  This is paper for the smooth function approximation by neural networks (NN).
Mathematical or physical functions can be replaced by NN models through
regression. In this study, we get NNs that generate highly accurate and highly
smooth function, which only comprised of a few weight parameters, through
discussing a few topics about regression. First, we reinterpret inside of NNs
for regression; consequently, we propose a new activation function--integrated
sigmoid linear unit (ISLU). Then special charateristics of metadata for
regression, which is different from other data like image or sound, is
discussed for improving the performance of neural networks. Finally, the one of
a simple hierarchical NN that generate models substituting mathematical
function is presented, and the new batch concept ``meta-batch" which improves
the performance of NN several times more is introduced.
  The new activation function, meta-batch method, features of numerical data,
meta-augmentation with metaparameters, and a structure of NN generating a
compact multi-layer perceptron(MLP) are essential in this study.
