Machine learning-based Bayesian regularization algorithm for thermal analysis of tri-hybrid nanofluid flow over a stretched sheet

Tri-hybrid nanofluids, which consist of three different nanoparticles dispersed in a base fluid, have shown excessive potential as a new generation of thermal materials because of their exceptional heat transfer properties. These fluids are particularly useful for next-generation thermal systems, such as microfluidic cooling devices, solar collectors, aerospace heat exchangers, and nuclear reactor cooling systems. In this article, the heat transfer characteristics of a Williamson-type tri-hybrid nanofluid over a bidirectional stretching sheet under the influence of thermal radiation, magnetic field, and porosity are explored. The main objective is to build and test an effective hybrid framework that combines conventional numerical methods with artificial intelligence to reliably forecast the flow and thermal properties of complicated nanofluid systems. The primary objective is to develop and validate a hybrid numerical-machine learning algorithm that integrates MATLAB's BVP4C solver and an Artificial Neural Network with Bayesian Regularization (ANN-BRA) for estimating velocity and temperature distributions under varying physical parameters. The partial differential equations governing (PDEs) are transformed to ordinary differential equations (ODEs) via similarity variables and numerically resolved to train the ANN. This is the first use of ANN-BRA trained on BVP4C-generated data to simulate tri-hybrid nanofluids inside a Williamson fluid framework. The ANN-BRA model obtains exact regression (R = 1) and a mean squared error (MSE) less than 10⁻¹¹, which reflects high precision and generalization. Outcomes indicate that an increased magnetic field ($M$) and porosity ($\varphi$) decrease the flow velocity, while an elevated volume fraction of nanoparticles (${\varphi }_n$) strengthens thermal boundary layers. The Eckert number ($\mathrm{Ec}$), Biot number ($\mathrm{Bi}$), and thermal radiation ($\mathrm{Rd}$) are indicated to have strong influences on heat transfer rates. This work presents both numerical and physical understanding of tri-hybrid nanofluid behavior and a useful modeling methodology to optimize practical thermal engineering applications.