## 2 Materials and Methods  
The mechanical system \(\mathbf{\Omega}\) was considered. On the part of the boundary of \(\mathbf{\Omega}\), displacements \(\mathbf{u^{0}}\) are known. On the rest of the boundary, acting forces are known. Under the influence of forces, the mechanical system deforms. The motion of system \(\mathbf{\Omega}\) is defined by displacement \(\mathbf{u}\) in time \(t\): \(\mathbf{u}(\mathbf{x},t)\), where \(\mathbf{x}\in\mathbf{\Omega}\). In the hybrid simulation, the mechanical system is divided into analytical subsystem \(\mathbf{\Omega^{A}}\) and physical subsystem \(\mathbf{\Omega^{P}}\), as presented in Figure 2, where \(\mathbf{\Omega^{A}}\cup\mathbf{\Omega^{P}}=\mathbf{\Omega}\). The division between both subsystems is defined by interface \(\mathbf{I}\). Boundary conditions for interface boundaries of subsystems: \(\mathbf{I^{A}}\) and \(\mathbf{I^{P}}\) are defined by actuators and force sensors. On the boundary \(\mathbf{I^{A}}\), forces are known (measured by the sensors), and on the boundary \(\mathbf{I^{P}}\), displacements are known (imposed by the actuators) [2; 3; 13]. The equation of motion of mechanical system \(\mathbf{\Omega}\) is as follows:  
\[(\mathbf{M^{A}}+\mathbf{M^{P}})\ddot{\mathbf{u}}(t)+(\mathbf{C^{A}}+\mathbf{C^{P} })\dot{\mathbf{u}}(t)+(\mathbf{K^{A}}(t)+\mathbf{K^{P}}(t))\mathbf{u}(t)= \mathbf{F}(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are mass, damping, and stiffness matrices, respectively. Superscript \(\mathbf{A}\) denotes an analytical subsystem and superscript \(\mathbf{P}\) physical subsystem. If the system is not linear elastic, the stiffness matrix is time-dependent. \(\mathbf{F}\) is the vector of excitation forces [3]. Because forces developed in the physical subsystem are measured, Equation (1) takes the form:  
\[\mathbf{M^{A}}\ddot{\mathbf{u}}(t)+\mathbf{C^{A}}\dot{\mathbf{u}}(t)+\mathbf{ K^{A}}(t)\mathbf{u}(t)+\mathbf{r^{P}}=\mathbf{F}(t) \tag{2}\]  
where the vector of measured forces \(\mathbf{r^{P}}\) corresponds to the sum of inertia, damping, and stiffness forces:  
\[\mathbf{r^{P}}=\mathbf{M^{P}}\ddot{\mathbf{u}}(t)+\mathbf{C^{P}}\dot{\mathbf{ u}}(t)+\mathbf{K^{P}}(t)\mathbf{u}(t). \tag{3}\]  
Normally in hybrid simulation, Equation (2) is solved step-after-step using explicit or implicit integration schemes. These algorithms in detail can be found in [5]. The artificial neural network, to substitute the finite element model, to be built and trained, should approximate the following function\(f\):  
\[\mathbf{u^{I}_{i+1}}=f\big{(}\mathbf{r^{P}_{i}},\mathbf{F}_{i+1},\mathbf{u^{I }_{i}},\mathbf{u^{I}_{i-1}},\mathbf{u^{I}_{i-2}}\big{)} \tag{4}\]  
where \(\mathbf{u^{I}}\) is the displacement vector of the interface \(\mathbf{I}\). The five inputs to the ANN are result of the fact that in integration schemes for hybrid simulation, the displacement vector of the whole system for time step \(i+1\) is computed from (a) measured forces in the physical subsystem in time step \(i\), (b) excitation force vector for time step \(i+1\), (c) displacements of the whole system in three previous discrete states of time before \(i+1\) (sufficient approximation of velocity and acceleration of the system), (d) system matrices (mass, damping, and stiffness), where the stiffness matrix of the physical subsystem is approximated (at least roughly). As one can see from Equation (4), the ANN does not compute the displacements of the whole system. In order to improve computational efficiency, only displacements of the interface \(\mathbf{I}\) are calculated. This information is sufficient to perform hybrid simulation, and after it, the complete results can be easily reproduced numerically, knowing vectors \(\mathbf{F}(t)\) and \(\mathbf{r^{P}}(t)\). The ANN is built based on the offline FE model from which training data is generated as a set of samples. Each sample consists of a set of five input values and the correct output value (displacement of the interface for the next iteration).