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Page 8 of 18 Li et al. J. Mater. Inf. 2025, 5, 29 https://dx.doi.org/10.20517/jmi.2024.103

Random forest
Random forest (RF) is an ensemble learning method that integrates multiple decision trees for regression or
classification tasks. Each tree is trained on a bootstrap sample from the original dataset, and at each split,
only a randomly selected subset of features is considered.

Extreme gradient boosting
Extreme gradient boosting (XGBoost) is a powerful ensemble technique built upon the principles of
gradient boosting. It incrementally constructs an ensemble of weak learners, usually decision trees, where
each new tree attempts to correct the residual errors of the previously combined models.

SR algorithm
We select three SR methods as our candidate approaches. These methods cover typical algorithms in SR,
such as the Transformer model architecture, reinforcement learning, and genetic algorithms, which are
representative among numerous SR algorithms.

E2E
The E2E algorithm is designed to directly predict the full mathematical expression, including constants. It
leverages a Transformer-based architecture and directly decodes the hidden formula behind the data
through the input data. The advantage of this method is that it can pre-train the Transformer model,
establishing the mapping relationship between the input data and the output formula in advance, and then
directly predict in downstream tasks. However, it is prone to the problem of poor generalization ability.

Deep symbolic optimization
Deep symbolic optimization (DSO) uses recursive neural networks to generate symbolic expressions,
optimizing them based on the error between the expression and the data and using a reward strategy to
stabilize training. The reinforcement learning iterative process in DSO is shown in the Reinforcement
Learning section of Figure 1A.

uDSR
uDSR combines multiple SR-solving strategies for better performance, including recursive problem
simplification, neural-guided search, and GP. The uDSR algorithm first breaks down the problem of the
explosive combination of variables and operators into multiple low-dimensional subproblems, then uses
reinforcement learning to generate an initial population (Reinforcement Learning section of Figure 1A),
and finally uses a genetic algorithm for the evolution of the population to obtain more general formulas
(Genetic Programming section of Figure 1A).

Evaluation metric
SR4Real dataset evaluation metric
2
We use both the coefficient of determination (R ) score on the test set and the normalized edit distance
(NED) measure as evaluation metrics for the SR4Real benchmark. Here, R reflects how well the predicted
2
formula fits the test data, as expressed by

(1)