algorithm are employed from <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12065_2024_1007_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$-10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12065_2024_1007_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>. The proposed technique's accuracy, reliability, and validity are demonstrated by consistently matching the results obtained using NDsolve from mathematics built-in function as a reference solution. Additionally, statistical analyses including absolute and mean squared errors are conducted to confirm further the precision and accuracy of the physics-informed neural networks. It is observed that the absolute errors and mean square error between the reference solution and proposed technique range from <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12065_2024_1007_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12065_2024_1007_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-9}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>9</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and range from <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12065_2024_1007_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-6}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12065_2024_1007_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{-11}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>-</mo> <mn>11</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> for distinct cases. The novelty of this study highlights the importance of physics-informed neural networks for a balanced diet rich in vitamins for reducing the risk of deadly diseases, particularly cancer and the potential of machine learning algorithms in modelling and analyzing complex biological systems.</p>

An intelligent approach for analyzing the effects of normal tumor immune unhealthy diet model through unsupervised physics informed neural-networks integrated with meta-heuristic algorithms

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发表日期：

2025年1月25日

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Muhammad Naeem Aslam,
Nadeem Shaukat,
Muhammad Sarmad Arshad,
Muhammad Waheed Aslam,
Javed Hussain

摘要

This paper presents the development of integrated intelligent computing using physics-informed neural networks to solve the mathematical normal-tumor immune-unhealthy diet model. The model considers vitamin intervention as a moderating factor within one day. A nonlinear activation function called sigmoid was used for the model across three different scenarios to define the fitness or error function. For computing the optimized biases and weights of physics-informed neural networks, hybridization of heuristic algorithms, particularly particle swarm optimization and neural networks algorithm are employed from $-10$ $- 10$ to $10$ $10$ . The proposed technique's accuracy, reliability, and validity are demonstrated by consistently matching the results obtained using NDsolve from mathematics built-in function as a reference solution. Additionally, statistical analyses including absolute and mean squared errors are conducted to confirm further the precision and accuracy of the physics-informed neural networks. It is observed that the absolute errors and mean square error between the reference solution and proposed technique range from ${10}^{-2}$ $10^{- 2}$ to ${10}^{-9}$ $10^{- 9}$ and range from ${10}^{-6}$ $10^{- 6}$ to ${10}^{-11}$ $10^{- 11}$ for distinct cases. The novelty of this study highlights the importance of physics-informed neural networks for a balanced diet rich in vitamins for reducing the risk of deadly diseases, particularly cancer and the potential of machine learning algorithms in modelling and analyzing complex biological systems.