Hardenability is a critical parameter for evaluating the service performance of high-strength steel components such as gears and shafts^[1]. It reflects the ability of steel to develop uniform hardness and microstructural characteristics throughout the cross-section during quenching heat treatment, thereby directly influencing the mechanical properties and service life of the components^[2,3]. At present, steel hardenability is most commonly assessed using the Jominy end-quench test. Originally proposed by Jominy and Boegehold^[4], this standardized method characterizes hardenability by measuring the hardness distribution along the length of a cylindrical specimen subjected to water cooling at one end, yielding the corresponding end-quench curve^[5,6]. Despite its widespread adoption and high degree of standardization, the Jominy test is prone to experimental uncertainties arising from oxide scale formation, microstructural inhomogeneity, and operator-dependent variability. Moreover, the testing procedure is relatively labor-intensive and time-consuming, limiting its suitability for rapid and cost-effective evaluation of steel hardenability in practical applications^[7,8].

To enable rapid evaluation of the hardenability of steel materials, the literature has proposed a wide range of empirical and semi-empirical models, along with various predictive equations^[9-11]. However, these approaches generally depend on restrictive assumptions and simplified parameter relationships, which limit their ability to capture the complex, nonlinear interactions inherent in multi-component steel systems. In recent years, the rapid advancement of machine learning techniques has led to significant progress in data-driven prediction and design of materials properties^[12-14]. Among these approaches, federated learning has emerged as a promising distributed learning strategy that effectively balances predictive performance and data privacy, making it particularly attractive for industrial applications^[15,16]. To enable accurate prediction of defects in sheet metal forming processes, da Silveira Dib et al. proposed a federated learning-based framework integrated with digital envelopes, which effectively predicted defects in sheet metal forming^[17]. To accurately predict the welding quality of friction stir welding, Chakraborty et al. developed a federated learning-based monitoring system that effectively predicted weld quality^[18]. Similarly, Wu et al. proposed a three-layer federated learning framework for the collaborative training of deep learning models, which was successfully applied to quality defect detection in construction projects^[19]. These findings demonstrate that federated learning enables collaborative modeling across companies and production lines while preserving data security, and that its range of applications in industrial environments is steadily expanding.

In this work, experimental hardenability data of gear steel from four steel companies are collected, and the J9 value is used as the target performance indicator for modeling and prediction. The methodological advance of this work does not lie in introducing a new elementary neural-network operator. Instead, it develops a federated regression strategy for hardenability, in which the network architecture, regularization strategy, and server aggregation rule are jointly designed for gear steel data from multiple sources. Specifically, efficient channel attention (ECA) is used to reweight coupled alloying element channels, preactivation residual blocks are employed to stabilize learning from moderately sized tabular datasets, dual-noise regularization is calibrated to composition measurement fluctuations and optimization uncertainty, and hierarchical feature aggregation is introduced to combine element-wise sensitivity with nonlinear alloying interactions. In addition, an adaptive sample loss aggregation rule is designed to mitigate non-independent and identically distributed (non-IID) client drift, extending beyond conventional aggregation relying on sample size. This strategy protects clients’ raw data and improves the robustness of J9 hardenability prediction under heterogeneous industrial data distributions.

The overall workflow of the proposed federated learning strategy is illustrated in Figure 1. In the traditional approach, models are trained independently by each client using only local datasets; however, predictive performance is frequently hindered by limited data volumes and heterogeneous sample distributions. To overcome these constraints, the federated learning strategy is employed, where local models are uploaded to a central server for parameter aggregation. This process updates the global model, which is subsequently redistributed to the clients to achieve significantly enhanced predictive accuracy. Crucially, this strategy precludes direct data exchange between clients, thereby ensuring the security of the clients’ raw data.

The dataset is divided into training and testing sets at an 8:2 ratio, and the data distributions of the four clients are illustrated in Figure 2. The results indicated that for each client, both the training and testing sets generally follow an approximately normal distribution. This suggests that the data-splitting process preserves good statistical consistency, which is conducive to stable model training and reliable performance evaluation. Nevertheless, notable differences remain among the four clients in terms of data range and distribution characteristics. Specifically, the J9 hardness values are primarily distributed within the range of 28-44 HRC; however, substantial variations are observed across clients with respect to mean values, distribution widths, and sample densities. These discrepancies reflect the typical characteristics of multi-source heterogeneous data. The detailed statistical evaluation of each dataset is presented in Supplementary Table 2.

To better accommodate the modeling requirements of multi-source heterogeneous data within a federated learning strategy, this study proposed a novel MRAN-J9 model. The overall network architecture is illustrated in Figure 3A. The model is composed of four principal components: the Input Layer, the Residual Block Area, the Feature Aggregation Layer, and the Output Layer. In the Input Layer, multi-dimensional feature data are received and normalized. Within the Residual Block Area, residual connections are introduced to mitigate gradient vanishing during deep network training, thereby enhancing the model’s ability to capture complex nonlinear feature representations. The Feature Aggregation Layer fuses features extracted from different network layers, further strengthening the comprehensive modeling of salient information. Finally, the Output Layer generates the prediction of the target variable. Compared with conventional machine learning models, the MRAN-J9 model demonstrates clear advantages in feature representation capacity and generalization performance, making it more suitable for meeting the accuracy and stability requirements of practical production-line applications.

During the model evaluation phase, 20% of the data from each client is reserved as a testing set to assess predictive performance. A comparison of the prediction performance of the MRAN-J9 model and eight conventional machine learning models across datasets from four clients is presented in Figure 3B. The eight traditional machine learning models are trained independently using local data. The results indicate that MRAN-J9 consistently achieves superior predictive performance across all clients, which outperforms the remaining eight machine learning models, as reflected by the highest R² and the lowest RMSE, the blue five-pointed star indicates the lowest RMSE result. These results demonstrate enhanced fitting capability and more stable predictive behavior (Client 1: R² = 0.79, RMSE = 1.32 HRC; Client 2: R² = 0.68, RMSE = 1.55 HRC; Client 3: R² = 0.68, RMSE = 1.67 HRC; Client 4: R² = 0.72, RMSE = 1.41 HRC). The detailed predictive results for the other eight machine learning models are summarized in Supplementary Table 3. Collectively, the results indicate that the MRAN-J9 model exhibits significant advantages in multi-client, multi-distribution data environments and effectively improves overall prediction performance. In addition to traditional machine learning, representative deep learning backbones [1-dimensional convolutional neural network (1D-CNN), multilayer perceptron (MLP), and squeeze-and-excitation multilayer perceptron (SE-MLP)] trained solely on local data are also evaluated. As detailed in Supplementary Table 4, the MRAN-J9 model consistently surpasses these modern deep learning baselines even before federated aggregation, and the gap widens when the federated strategy is applied, confirming the necessity of both the customized architecture and the collaborative training paradigm.

The parameter optimization process of the MRAN-J9 model for four clients within the federated learning strategy, together with the predictive performance of the corresponding optimal-parameter models, is presented in Figure 4. In each global communication round, the central server aggregates client-updated model parameters to update a global model, which is subsequently broadcast back to the clients to enable collaborative training while keeping raw data local. Each client then performs iterative local updates of the MRAN-J9 parameters on its own dataset, progressively converging to a client-specific optimum. For client 1, Figure 4A₁ shows the optimization results over 100 global communication rounds, Figure 4A₂ reports the prediction performance of the optimal model (R² = 0.88, RMSE = 1.03 HRC), and Figure 4A₃ depicts the frequency distribution of the predicted values. Figure 4B-D present the parameter optimization process and the corresponding prediction performance for clients 2-4 (Client 2: R² = 0.80, RMSE = 1.25 HRC; Client 3: R² = 0.84, RMSE = 1.19 HRC; Client 4: R² = 0.83, RMSE = 1.34 HRC), respectively. Furthermore, the proportions of prediction errors falling within the ± 2 HRC range for the four clients are 94.39%, 92.99%, 89.72%, and 94.85%, respectively. Across all clients, predictions agree closely with the ground truth, with most points clustering around the 1:1 line, indicating that MRAN-J9 delivers consistent and robust predictive performance under the federated learning strategy. Beyond predictive accuracy, a SHAP feature-importance analysis is conducted on the datasets of all four clients. The results reveal that the top six most influential features (Mn, Cr, C, Ti, Si, and B) are identical across all clients. Notably, these elements are widely recognized in physical metallurgy for their significant influence on steel hardenability. This consistency effectively validates the model’s physical reliability. The detailed results are presented in Supplementary Figure 1.

To further clarify the source of the performance improvement, two levels of controlled comparisons are conducted. First, architectural ablation methods are performed under a purely local setting. The residual connection, ECA attention, noise regularization, and feature aggregation layer are removed individually while the remaining training settings are kept unchanged. As summarized in Supplementary Table 5, the full MRAN-J9 model achieved an average R² of 0.72 across the four clients. After removing the residual connection, ECA attention, noise regularization, and feature aggregation layer, the average R² decreased to 0.66, 0.67, 0.67, and 0.66, respectively. These results indicate that each architectural module contributes to the predictive capability of MRAN-J9 under the purely local setting.

To further distinguish the contribution of the federated aggregation strategy from the backbone architecture, benchmarks for both FedAvg and FedProx are additionally established under the identical MRAN-J9 network. The detailed ablation comparison is summarized in Supplementary Table 6. The results indicate that the adaptive weighting and robust preprocessing, rather than the proximal term alone, are the primary drivers of the performance enhancement, confirming the rationality of the federated learning design.

The results presented in Figure 4 demonstrate that the MRAN-J9 model trained under the federated learning strategy can effectively predict the test-set data of all four clients, as illustrated in Figure 5A. High predictive accuracy and good stability are maintained across different client data distributions. To further evaluate the generalization capability of the MRAN-J9 model, an additional validation dataset containing 755 samples is constructed. This validation dataset is derived from a different source than the original datasets of the four clients. The prediction results on the validation dataset are shown in Figure 5B, where an R² of 0.88 and an RMSE of 0.99 HRC are achieved. Additionally, the model demonstrates excellent practical accuracy, with 93.38% of the errors strictly confined within the ± 2 HRC tolerance. These results indicate that the MRAN-J9 model can maintain robust predictive performance when applied to new data not used during training. Moreover, the findings demonstrate that the federated learning strategy effectively mitigates overfitting while fully leveraging multi-source data information, thereby significantly enhancing the model’s generalization capability and practical applicability.

In traditional centralized machine learning, distributed data are aggregated at a central server for model training^[24-26]. In contrast, federated learning enables the construction of shared models among multiple participants in a secure, efficient, and regulation-compliant manner without requiring data to leave their respective domains^[27,28]. As demonstrated by the results in Figures 4 and 5, the MRAN-J9 model trained under the federated learning strategy effectively adapts to heterogeneous data distributions across different clients while exhibiting strong generalization performance on an external independent validation dataset. These findings further validate the advantages of federated learning for collaborative modeling involving multi-source heterogeneous data.

With the continued advancement of federated learning theory and techniques, its application scope has been progressively extended to a wide range of domains, including intelligent healthcare^[29], recommendation systems, smart cities^[30], finance and insurance, and edge computing^[31]. In response to the increasing complexity of application requirements, federated learning algorithms and system architectures are being actively investigated to achieve improved trade-offs among data security, communication efficiency, and model performance. By facilitating cross-institutional collaboration via secure connection platforms, federated learning is driving data collaboration paradigms toward more legal, compliant, and sustainable practices. In this context, the MRAN-J9 model proposed in this study provides effective technical support for deploying federated learning in industrial application scenarios.

In this work, an MRAN-J9 model suitable for federated learning strategy is proposed and systematically validated using data from four clients. The results demonstrate that the MRAN-J9 model trained under the federated learning strategy effectively adapts to the heterogeneous data distribution characteristics of different clients, achieving superior predictive performance on each client’s test set when compared with non-federated learning modeling approaches. Furthermore, high predictive accuracy is maintained on a completely independent external validation dataset (R² = 0.88, RMSE = 0.99 HRC), thereby demonstrating the model’s strong generalization capability and stability. These findings further confirm the feasibility and effectiveness of federated learning in enabling collaborative modeling across multiple data owners while protecting client raw data and complying with regulatory requirements. By integrating MRAN-J9 into the federated learning strategy, latent feature information from multi-source data is more fully exploited, and the limitations of single-source modeling are alleviated. This integrated approach offers a new paradigm for researching the properties of gear steel, facilitating broader application deployment across environments involving multiple enterprises and production lines.