Journal of Babylon University/Pure and Applied Sciences/ No.(1)/ Vol.(26): 2018 77 The Essential Order of       Approximation Using Regular Neural Networks Eman S. Bhaya Omar A. Al-sammak University of Babylon, College of Education for Pure Sciences, Department of Mathematics emanbhaya@itnet.uobabylon.edu.iq omar.alhelly@itnet.uobabylon.edu.iq Abstract This paper is concerning with essential degree of approximation using regular neural networks and how a multivariate function in    spaces for  can be approximated using a forward regular neural network. So, we can have the essential approximation ability of a multivariate function in    spaces for  using regular FFN. Keywords. Neural network approximation, Modulus of smoothness,    Spaces, best approximation. خلصة ال المتعددةدوالمكن تقريب الف ية المنتظمة ، وكيكة العصبيسي بأستخدام الشبلساحث درجة التقريب ا في هذا الب درسنا ت في فضاء ا المتغير   عندما ى مبرهناتحصول عم ال بامكانناذلكمية المنتظمة ، وكلماكة العصبية ا بأستخدام الشب ت في فضاء اب المتعددة المتغيرة ونظرية تكافؤ لمتقري مباشرة وعكسي   عندما لماميةكة العصبية ا بأستخدام الشب المنتظمة . لمفتاحية :ت اما الكمس العصبية ،مقيا الشبكة ا تقريبءات لنعومة ، فضا   فضل .تقريب ال ، ال 1. Introduction "Various papers on feasibility of approximation by forward neural networks have been made in past years (see [Cardaliaguet & Euvrard,1992; Chen, 1995; Chen, 1994; Chui, 1992; Cybenko, 1989; Gallant, ,1992; Hornik, 1989; Hornik, & Stinchombe, 1990; Leshno et al., 1993; Mhaskr & Michelli,1992]. The most important result among these papers is that : If we have a continuous function with multivariable and compact domain subset of   there exist a feed forward neural networks (FNNs) as an approximation for it . By a sigmoidal we can be approximated arbitrarily well . A three-layer of the FNNs with  input units and one hidden and one output units can be mathematically expressed as     ∑   (〈    〉  )          where         is the threshold,              are connection weights of neuron  in the hidden layer with input neurons,    are the connection strength of neuron  with the output neuron, and  is the sigmoidal activation function used in the network. In this paper we prove direct and inverse estimation and saturation problem for the approximation of multivariate function in    spaces for  using a forward regular neural network .