## 3 Numerical Uncertainty Analysis
### Evaluation of the Results
#### 3.2.3 Natural Frequency  
In terms of the natural frequency of the single-mass oscillator, the difference between the identified and the true value is used to evaluate the correctness of a trained model. The true eigenfrequency of an oscillator from the training data set can be calculated using the following equation:  
\[f_{true}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}. \tag{12}\]  
Since the mass, \(m\), and stiffness, \(k\), are randomly generated, the true eigenfrequency from the training set shows a symmetrical distribution in the histogram Figure 7.  
Figure 7: Distribution of the natural frequency from training data set.  
The eigenfrequencies \(f_{rec}\) of the reconstructed data were obtained from the peaks of the spectra of the acceleration signals. The natural frequency difference reads  
\[\Delta f=f_{rec}-f_{true}. \tag{13}\]  
The distribution of \(\Delta f\) with the initial state \(\mathbf{x_{1}^{\star}}(0)\) was evaluated in order to identify the dependency of natural frequency on the polynomial order of the candidate functions and noise intensity, because most reconstructions of the signals with \(\mathbf{x_{1}^{\star}}(0)\) succeed, as discussed in Section 3.2.2. In Figure 8, the natural frequencies of reconstructed accelerations with both methods have similar distributions up to polynomial order 4. With stronger noise, the error in the natural frequency increases continuously without a significant change in the range between quantiles. In the simulation with \(\epsilon_{N}=0.01\) and \(ord=4\), the medians of \(\Delta f\) for the LSST and LSPL methods are equal to \(-0.0035\) and \(-0.0034\), respectively. The results of the LSST method in polynomial order 5 differ obviously from the other groups of data, with a larger error in the median and wider range between the 0.75 and 0.25 quantiles. Its median reaches \(-0.0158\) with \(\epsilon_{N}=0.01\). This phenomenon corresponds well with Figure 6a,b regarding the divergence ratio of both methods with the initial state \(\mathbf{x_{1}^{\star}}(0)\), which means the missing samples of divergent reconstructions cause an additional error to the distribution of the natural frequency.  
In order to reduce the complexity of the detailed comparison of the natural frequency, in the following analysis, only the results of the training data sets \(Ord=3\) and \(\epsilon_{N}=0.005\) are shown as an example. The natural frequency differences \(\Delta f\) of both methods show a similar distribution for the initial condition \(\mathbf{x_{1}^{\star}}(0)\) as shown in Figure 9a,b. The maximum deviation corresponds to the resolution of the frequency vector after the fast Fourier transformation, i.e., 0.01 Hz. With the initial value \(\mathbf{x_{2}^{\star}}(0)\), the distribution of the natural frequencies is flatter and wider when using LSST instead of using LSPL; see Figure 9c. Furthermore, the peak in Figure 9c is underestimated from the true value zero by 0.01 Hz. Despite the missing reconstructed signals with the LSSL method and the initial state \(\mathbf{x_{2}^{\star}}(0)\), a conclusion can be made that the LSPL method has a more precise and general prediction of the natural frequency of oscillators than LSST, especially with a large initial state.  
Figure 8: Median and quantile of error in frequency with the initial state \(\mathbf{x_{1}^{\star}}(0)\) and the two methods. (**a**) LSST and (**b**) LSPL.