## 4 Discussion  
The provided two numerical examples and one experimental example proved that ANNs could be utilized in the hybrid simulation to substitute the finite element model, to speed-up the computations.  
In the first numerical example, two-layer ANN with only eight hidden neurons could be utilized, providing the high accuracy of the results. About 50% of computational time reduction was achieved in a PC simulation. If taking this as a measure of model order reduction, one could expect that in real-time computations on given hardware, the time step of hybrid simulation could be reduced by half by utilizing ANN. The force \(F\) had two not constant parameters--amplitude \(A\) and frequency \(f\). By performing a set of simulations to generate the reference data (with discrete values of amplitude and linearly swept-frequency), the ANN could generalize it and provide accurate results for any values of \(f\) and \(A\) (within the training range), constant or not.  
In the second numerical example, a much more complex mechanical system was presented, containing 73 DOFs, while four of them make up the interface \(\mathbf{I}\). Therefore, the ANN got more inputs and outputs than in the first example. The ANN search algorithm resulted in 27 neurons in the hidden layer. The accuracy of the results obtained by ANN compared to the accurate FE model is satisfying, taking into account the reduction of computational time by five times.  
The third example (experimental) proved that ANNs could be utilized in real-time hybrid simulations. The ANN search algorithm was executed twice resulting in 2 and 20 hidden neurons, testifying strongly to non-deterministic (random initial weights, random division of reference data to training, test, and validation) character of the algorithm (it may also indicate that \(n=20\) was too low in this case). Implementing the ANNs allowed to reduce the time step of the simulation to 0.5 ms, which would be impossible without model order reduction with the utilized microcontroller (the same  
Figure 17: Time course of force F2: (**a**) load case 1, (**b**) load case 2.  
Figure 18: Elongation of the shock absorber: (**a**) load case 1, (**b**) load case 2.  
hardware was used by author in [3] to perform different hybrid simulation of system with 17 DOFs, and the lowest time step that could be achieved was 5 ms). Therefore, it is hard to measure the accuracy of the obtained results from the reduced models (the nonlinearity of the shock absorber cannot be simulated accurately). However, by comparing the results from two significantly different ANNs (presented in Figure 16), it can be said that they are very similar, which can lead to a conclusion about their accuracy. Furthermore, both ANNs demonstrated high accuracy in testing simulations during the execution of the searching algorithm. The ANN with 20 hidden neurons appeared to be more accurate, taking into account more linear regression of the elongation (Figure 16) in the first phase of loading the mechanical system.  
All the examples proved the important assumption that by generating reference data from a proper set of linear simulations, the ANN was able to generalize it and provide accurate results in hybrid simulations with the nonlinear physical subsystem.  
The conducted research is a starting point for future research on the development and applications of artificial intelligence techniques in hybrid simulations. The obtained results and conclusions drawn from them indicate the possibilities of future research regarding testing different architectures of neural networks and deep learning techniques in order to reduce the model order in hybrid simulations.