The design of integrated numerical computing through back-propagated neural networks with Levenberg-Marquard system (BNN-LMS) is presented to explore the fluid mechanics problems governing the system of heat transfer between two porous parallel plates of steady nanofluids (HTPSNF) under the stimulus of thermophoretic and Brownian motion. By introducing the similarity transformations, the original system model HTPSNF in terms of PDEs is converted to nonlinear ODEs. Strength of Homotopy Analysis Method (HAM) is utilized the governing equations of original model HTPSNF to obtain the data set. Reference collection for the suggest BNN-LMS scheme is originated in terms of various scenarios associated HTPSNF such as Porosity parameter, Schmidt number, Brownian parameter, viscosity parameter, Prandlt number and thermophoric parameter. To uphold the trueness of the suggest BNN-LMS, the validation, training and testing process of BNN-LMS are accomplished to govern the estimate solution of HTPSNF for various cases and evaluation with reference results. The comparative studies and performance analyses based on outcomes of MSE, error histograms, correlation and regression intimate the effectiveness and virtue of designed LMBNN technique. Mean Square Errors in the ranges of 10⁻⁰⁷ to 10⁻¹⁴ confirm the perfection of the presented methodology for the closed correspondence between suggested and reference results.

提出了通过带有 Levenberg-Marquard 系统 (BNN-LMS) 的反向传播神经网络进行集成数值计算的设计，以探索在激励下控制两个多孔平行稳态纳米流体板 (HTPSNF) 之间的传热系统的流体力学问题热泳运动和布朗运动。通过引入相似变换，原始系统模型 HTPSNF 在 PDE 方面被转换为非线性 ODE。强度同伦分析法（HAM）利用原始模型HTPSNF的控制方程得到数据集。建议 BNN-LMS 方案的参考集合起源于与 HTPSNF 相关的各种场景，例如孔隙率参数、施密特数、布朗参数、粘度参数、普兰特数和热泳参数。为保证所建议的 BNN-LMS 的真实性，完成了 BNN-LMS 的验证、训练和测试过程，以管理 HTPSNF 对各种情况的估计解决方案，并以参考结果进行评估。基于 MSE 结果、误差直方图、相关性和回归的比较研究和性能分析表明设计的 LMBNN 技术的有效性和优点。10 范围内的均方误差^-07到 10 ^-14证实了所提出的用于建议和参考结果之间封闭对应的方法的完善。