# Solid-State Lithium Battery Cycle Life Prediction Using Machine Learning
## 4 Discussion  
In this work, we successfully assembled 12 solid-state polymer batteries with cycle lives ranging from 70 to 213 cycles to build the training datasets. Then, we performed battery \(RUL\) prediction by using the symbolic regression method with 11 highly correlated features as model inputs. These 11 physical quantities, such as capacity, energy, and charge/discharge voltage differences, were designed and screened by analyzing their different degradation mechanisms and charging/discharging curves compared with liquid lithium batteries. After populations of breed, mutation, and evolution training, a high number of formulas were generated. Among these formulas, only those with high accuracies and low complexities can be explained for researchers to better understand the relationship of solid-state batteries to their characteristics. The six mathematical formulas that met the criteria of simplicity and accuracy are shown in Table 2. The first formula, X5, is the integral of the capacity and voltage charging curve of each battery (measured in units of mWh), which represents the amount of energy charged into the battery. This value is more than just capacity because it also takes inner resistance into account. The increase in impedance will affect the voltage value of each capacity point on the charge/discharge curve, thus affecting the integral energy. It could roughly predict the value of \(RUL\) with the simplest form. The second formula, min(X29, X5), is the smaller value between charging \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Computational**} & \multicolumn{4}{c}{**Value**} \\ \cline{2-5}  & **SVR** & **GPR** & **EN** & **SR** \\ \hline Training time & 0.1800 s & 3.1527 s & 0.0021 s & 13.2677 s \\ Inference time & 0.0023 s & 0.0078 s & 0.0013 s & 0.0017 s \\ Total used memory & 0.5975 GB & 0.5999 GB & 0.5963 GB & 0.6346 GB \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The computational characteristics for the SR model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{\(R^{2}\)} & \multirow{2}{*}{**RMSE**} & \(RUL_{\text{Error}53}\) & \(RUL_{\text{Error}105}\) & \(RUL_{\text{Error}179}\) \\  & & & (\(RUL_{\text{True}}\) = 157) & (\(RUL_{\text{True}}\) = 105) & (\(RUL_{\text{True}}\) = 31) \\ \hline SVR & 0.81 & 26.50 & 2 & 6 & 27 \\ GPR & 0.83 & 24.64 & 23 & 7 & 13 \\ EN & 0.69 & 33.27 & 30 & 8 & 27 \\ SR & 0.91 & 18.17 & 4 & 4 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of \(RUL\) prediction results of GPR, SVR, EN, and SR (\(RUL_{\text{Error}}\) = \(RUL_{\text{prediction}}\) – \(RUL_{\text{True}}\)!). time and integral energy. The charging time is proportional to the capacity because of the constant current charging operation. This formula shows that the battery capacity and energy will decrease with the decrease in its _RUL_. The third formula is the battery discharge capacity (measured in units of mAh) multiplied by a factor of 0.820, and it could basically predict the value of _RUL_. It proves that capacity degradation also takes up a major factor for the aging of solid-state batteries. The fourth formula, min (X20, 0.798 \(\times\) X1), takes the smaller value between our proposed kurtosis feature and the discharge capacity multiplied by a factor. X20 is a measure of polarization degree, which is the kurtosis value of the charge/discharge voltage difference sequence. In addition, X20 monotonously decreases with the cycle numbers, as shown in Figure 3b, because the overpotential caused by polarization concentrates at the beginning and the end of the charge/discharge process in the early battery cycles, and then gradually fills the whole charge/discharge process. X1 is a measure of capacity degradation, as we discussed in the third formula. Taken together, it proves that capacity degradation and impedance evolution are both highly correlated factors in solid-state battery aging, which is consistent with our previous conjecture.