# Solid-State Lithium Battery Cycle Life Prediction Using Machine Learning
## 3 Results  
The battery dataset was collected when the discharge capacity of each battery decreased to 80% of the initial discharge capacity. The capacity of the batteries ranged from 40 to 60 mAh and the cycle lives ranged from 70 to 213 cycles. Figure 6 shows the discharge capacity as a function of cycle number for the whole cycle life, where the color denotes different batteries. The capacity fade speeds were different in the first few cycles and there were even some noisy fluctuations in the capacity fading curves.  
We built a symbolic regression model using a series of descriptors extracted from the full voltage curves of each cycle. The specific feature selection and hyperparameter setup are discussed in the above section. After several populations of training, an optimized expression generated by our SR model outputted predicted values approaching the ground-truth _RUL_ values in our training battery data. The formulas generated by our SR model can be found in Table 2. In addition, the pair plot for the actual _RUL_ of the test batteries and the predicted _RUL_ inferred by the fourth formula in Table 2 is shown in Figure 7.  
Figure 5: Structure of symbolic regression model.  
The explanations of each symbol shown in Table 2 are as follows: X1 denotes charge-specific capacity; X5 and X6 represent the integrals of capacity and voltage charging/discharging curves, respectively; X20 is the kurtosis of difference between the charging and discharging curve; X21 describes the previous locally convex/concave X coordinate of the voltage and capacity charging curve; X24 denotes the latter locally convex/concave Y coordinate of the voltage and capacity charging curve; X29 is the charging time.'min()' means taking the minimum value of elements.'max()' means taking the maximum value of elements. \(R^{2}\) can be calculated by using Equations (5) and (6). The complexity is also the number of operational symbols that can measure the complexity of formulas in the case of overfitting.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Formulas** & \(\mathbf{R^{2}}\) & **Complexity** \\ \hline X5 & 0.7482 & 0 \\ \hline min(X29, X5) & 0.7730 & 1 \\ \hline
0.820 \(\times\) X1 & 0.8768 & 1 \\ \hline min(X20, 0.848 \(\times\) X1) & 0.9093 & 2 \\ \hline max(X1, \(\sqrt{\text{X24}})\times 24\) & 0.7884 & 3 \\ \hline \(\text{X5}\times\text{max}(\sqrt{\text{X21}},\text{X6}-(\text{X24}/(\text{X21})))\) & 0.9307 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The six mathematical formulas generated by the SR model.  
Figure 6: Cycle performance of \(\text{LFP}/\text{NCM}\) solid-state lithium polymer batteries: discharge capacity decreases with cycle numbers.  
Figure 7: The pair plot for predicted/actual _RUL_.  
The final equation obtained from the SR model is expressed as follows:  
\[\text{min}(\text{X20},\ 0.848\times\text{X1}) \tag{8}\]  
where X20 denotes the kurtosis of difference between the charging and discharging curve, and X1 represents the charge-specific capacity.  
We also conducted a comparison of predicted results and errors with the other three machine-learning methods coming from recent excellent related studies in the literature: support vector regression (SVR), Gaussian process regression (GPR), and elastic net (EN), as shown in Table 3. The \(RUL_{\text{prediction}}\) results for the SR model were calculated by the fourth formula in Table 2, and the other three were calculated by the methods in references [4,22,24], respectively. The computational characteristics for the four models are shown in Table 4. The explanation of each characteristic is as follows. Training time is the whole time for the training of our SR model, including 5 generations to breed, mutate, and evolve. Inference time is the time taken by the model from receiving the input battery feature parameters to calculating the predicted \(RUL\) values. Total used memory is the memory containing the model parameters and formulas of every generation.