<p>This study outlines the implementation and analysis of an improved Physics-Informed Neural Network (PINN) for modeling complex heat transfer in fluid dynamics, particularly comparing the properties of a base fluid with those of a Nanofluid. The PINN incorporates the governing partial differential equations (PDEs) for fluid dynamics in a magnetic field and porous material directly into the neural network’s loss function; ensuring solutions comply with physical principles. This approach acts as a robust alternative to traditional numerical solvers, especially in situations with limited data. To improve convergence and accuracy, the model’s architecture has been changed in many ways: Added Residual Connections: A new residual connection is used to make gradient propagation better. The training process includes adaptive weighting for boundary condition loss, gradient clipping to avoid instability, and an early stopping mechanism. The simulation shows that the model can predict dimensionless profiles for axial velocity (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation>), angular velocity (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(g\)</EquationSource> </InlineEquation>), temperature (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\theta\)</EquationSource> </InlineEquation>), and concentration (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varphi\)</EquationSource> </InlineEquation>) in a certain radial area. The study demonstrates how nanoparticles alter the system’s thermo-physical properties by comparing simulations for two cases are a pure fluid (0% nanoparticle volume fraction) and a Nanofluid (4% nanoparticle volume fraction). The findings reveal that the enhanced PINN architecture can effectively approximate solutions of the coupled micropolar nanofluid equations in the cylindrical coordinate with training losses of 10<sup>–2</sup> and physical trends. Although the results are encouraging for this type of problem, more verification with known numerical solvers and experimental data has to be done before one can make statements about the wider engineering problems. The model’s lowest final training loss is about 11.30 when the magnetic parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(M\)</EquationSource> </InlineEquation> is set to 1.0 and the porous parameter <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(Kp\)</EquationSource> </InlineEquation> is set to 0.1, which shows that the model is stable. In these conditions, the nanofluid’s temperature rises by about 7.05% and its axial velocity changes by about 14.11% compared to the pure fluid. These findings suggest that the enhanced PINN design can resolve the nanofluid effect in this geometry with reasonable accuracy, which may be useful as a complementary device to multi-physics heat transfer analysis. More general propositions about the solution of general engineering problems however, demand a very large amount of validation under a variety of conditions and a direct test against experimental data as well as an existing set of numerical procedures.</p>

Physics-informed neural networks (PINN) for heat transfer: a comprehensive analysis

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发表日期：

2026年4月10日

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Hammam M. Abdelaal,
Mohamed Wahba,
H. A. El-dawy

摘要

This study outlines the implementation and analysis of an improved Physics-Informed Neural Network (PINN) for modeling complex heat transfer in fluid dynamics, particularly comparing the properties of a base fluid with those of a Nanofluid. The PINN incorporates the governing partial differential equations (PDEs) for fluid dynamics in a magnetic field and porous material directly into the neural network’s loss function; ensuring solutions comply with physical principles. This approach acts as a robust alternative to traditional numerical solvers, especially in situations with limited data. To improve convergence and accuracy, the model’s architecture has been changed in many ways: Added Residual Connections: A new residual connection is used to make gradient propagation better. The training process includes adaptive weighting for boundary condition loss, gradient clipping to avoid instability, and an early stopping mechanism. The simulation shows that the model can predict dimensionless profiles for axial velocity ( \(f\) ), angular velocity ( \(g\) ), temperature ( \(\theta\) ), and concentration ( \(\varphi\) ) in a certain radial area. The study demonstrates how nanoparticles alter the system’s thermo-physical properties by comparing simulations for two cases are a pure fluid (0% nanoparticle volume fraction) and a Nanofluid (4% nanoparticle volume fraction). The findings reveal that the enhanced PINN architecture can effectively approximate solutions of the coupled micropolar nanofluid equations in the cylindrical coordinate with training losses of 10^–2 and physical trends. Although the results are encouraging for this type of problem, more verification with known numerical solvers and experimental data has to be done before one can make statements about the wider engineering problems. The model’s lowest final training loss is about 11.30 when the magnetic parameter \(M\) is set to 1.0 and the porous parameter \(Kp\) is set to 0.1, which shows that the model is stable. In these conditions, the nanofluid’s temperature rises by about 7.05% and its axial velocity changes by about 14.11% compared to the pure fluid. These findings suggest that the enhanced PINN design can resolve the nanofluid effect in this geometry with reasonable accuracy, which may be useful as a complementary device to multi-physics heat transfer analysis. More general propositions about the solution of general engineering problems however, demand a very large amount of validation under a variety of conditions and a direct test against experimental data as well as an existing set of numerical procedures.