  To combine a feedforward neural network (FNN) and Lie group (symmetry) theory
of differential equations (DEs), an alternative artificial NN approach is
proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs).
Introducing the Lie group expressions of the solution, the trial solution of
ODEs is split into two parts. The first part is a solution of other ODEs with
initial values of original IVP. This is easily solved using the Lie group and
known symbolic or numerical methods without any network parameters (weights and
biases). The second part consists of an FNN with adjustable parameters. This is
trained using the error back propagation method by minimizing an error (loss)
function and updating the parameters. The method significantly reduces the
number of the trainable parameters and can more quickly and accurately learn
the real solution, compared to the existing similar methods. The numerical
method is applied to several cases, including physical oscillation problems.
The results have been graphically represented, and some conclusions have been
made.

Numerical Analysis;Numerical Analysis