Modeling on magnetohydrodynamic Stokes flow using machine learning and curve fitting

Abstract In this study, neural network (NN) and curve fitting modeling of fluid flow characteristics of the magnetohydrodynamic (MHD) Stokes flow in a lid-driven cavity are utilized. Firstly, the MHD Stokes flow equations are numerically solved by the method of approximate particular solution for the variations of Hartmann number $$M\in [1,120]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>120</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> and the inclination angle $$a \in [0, \pi ]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . The essential data for modeling are extracted from the numerical results. The inputs are M and a , and the outputs are the infinity norm of stream function $$\psi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> , v velocity component, vorticity $$\omega$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> and the minimum value of u velocity. In modeling of these outputs, the distinct curve fitting functions are examined. NN is employed for different layer numbers and data partitions. It is obtained that the increase in the number of the hidden layers gives less error and locally weighted quadratic regression fit captures the best behavior in curve fitting. The usage of modeling allows us to be independent from the repeated numerical calculations. The capability of trilayer NN for modeling $$\psi ,u,v,\omega$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ψ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> in the entire region is also shown.