What question did this study set out to answer?

This work aims to investigate the thermal behavior of nanoparticles in magnetized chemically reacting fluids using neural networks optimized by the Levenberg-Marquardt technique.

April 27, 2026Open Access

Levenberg–Marquardt Optimized Stochastic PINNs for Magnetized Radiative Transport of Motile Nanoparticles in Synchronous Chemically Reacting Fluids with Energy Storage Applications

Key Points

This work aims to investigate the thermal behavior of nanoparticles in magnetized chemically reacting fluids using neural networks optimized by the Levenberg-Marquardt technique.
Utilized Stochastic Physics-Informed Neural Networks (PINNs) with Levenberg-Marquardt optimization for computation.
Transformed governing equations into coupled ordinary differential equations and solved numerically.
Trained a neural network in MATLAB using datasets generated, assessed through Mean Squared Error and regression analysis.
Achieved mean squared error values as low as 10⁻¹¹, indicating high accuracy.
Demonstrated strong consistency between numerical solutions and NN-BLMS, with absolute errors ranging from 10⁻⁷ to 10⁻⁴.
Revealed significant correlations between magnetic and viscoelastic factors and thermal dynamics in fluids, enhancing energy transfer systems.

Abstract

• Nanoparticles in Magnetized Chemically Reacting flow is discussed. • Stochastic Physics-Informed Neural Networks are considered. • Thermal liquid flowing with radiation factor is studied. • Levenberg-Marquardt Optimization is used for computation. • Flow behavior is discussed through graphs and tables. In the present article, the thermal motility of tiny particles in magnetized chemically reacting upper-convective Maxwell fluid (CRUC-MF) is discussed by using the technique of neural network backpropagated with Levenberg Marquardt scheme (NN-BLMS). Research on non-Newtonian fluids under diverse physical conditions is extensive, owing to the importance of enhancing the thermal properties of industrial fluids and materials to improve the efficiency of various equipment. This work specifically investigates the thermal behavior of nanoparticles in a chemically reacting, magnetized upper-convected Maxwell fluid near a stagnation point, utilizing NN-BLMS to better understand the effects of radiation on heat exchangers, cooling technologies and energy transfer systems. The governing equations of the CRUC-MF are transformed into a set of coupled ordinary differential equations through similarity transformations, which are then solved numerically with high accuracy using the ND Solve method. The resulting dataset is exported to MATLAB, where a neural network model is trained using the Levenberg-Marquardt technique. Performance is checked using Mean Squared Error (MSE), regression analysis and error histograms, which demonstrate how effectively the NN-BLMS represents the system dynamics. The findings indicate that there is a high level of consistency between numerical and NN-BLMS solutions and mean squared error values are as small as 10⁻¹¹ and good regression performance (R ≈ 1). In all the scenarios, the absolute error is within the range of 10⁻⁷ to 10⁻⁴, which proves high accuracy, stability and reliability of the proposed approach. The work provides insightful information on how magnetic and viscoelastic aspects impact the fluid's thermal and flow characteristics, with implications for enhancing industrial heat transfer systems.

Levenberg–Marquardt Optimized Stochastic PINNs for Magnetized Radiative Transport of Motile Nanoparticles in Synchronous Chemically Reacting Fluids with Energy Storage Applications

Key Points

Abstract

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