  This paper deals with the following important research question.
Traditionally, the neural network employs non-linear activation functions
concatenated with linear operators to approximate a given physical phenomenon.
They "fill the space" with the concatenations of the activation functions and
linear operators and adjust their coefficients to approximate the physical
phenomena. We claim that it is better to "fill the space" with linear
combinations of smooth higher-order B-splines base functions as employed by
isogeometric analysis and utilize the neural networks to adjust the
coefficients of linear combinations. In other words, the possibilities of using
neural networks for approximating the B-spline base functions' coefficients and
by approximating the solution directly are evaluated. Solving differential
equations with neural networks has been proposed by Maziar Raissi et al. in
2017 by introducing Physics-informed Neural Networks (PINN), which naturally
encode underlying physical laws as prior information. Approximation of
coefficients using a function as an input leverages the well-known capability
of neural networks being universal function approximators. In essence, in the
PINN approach the network approximates the value of the given field at a given
point. We present an alternative approach, where the physcial quantity is
approximated as a linear combination of smooth B-spline basis functions, and
the neural network approximates the coefficients of B-splines. This research
compares results from the DNN approximating the coefficients of the linear
combination of B-spline basis functions, with the DNN approximating the
solution directly. We show that our approach is cheaper and more accurate when
approximating smooth physical fields.
