Page 54 - Read Online

Li et al. J. Mater. Inf. 2025, 5, 29 https://dx.doi.org/10.20517/jmi.2024.103 Page 13 of 18

Table 5. Prediction performance of six regression methods on the test set with Shore hardness as the dependent variable
Algorithm Mean RMSE Std RMSE Mean MAE Std MAE
MLR 2.690 0.677 2.189 0.935
SVR 2.890 1.360 2.547 1.118
RF 3.038 1.324 2.481 0.972
XGBoost 3.326 1.449 2.687 1.039
DSO 2.253 1.043 1.814 0.848
uDSR 2.441 1.128 1.771 0.534

We performed a 7:1:2 split for training, validation, and testing on 37 data points, and conducted five-fold cross-validation. The results presented
above are the outcomes of five runs. RMSE: Root mean square error; MAE: mean absolute error; MLR: multiple linear regression; SVR: support
vector machine regression; RF: random forest; XGBoost: extreme gradient boosting; DSO: deep symbolic optimization; uDSR: unified deep
symbolic regression.

Table 6. Prediction performance of six regression methods on the test set with fracture elongation as the dependent variable
Algorithm Mean RMSE Std RMSE Mean MAE Std MAE
MLR 0.076 0.01 0.055 0.007
SVR 0.071 0.007 0.063 0.006
RF 0.072 0.005 0.058 0.006
XGBoost 0.073 0.006 0.057 0.004
DSO 0.061 0.006 0.050 0.004
uDSR 0.066 0.009 0.053 0.007

We performed a 7:1:2 split for training, validation, and testing on 37 data points, and conducted five-fold cross-validation. The results presented
above are the outcomes of five runs. RMSE: Root mean square error; MAE: mean absolute error; MLR: multiple linear regression; SVR: support
vector machine regression; RF: random forest; XGBoost: extreme gradient boosting; DSO: deep symbolic optimization; uDSR: unified deep
symbolic regression.

Table 7. The formula obtained for MLR, DSO, and uDSR under the hardness task
Algorithm Formula
MLR Hardness = 26.775·ω + 6.505·μ + 11.397·XLD - 20.073·LC + 33.808
a
1
3
DSO Hardness = (27.154XLD + 7.186) (exp (μ ) - 0.641)
3
uDSR Hardness = XLD·exp (μ + 3.000) + XLD
2
For the definitions of the variables, please refer to Table 2. The constant coefficient has been rounded to three decimal places. MLR: Multiple
linear regression; DSO: deep symbolic optimization; uDSR: unified deep symbolic regression; XLD: crosslink density.
crosslinking. Furthermore, the relationship between Fe and XLD is negative, while it is positive with
plasticizer content.

We observe from the regression formula for Hardness in Table 7 that, in the formula obtained from DSO,
the term (exp(μ ) - 0.641) can be treated as a constant, given that the range of μ is approximately 0 to 0.1.
3
3
This approximation allows the second formula to align well with chemical prior knowledge. However, in the
MLR model, the coefficient for the double-bond carbon is positive, which contradicts the chemical prior
knowledge. This suggests that the relationship between the content of double bonds and hardness should
not be modeled as linear.

From the regression formula of Fe presented in Table 8, it is noted that, in the formula acquired through
DSO, the second term (-ω ·ω ·μ ·exp(-ω ) - ω ) is significantly smaller than the term (exp(exp(ω ))). As a
4
2
1
4
4
2