## 2 Materials and Methods  
In practice, mass operators are built using accelerations and stresses or loads. In this case, the accelerations, loads, and stresses were computed using the finite element method [11; 12]. Once the mass operators were built and verified, they were deployed during the test to compute stresses and loads based on measured accelerations. In this paper, only ANNs are considered for creating mass operators and MATLAB 2018b (Mathworks, Natick, MA, USA) was used to create and train the proposed ANN. A prediction of the structure's response is indeed provided by the finite element analysis (FEA) data. The FE model in this specific case needed to comprise two main aspects. One aspect was the accurate modeling of NIRSpec's ceramic bench, as this is the instrument being designed by AIRBUS and is one of the most sensitive parts. The other aspect was the compliance of the surrounding structure, i.e., the structural elements onto which NIRSpec was mounted. The latter aspect was addressed by conducting a coupled load analysis (CLA), where NIRSpec was considered via a standard FE model and the remaining ones, for instance, the instrument module, the optical telescope, and space craft elements, were represented through stiffness representative super-elements. From this CLA, only the forces and moments acting on NIRSpec were derived. In order to have the full picture, phase information was considered as well in order to depict the dynamical compliance of the overall structure. Next, the interface load input was condensed by only considering frequency steps in the vicinity of peaks in direct response as well as in cross-response. This condensation reduced the input size from roughly 42,000 frequency support points down to 700 (1.7%). This, evidently, reduced the computational efforts on our detailed FE model in terms of stress calculation and post-processing considerably, thereby allowing detailed investigation at mechanically interesting frequency ranges to address the first aspect of our FE approach, namely the detailed stress prediction on our ceramic bench. However, to use FE models for deriving predictions, one has to assume damping. This highlights the major contributor to potential discrepancies, together with overall system nonlinearities stemming from interface mechanics, secondary structures like harnesses, implemented damper elements, and the like. This infers that the real-life physical state of the structure, namely the interface forces and stresses, needs to be predicted live during the vibration test based on the actual response of the system. Only then will it be possible to adequately adapt the testing level to protect the structure. Unfortunately, only a limited set of data about the state of the structure is available, such as the measured accelerations at discrete locations on the structure [2]. From these accelerations, the stresses or forces working in the tested structure need to be derived using a dedicated method, such as mass operators or ANNs, which is the subject of this investigation. Any method must meet the following requirements:  
* Robustness with regard to natural frequency shifts during testing as compared to the ones computed with FEA. * Fast deployment during vibration testing in order to react adequately to the resulting responses; the effort of the post-processing model during the test must be small. * Accuracy for stresses and the interface forces. In this investigation, the von Mises yield criterion was used to monitor the state of the NIRSpec module. * Fast training, data acquisition, processing, and configuration. * Robustness with regard to the lack of sensors. As the number of available sensors during the test is restricted, the method must be accurate with a limited number of sensors and, at the same time, potentially inconveniently positioned ones [2; 3]. Artificial neural networks (ANNs) mimic the human brain in its mechanisms to transfer data from one neuron to another (see Figure 2).