## 2 Experimental Data Analysis
### Data Analysis  
The ARBIN battery testing system establishes operating conditions by regulating the current magnitude per second. Excluding the effects of electromagnetic interference and sampling errors, the calculation of SOE using the discrete power integration method does not introduce cumulative errors under ideal conditions.  
\[SOE_{t}=SOE_{t-1}+\frac{\Delta t\times U_{t}\times I_{t}}{3600\times E} \tag{1}\]  
Here, SOE\({}_{t}\) is the value of SOE at time step t, \(\Delta\)t is the sampling time, and U\({}_{t}\) and I\({}_{t}\) represent voltage and current at time step t, respectively. E denotes the battery energy, which is acquired from the energy calibration experiment. The SOE is calibrated based on the average discharge energy measured over three repeated cycling tests.  
The relationships between four multiphysics features and SOE are shown in Figure 3, measured at 25 \({}^{\circ}\)C with a preload force of 500 N. By comparing the characteristic curves of the different operating conditions, the advantages of the mechanical force feature in SOE prediction can be summarized as follows.  
1. The voltage and temperature exhibit a strong dependence on the periodically varying current. Both parameters fluctuate significantly with the current within a localized range. In contrast, the mechanical force remains relatively stable in both local variations within a single operating condition and overall variations across different operating conditions.
2. Conventional methods utilize voltage, current, and temperature as model inputs to predict the SOE of LFP batteries. However, similar to SOC estimation, these methods suffer from the issue of the voltage plateau, which leads to reduced estimation accuracy. Figure 3(a2-d2) show the overall trend of voltage with a slow decrease in the range of 0.3 to 0.9 SOE, although it fluctuates periodically with the current. In contrast, the curve of mechanical force shows a significant change with an overall decreasing trend, but a reverse peak occurs in the range of 0.3 to 0.6 SOE.  
Figure 2: Test procedure with a preload force of 500 N.  
The raw measurements need to be preprocessed to minimize the impact of noise and outliers on the accuracy of model estimation. The multidimensional data are initially smoothed using a fixed window of size 50. For each feature, the precedent 49 data are smoothed using a variable window. The average is placed to the last unit of each window. As shown in Figure 3(a2-d2), mechanical force varies over a range of approximately one hundred times the voltage variation. The min-max method is employed to uniformly normalize the four features to the range of [0, 1], which reduces the scale sensitivity of the model across different features and accelerates the convergence speed. The equation is as follows:  
\[{x^{\prime}}_{i}=\frac{x_{i}-x_{i,min}}{x_{i,max}-x_{i,min}} \tag{2}\]  
where xi represents the original feature data, xi,max and xi,min denote the maximum and minimum values of the feature, respectively, and \({x^{\prime}}_{i}\) denotes the normalized value of the feature.  
Figure 3: Curves of current, temperature, voltage, and mechanical force under four operating conditions: (**a1,a2**) DST; (**b1,b2**) BJDST; (**c1,c2**) FUDS; (**d1,d2**) US06.  
In order to more effectively capture localized temporal dependencies within the time-series data, a sliding window of size n is employed to input data into the deep learning model. The value of SOE at the k-th time step is estimated using (k - n + 1) previous inputs of current, voltage, temperature, and mechanical force. The window slides incrementally to cover the entire dataset. The SOE values for the first (n - 1) time steps are estimated based on data from a single time step. The value of n is set to 55 in this study. Compared to prediction methods using a single time step, the estimation accuracy of SOE is improved with the addition of a window prediction method. The sliding window is shown in Figure 4.