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Li et al. J. Mater. Inf. 2025, 5, 29 https://dx.doi.org/10.20517/jmi.2024.103 Page 11 of 18

uDSR outperforms, while E2E lags behind. In the domain with a wide data threshold range, DSO shows
greater adaptability, and E2E and uDSR perform moderately. Overall, DSO and uDSR exhibit enhanced
adaptability and performance across diverse complex scenarios, while E2E is susceptible to performance
fluctuations and reductions in certain complex settings. We conduct quantitative experiments under the
Num Noise Dummy conditions, the details and results of which are presented in the “Quantitative
experiments under the Num, Noise, and Dummy conditions in SR4Real” section of the Supplementary
Materials and Supplementary Figure 2. Additionally, we evaluate the approximate computational time
overhead of SR4Real, the results of which are presented in the “Computational time overhead of SR4Real”
section of the Supplementary Materials and Supplementary Table 6.

In the evaluation of NED, DSO is significantly superior to E2E and uDSR, with the smallest average NED
value, indicating that when considering a comprehensive range of situations, DSO has the lowest average
deviation from the true formula. In the ops scenario with a large number of formula operands, DSO
performs excellently. In contrast, E2E has a relatively large NED value. DSO can more effectively handle
complex formula structures to approach the true formula. In the noise scenario where the data contains
noise, DSO demonstrates outstanding noise resistance. Its NED value is much lower than that of E2E and
uDSR, which strongly proves that DSO can still maintain a relatively close distance to the true formula
under noise interference, with its stability and accuracy being well manifested. Our discussion of the
phenomenon that DSO is slightly less accurate than uDSR but NED significantly outperforms uDSR is
discussed in the “Discussion of Performance Difference Between Different Symbolic Regression Methods”
section of the Supplementary Materials.

Therefore, we believe that DSO has a better average performance in scenarios with various data
characteristics, including fitting accuracy and the authenticity of the regression formula, and thus has the
potential for application in experimental datasets.

SR method performance on aging experimental dataset
Due to the subpar performance of the E2E method on the SR4Real dataset, we select the top two SR
methods, DSO and uDSR, to conduct verification on the experimental data of material aging. To evaluate
the advantages of SR, we compare it with four methods: linear regression based on variable screening, SVR,
RF, and XGBoost. The selection of linear regression based on variable screening is because we aim to
compare the performance differences between the linear regression with variable screening involving expert
knowledge and SR. Hence, we perform a SHapley Additive exPlanations (SHAP) analysis on the
experimental data of rubber aging, and the results are shown in Figure 3. Furthermore, RF, SVR, and
XGBoost are included because, despite SR’s advantages in producing interpretable expressions, this study
seeks to assess the accuracy differences between SR and machine learning regression methods that are
suitable for small-sample datasets. We add more experimental results of regression models in the
“Additional Regression Method Details and Results” section of the Supplementary Materialss and
Supplementary Tables 7 and 8.

A total of eleven variables are subjected to SHAP analysis, resulting in two sets of results. Horizontally,
samples numbered from 0 to 36 are presented. Each feature of every sample will have a SHAP value, and
finally, the importance of each feature is obtained by averaging the Shapley values of each sample.

It can be seen from Figure 3 that the feature importance of XLD, antioxidant content and plasticizer content
is significant. This phenomenon is also consistent with the order of the correlation magnitudes between
these three features and Shore hardness in the correlation analysis. This will be elaborated on in the final