  One of the most common and universal problems in science is to investigate a
function. The prediction can be made by an Artificial Neural Network (ANN) or a
mathematical model. Both approaches have their advantages and disadvantages.
Mathematical models were sought as more trustworthy as their prediction is
based on the laws of physics expressed in the form of mathematical equations.
However, the majority of existing mathematical models include different
empirical parameters, and both approaches inherit inevitable experimental
errors. At the same time, the approximation of neural networks can reproduce
the solution extremely well if fed with a sufficient amount of data. The
difference is that an ANN requires big data to build its accurate approximation
whereas a typical mathematical model needs just several data points to estimate
an empirical constant. Therefore, the common problem that developer meet is the
inaccuracy of mathematical models and artificial neural network. An another
common challenge is the computational complexity of the mathematical models, or
lack of data for a sufficient precision of the Artificial Neural Networks. In
the presented paper those problems are addressed using the integration of a
mathematical model with an artificial neural network. In the presented
analysis, an ANN predicts just a part of the mathematical model and its weights
and biases are adjusted based on the output of the mathematical model. The
performance of Integrated Mathematical modeling - Artificial Neural Network
(IMANN) is compared to a Dense Neural Network (DNN) with the use of the
benchmarking functions. The obtained calculation results indicate that such an
approach could lead to an increase of precision as well as limiting the
data-set required for learning.

Neural and Evolutionary Computing