# Solid-State Lithium Battery Cycle Life Prediction Using Machine Learning
## 2 Materials and Methods
### Feature Selection  
Considering the lack of cycle-by-cycle impedance spectroscopy in our dataset and the impracticality and expensive cost of real-time impedance testing, we captured the impedance evolution from the voltage-time curve. Luckily, the voltage-time curve did reflect the battery aging information caused by impedance increases, and similar approaches have been proved efficient in previous work [35]. Due to the existence of internal impedance, the electrode potential deviated from the equilibrium potential when the current passed through the electrode. This phenomenon is called polarization, which produces overpotential, as shown in Figure 2. The result of polarization is that the terminal voltage of a battery is lower than the equilibrium potential when the battery is discharging and higher than the equilibrium potential when the battery is charging. Therefore, the working voltage curve of the battery becomes the superposition of polarization information inside the battery. For example, [PERSON] et al. proved the efficiency of quasi-polarization, defined as half of the difference in the averaged value between the charge and discharge voltage curve. Quasi-polarization monotonically increases with cycle numbers and roughly estimates the evolution of polarization. We also established a measure of polarization degree, which is the kurtosis of the charge/discharge voltage difference sequence (gray area), as shown in Figure 3a. Kurtosis is a common mathematical feature, which can be used to measure the steepness of the probability distribution of random variables. Its calculation formula is shown in Equation (4). The larger the kurtosis value, the more concentrated the probability distribution. The smaller the kurtosis value, the more uniform the probability distribution. As the overpotential caused by polarization concentrates at the beginning and end of the charge/discharge process in the early battery cycles, and then gradually fills the whole charge/discharge process, we proposed that the kurtosis value would decrease with the cycle numbers. As shown in Figure 3b, taking 6# battery as an example, its kurtosis value did monotonously decrease with the cycle numbers, which proves the correctness of our assumption. A correlation coefficient (\(\rho\)) was introduced to examine the correlation between features and the _RUL_[36]:  
\[\rho=Cor\Big{(}X_{feature},RUL\Big{)}=\frac{cov\Big{(}X_{feature},RUL\Big{)}}{ \sqrt{Var\Big{(}X_{feature}\Big{)}Var\Big{(}RUL\Big{)}}} \tag{1}\]  
Figure 2: A typical discharge curve of a solid-state battery. where \(X_{feature}\) denotes one of features we extracted from our dataset, '_Cov_' stands for the covariance between variables, '_Var_' represents the variance in the data, and _RUL_ represents the cycle numbers of the remaining useful life [37]. The _RUL_ for the \(i\)th cycle is calculated by  
\[RUL_{i}=n+1-i \tag{2}\]  
where \(n\) is the total number of battery charge and discharge cycles, and \(i\) is the \(i\)th charge and discharge cycle we chose to study. Equation (1) means the present cycle is also included in the remaining useful cycles. Based on the above discussion of degradation mechanisms and analysis of charge/discharge curves, 30 features were selected for the _RUL_ prediction (Supplementary Table S1). Some features represent the fading of capacity. The other features represent the increase in internal resistance. Furthermore, as shown in Figure 4, by sorting their coefficients of association between feature value and the _RUL_, 11 important features were retained, and they are described in detail below:  
Feature 1-2: Charge/discharge time. Time of constant current charge/discharge progress. The duration indicates the capacity by which the battery can be charged. Hence, this phenomenon can reflect the battery degradation process. Feature 3-4: Charge/discharge capacity. The total capacity of constant current charge/discharge progress.