## 2 Principle of the Proposed Hybrid Seismic Analysis  
The hybrid seismic analysis proposed in this paper is inspired by the hybrid simulation developed by [PERSON] et al. [15; 16] for real-time pseudo-dynamic testing. The target structure is divided into a numerical model and a physically tested model. The latter one may be a structural component or a group of components as a sub-structure. It is an efficient and most economical alternative to testing an entire structure. The control and the limitation of the numerical error require the choice of an appropriate numerical integration method [17]. [PERSON] et al. [18] proposed a mixed implicit-explicit direct integration method that incorporates the physically measured restoring force into the numerically solved second order equation of motion. It is a predictor-one-corrector displacement method based on the operator splitting technique proposed by [PERSON] et al. [19] and the implicit Newmark-\(\beta\) method [20]. It was proven that this method (referred to as the OpS method in the remainder of this paper) is unconditionally stable for pseudo-dynamic testing of structures with softening-type nonlinearities.  
Instead of measuring the restoring force developed in the tested physical model, the hybrid seismic analysis proposed in this paper evaluates it using an MLM, previously trained to simulate the hysteretic behavior of the target component or group of components (here, isolation layer). The commonly used OpS method is selected as a suitable numerical integration algorithm for NTHA. Figure 1 summarizes the principle of the proposed hybrid seismic analysis. Assuming a building structure and an input ground motion, the predictor displacement vector is first evaluated at the current time step following the algorithm of the OpS method. Restoring forces of structural components with already mastered hysteresis behavior, such as reinforced concrete or steel columns and beams, are evaluated using available validated analytical models. Some existing or newly developed materials, components, or devices may have an excessively complicated analytical model and/or still not-exhaustive understanding of their true behavior among researchers and practitioner communities. Instead of adopting simplified analytical models based on many assumptions, an MLM previously trained by available experimental data can capture the true behavior of the target component or group of components and then be used to make predictions within the time integration algorithm. It would predict the component restoring force at the next time step, based on the displacement time history and the current predictor displacement. Then, the full seismic response at the next time step is computed as performed in the OpS method. The same process is repeated till the end of the analysis.  
The concept of the proposed hybrid seismic analysis is general to any structure and relies on developing MLMs beforehand. The development of MLMs from experimental data is nonrepetitive since it can be saved and reused in any new structure containing the same target components. For instance, the same isolation device can be used in different buildings but with different configurations and numbers. New experimental data of the same nature may be used to update the MLM, though.  
In addition to the numerical error of the selected integration method, making predictions by an MLM introduces an inevitable uncertainty. It is due to the epistemic uncertainty of the experimental data used for the training process and the aleatory uncertainty of ANNs explained further in this paper. Since artificial data are generated in this study from NTHAs, only the intrinsic uncertainty of MLMs needs to be dealt with.