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Liu et al. J. Mater. Inf. 2026, 6, 18 Page 7 of 29

depth of each tree; min_samples_split specifies the minimum number of samples required for node splitting;
and min_samples_leaf defines the minimum sample size required for leaf nodes. For GBRT, n_estimators
determines the number of boosting iterations and model capacity. The learning rate, as a key regularization
parameter, controls the step size of each iteration and works in conjunction with n_estimators - smaller
learning rates require more trees, yielding more stable models with superior generalization. max_depth
limits the complexity of individual weak learners by capping DT depth, ensuring gradual learning
progression. min_samples_split acts as a node-level regularization constraint, preventing the model from
learning noise by requiring a minimum number of samples for splitting. The main hyperparameters and
their adopted values for the 12 ML models used in this work are summarized in Table 3. For model
performance comparison, the hyperparameters of all models were set to their default values.

Effective optimization of these hyperparameters balances model expressiveness and computational efficiency.
Excessively high learning rates may cause premature convergence, while insufficient tree depth can restrict
the model’s ability to capture nonlinear relationships. To systematically explore the hyperparameter space, a
grid search strategy was employed to evaluate combinations of values within predefined ranges. This method
identified configurations with high validation performance and strong robustness, indicated by less than a
5% drop between training and testing performance. The grid search protocol thus proves effective in building
reliable and generalizable models through structured exploration of the parameter space.

In addition to point predictions, estimating the reliability of model outputs is crucial, particularly for
material design under extreme conditions. This work leveraged the ensemble nature of ML algorithms to
quantify predictive uncertainty. For the RF model (used for ductility prediction), the prediction uncertainty
for a given input x was estimated by calculating the standard deviation (σ) of predictions from all N
individual DTs. The 95% confidence interval (CI) was defined as y(x) ± 1.96σ(x). For the GBRT model (used
−
for strength prediction), a similar variance-based approach using quantile estimates was employed to
determine confidence bounds.

Multi-objective optimization algorithm
Multi-objective optimization algorithms address the trade-off issues inherent in traditional single-objective
design methods .
[43]

In this work, a multi-objective optimization system for titanium alloys was established based on the NSGA-II
framework. The system defines a 9-dimensional alloy element space as the design variables, with
dual-objective optimization functions aimed at maximizing impact ductility and ductility. Domain
knowledge-based ML models were integrated into the NSGA-II algorithm, which adopts an elitist strategy
and is initialized using Latin hypercube sampling to ensure uniform coverage of the design space. The
evolutionary search is driven by simulated binary crossover (crossover probability = 0.9; distribution index =
15) to promote global exploration, combined with polynomial mutation (mutation probability = 0.25;
distribution index = 15) to enhance local refinement. An innovative constraint-handling mechanism is
introduced, incorporating component normalization X = X · Í 0.3 and titanium compensation X′ =
ˆ
i
i
i
X· . Engineering constraints (g = Mo - 2.5 ≤ 0, g = Al - 8.5 ≤ 0) are integrated into the Pareto
0.3 − (0.695 − )
1
i
eq
2
eq
Í

dominance relationship. Non-dominated sorting stratifies solutions based on Pareto dominance i,j:f k (i) ≺ f ,
(j)
∀
k
while crowding distance is used to quantify solution diversity. Elite retention balances convergence and
exploration via arg max (rank , -d ), with termination at 500 generations or upon adaptive stagnation
∈ ∪ i i
detection, specifically when ΔPF < 1% for 20 consecutive generations. This framework facilitates cross-scale
mapping from material genes to macroscopic performance through data-driven optimization and
physics-informed model reduction. The integration of constraints synergizes metallurgical principles with
optimization theory, while maintaining computational efficiency for populations of 500 individuals. The