Neural Network Learns to Predict Heat Transfer in Elastic Nanofluid Flow Over an Electromagnetic Surface

A team of researchers in India, Pakistan, Turkey, Azerbaijan and Saudi Arabia has combined a classical fluid-dynamics solver with a Kolmogorov-Arnold neural network to predict how a viscoelastic Boger nanofluid transfers heat and mass as it flows over a Riga surface, an actuator that generates a streamwise Lorentz force. The study, published in Frontiers in Physics, reports that the trained network reproduces three key engineering quantities, the skin friction coefficient, the Nusselt number and the Sherwood number, with coefficients of determination above 0.999 on both validation and test data. The work is a computational modeling study rather than an experiment, and its claims rest entirely on numerical simulation of transformed governing equations.

The physical setting is stagnation point flow of a steady, incompressible, two-dimensional Boger nanofluid over a stretching sheet. Boger fluids are dilute polymer solutions that behave like Newtonian solvents with an added elastic component, which makes them useful models for polymer extrusion, fiber spinning and coating processes where elasticity must be controlled alongside heat transfer. The Riga surface adds an electromagnetic body force through an array of magnets and electrodes, while the surrounding medium is treated as porous. The model also incorporates a heat source or sink, Soret and Dufour cross-diffusion effects, and a first-order chemical reaction in the concentration field.

To make the problem tractable, the authors used similarity transformations to reduce the governing partial differential equations for continuity, momentum, energy and concentration into a set of ordinary differential equations with the corresponding boundary conditions. These were solved with a fourth-fifth order Runge-Kutta-Fehlberg scheme combined with the shooting technique, using an error tolerance of 10 to the power of minus 6 and a step size of 0.001. The team validated their solver against previously published skin friction values, reproducing reference results to within roughly one part in ten thousand across the tested parameter range, and carried out grid independence, domain independence and convergence checks showing that the target quantities stabilize as the mesh is refined and the far-field boundary is extended.

The reported parameter trends follow from the structure of the momentum, energy and species equations. Increasing the velocity ratio parameter raises the velocity profile, because when the free-stream velocity exceeds the stretching velocity the favorable momentum balance produces a conventional boundary-layer structure and steepens the wall velocity gradient. Raising the solvent fraction parameter, which measures the Newtonian solvent contribution in the polymer-solvent mixture, also increases velocity, since a larger solvent share reduces the influence of elastic resistance and lets the fluid respond more readily to stretching. In contrast, increasing the relaxation time ratio strengthens the elastic contribution of the Boger fluid, suppressing velocity within the boundary layer as the fluid exhibits a stronger memory effect.

The electromagnetic and porous effects act in opposition. A larger modified Hartmann number strengthens the Lorentz forcing generated by the Riga surface and increases the velocity profile, while a larger porous parameter strengthens Darcy resistance and decreases it. The authors highlight this competition as the study’s central physical insight: in a Boger fluid, elastic relaxation and the solvent fraction can partially offset porous resistance, producing a nonlinear boundary-layer response that they argue is invisible when these parameters are examined in isolation. Streamline and isothermal contour plots for cases with and without the porous medium illustrate how porous resistance redistributes the local velocity field and the convective pattern of energy transfer.

On the thermal and concentration side, increasing the heat source parameter injects additional thermal energy into the fluid and raises the temperature profile, whereas a heat sink removes energy and suppresses it. A larger Dufour number, which quantifies how concentration gradients drive an additional heat flux, also enhances the temperature distribution. For mass transfer, a stronger chemical reaction consumes the diffusing species and lowers the concentration field, while a larger Soret number, representing thermally driven mass diffusion, enhances the concentration profile. At the level of the engineering coefficients, the skin friction coefficient falls as the porous parameter and solid volume fraction rise, the Nusselt number decreases with increasing Dufour number, and the Sherwood number drops as the Soret number and solid volume fraction increase.

The machine-learning component is a Kolmogorov-Arnold network, an architecture built on the Kolmogorov-Arnold representation theorem, which states that any continuous multivariate function can be written as a composition of univariate functions. Unlike conventional multilayer perceptrons, which apply fixed activation functions such as ReLU at nodes and combine them with linear weights, KANs place learnable spline-based functions on the network edges. The authors argue that this adaptive, potentially interpretable functional representation suits the smooth nonlinear relationships between the dimensionless flow parameters and the engineering responses, and helps avoid problems such as vanishing gradients and saturation.

The specific network used a 10-5-3 configuration: ten dimensionless input parameters, one hidden KAN layer of width five, and three outputs corresponding to the skin friction coefficient, Nusselt number and Sherwood number. Training data were generated by one-factor-at-a-time variation of the ten inputs across 300 levels each, yielding 9,000 numerical solutions from the RKF-45 shooting solver, or 3,000 samples per target. The data were split 70 percent for training, 15 percent for validation and 15 percent for testing with a fixed random seed of 42. The model was trained for 300 epochs with the Adam optimizer, a learning rate of 0.001, mean squared error loss and a batch size of 64, without early stopping or a learning-rate scheduler.

Performance was strong. The coefficient of determination on the test and validation sets was 0.999889 and 0.999854 for skin friction, 0.999897 and 0.999951 for the Nusselt number, and 0.999452 and 0.999509 for the Sherwood number, with mean squared errors between 0.000029 and 0.000158. Predicted values overlapped actual values in validation plots, and error histograms clustered near zero without long tails or skewness, which the authors interpret as evidence of unbiased, stable predictions without significant overfitting. Loss curves declined rapidly and then stabilized over the training epochs.

In comparisons on the same test split, the KAN substantially outperformed a multilayer perceptron and support vector regression on all three targets, and was competitive with XGBoost. Random Forest achieved the lowest errors on this dataset, with coefficients of determination up to 0.999985, and the authors are explicit that KAN’s advantage should be understood through its adaptive spline-based functional form and competitive surrogate accuracy rather than universal superiority over all machine-learning methods. The practical payoff is speed: evaluating 3,000 parameter cases took the numerical scheme about 3,034 seconds, roughly 1.01 seconds per case, while the trained KAN completed the same evaluation in about 0.064 seconds, an inference-phase speedup of roughly 47,668 times, excluding the offline cost of training. The authors note limitations, including the one-factor-at-a-time sampling design, which does not support feature-attribution or parameter-interaction analyses such as SHAP or permutation importance, and they propose multidimensional datasets with simultaneously varied parameters for future work. Within its scope, the study offers a fast surrogate for exploring multiparameter design spaces in applications such as viscoelastic coating and electrohydrodynamic pump control, where engineers need rapid evaluation of heat transfer targets without excessive wall shear stress.

Subject of Research: Kolmogorov-Arnold neural network surrogate modeling of electromagnetohydrodynamic Boger nanofluid stagnation point flow over a Riga surface with heat source/sink, Soret-Dufour, chemical reaction and porous medium effects

Article Title: Kolmogorov-Arnold neural network technique for predicting heat and mass transfer analysis of Boger nanofluid flow over a Riga surface

Article References: Vinutha, K., Madhukesh, J. K., Khan, U., Jagannatha, C. G., Adnan, Viharika, J. U., & Elattar, S. (2026). Kolmogorov-Arnold neural network technique for predicting heat and mass transfer analysis of Boger nanofluid flow over a Riga surface. Frontiers in Physics, 14, Article 1924652. https://doi.org/10.3389/fphy.2026.1924652

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1924652

Keywords: Kolmogorov-Arnold networks, Boger nanofluid, Riga surface, stagnation point flow, heat transfer, mass transfer, porous medium, Soret effect, Dufour effect, skin friction coefficient, Nusselt number, surrogate modeling