In the present article, we introduce a feedforward neural network with four layers by considering the neural network interpolation operators induced by smooth ramp functions proposed by Qian and Yu (Anal. Appl. 20(4), 791–813 (2022)) by means of the Lagrange polynomials of degree \(\mathfrak {s}\) . We determine the rate of approximation of these operators by means of the modulus of continuity for continuous functions in a compact interval of \(\mathbb {R},\) and establish the converse theorem by proving two crucial inequalities concerning the derivative of the operators, and using the Peetre’s K- functional and the Berens-Lorentz lemma. A Voronovskaja type asymptotic theorem is also discussed. We extend our aforementioned operators to the multivariate case and investigate the direct and converse results of approximation by these operators for multivariate continuous functions in a box type domain in \(\mathbb {R}^r\) . Furthermore, we study the pointwise and uniform convergence estimates in the iterated approximation and complex approximation for the multivariate operators.
