Fig. 4: Integration of dimensionless learning with symmetric invariant SINDy for identifying the Navier–Stokes equation with Reynold number.

a Original data are generated from parametric simulations. To achieve symmetric invariance, another set of transformed data is obtained by flipping the original data along y = x. b The original and transformed data are concatenated for symmetric invariant SINDy, which implicitly incorporates symmetric invariance into SINDy to ensure that symmetric invariant terms have the same coefficients. c The identified temporary governing equations for each simulation case were obtained by optimizing the symmetric invariant SINDy. Some of the coefficients are close to constant, while others vary depending on the simulation case. All the other candidate terms have zero coefficients. d Dimensionless learning is applied to identify an explicit expression for the varying coefficients. The parametric space to be explored includes five parameters. By incorporating dimensional invariance, we need to optimize basis coefficients γ and fitting coefficient β. e Substituting the discovered regression coefficients (\({{{{{{{\rm{1/Re}}}}}}}}\)) into the temporary governing equation. In this step, a consistent dimensionally homogeneous governing equation, which is identical to the Navier–Stokes equation in the vorticity form, is obtained.