Application of expert-augmented machine learning modeling in high-strength and high-conductivity copper alloy development
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摘要: 材料領域數據具有小樣本、噪聲大、維度高、關系復雜、專家知識豐富的特點. 利用專家知識增強機器學習建模效果具有必要性和可行性. 本文通過計算自變量與因變量之間的秩相關系數，來定量描述成分狀態因素與性能之間單調關系的強弱. 在模型訓練過程中，將秩相關系數加入到神經網絡損失函數，實時評估模型輸出與專家知識的相符程度，得到了專家知識增強的機器學習模型. 對訓練過程分析后發現，模型輸出的合理性有顯著提升，模型的輸入輸出規律與專家知識的相符程度達到了0.98以上（1.0為完全相符）. 基于所建模型，采用遺傳算法進行了關于強度和導電率的多目標優化，找到了滿足帕累托最優的高強高導銅合金成分并開展了實驗驗證. 實驗結果表明，強度在高達637 MPa的同時，導電率仍能保持在77.5% IACS（國際退火銅標準）的水平；導電率高達80.2% IACS的同時，強度仍能保持在600 MPa的水平. 強度和導電率的預測值與實際值誤差在5%以內.Abstract: Investigation into material data is frequently limited by small sample sizes, high noise levels, complex associations, high dimensionality, and the need for expert knowledge. To improve the effectiveness of machine learning modeling, incorporating expert knowledge is necessary. In this study, we assembled a dataset including 410 data points containing composition, condition, and property data, in which the state symbols of the copper alloy were recoded using the one-hot encoding method. Because of the substantial capacity of neural network algorithms for powerful nonlinear fitting, we employed these algorithms for modeling. The network structures of the strength and conductivity models were optimized to 21–55–70–1 and 21–50–65–1, respectively. After optimizing the network structure, expert knowledge was integrated into the neural network loss function. This approach quantitatively describes the strength of the monotonic relations between the status factors of components and performance by calculating the rank correlation coefficient between the independent and dependent variables. During model training, the rank correlation coefficient was incorporated into the neural network loss function to assess the similarity between the model output and expert knowledge in real-time. For instance, the relation in which strength increases with the hardening level was quantitatively expressed with a Spearman score, and these Spearman scores were added to the loss function. A machine learning model augmented by expert knowledge was trained using genetic algorithm-based optimization of network weights. After updating each network weight, orthogonal data were generated to evaluate the consistency between output data and expert knowledge. The Spearman correlation coefficients between the model input–output data and expert knowledge exceeded 0.98, and the R2 scores of the strength and conductivity models achieved on the test set were >0.90. Multiobjective optimization based on composition, condition, strength, and conductivity models was conducted using a genetic algorithm, and Pareto-optimal solutions were obtained and experimentally validated after 100 generations of iteration. Three sets of components were selected from the Pareto-optimal solutions and were empirically tested for validation. The results showed that the tensile strength had reached 637 MPa, while conductivity was maintained at 77.5% IACS (International annealing copper standard), and when the conductivity was 80.2% IACS, the tensile strength was 600 MPa. The relative errors between the experimental and predicted values were <5%. Microstructure images of three experimental sample sets demonstrated that coarse second phases were present in the as-cast structure; however, these structures were dissolved and redistributed after the solid solution, cold deformation, and aging processes. The precipitated particles distributed along the grain boundary had low strength and conductivity. Our analysis revealed that the Mg and Ti elements were detrimental to the increase in strength, while Fe and Sn effectively increased strength. Additionally, Fe had a lower impact on conductivity than Sn. The results of this study demonstrate that the three optimized components identified can satisfy the performance requirements of interconnected frameworks in ultralarge-scale integrated circuits.
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圖 1 不同網絡結構下強度和導電率模型評分. （a）強度模型；（b）導電率模型
Figure 1. Strength and conductivity model scores for different network structures: (a) strength model; (b) conductivity model
圖 3 專家知識增強的模型迭代過程. （a）強度模型；（b）導電率模型
Figure 3. Iterative process of expert-augmented model: (a) strength model; (b) conductivity model
圖 4 模型在測試集上的效果. （a）強度模型；（b）導電率模型
Figure 4. Performance of the model on the test dataset: (a) strength model; (b) conductivity model
圖 5 關于強度和導電率的多目標優化結果
Figure 5. Results of multiobjective optimization between strength and conductivity
圖 7 不同成分下樣品性能實驗值與預測值對比
Figure 7. Comparison of the experimental and predicted values of sample performance at different compositions
圖 8 1#合金組織演變規律. （a）鑄態；（b）成品態
Figure 8. Microstructure evolution of sample 1#: (a) as-cast condition; (b) finished product
圖 9 2#合金組織演變規律. （a）鑄態；（b）成品態
Figure 9. Microstructure evolution of sample 2#: (a) as-cast condition; (b) finished product
圖 10 3#合金組織演變規律. （a）鑄態；（b）成品態
Figure 10. Microstructure evolution of sample 3#: (a) as-cast condition; (b) finished product
表 1 成分質量分數與性能的描述統計
Table 1. Statistical description of the composition mass fraction (%) and property data
Variables Indicators Avg Std Min Max Count of
nonzero valuew(Mn) /% 0.026 0.11 0 0.5 32 w(Fe)/% 0.19 0.49 0 2.35 103 w(Ti)/% 0.16 0.67 0 3.2 61 w(Co)/% 0.07 0.29 0 1.9 45 w(P)/% 0.02 0.05 0 0.2 159 w(Zr)/% 0.01 0.05 0 0.4 64 w(Sn)/% 0.86 2.06 0 10 208 w(Cr)/% 0.05 0.17 0 1.02 74 w(Zn)/% 0.67 3.06 0 22.46 83 w(Mg)/% 0.04 0.12 0 0.7 101 w(Si)/% 0.15 0.26 0 0.925 120 w(Ni)/% 1.6 3.59 0 21 174 w(Ag)/% 0.002 0.02 0 0.2 12 w(Al)/% 0.05 0.42 0 3.5 14 w(Te)/% 0.0003 0.003 0 0.02 12 UTS/MPa 623 211 248 1450 410 EC/%IACS 55 27 3 102 410 
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表 2 銅合金狀態代號的編碼映射表（部分）
Table 2. Coding schedule of copper alloy condition symbols (partial)
Material designation and
its meaningFeatures and their values after recoding Hardened level/% Immediate quenching Precipitation hardening Order hardening Stress relieving Annealed H00 1/8 Hard 5 0 0 0 0 0 H01 1/4 Hard 10 0 0 0 0 0 H04 Hard 37.1 0 0 0 0 0 TM00 heat-treated, 1/8 Hard 5 1 0 0 0 0 TM01 heat-treated, 1/4 Hard 10 1 0 0 0 0 TM06 heat-treated, extra hard 50 1 0 0 0 0 TR01 precipitation hardening, stress
relieving, 1/4 Hard10 0 1 0 1 0 HT04 order-hardening, Hard 37.1 0 0 1 0 0 O Annealed 0 0 0 0 0 1
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表 3 校驗數據生成和平均Spearman系數計算示例(H: 硬化程度; P_EC: 導電率預測值)
Table 3. A demo for checking data generation and calculating the average Spearman correlation coefficient values (H: Hardened level; P_EC: Predicted value of EC)
No. Ni mass fraction/% Si mass fraction/% Ti mass fraction/% ··· H/% P_EC/%IACS Absolute value of spearman
scores between H and P_EC1 0 0.1 0.1 ··· 0 80 0.8 2 0 0.1 0.1 ··· 15 85 3 0 0.1 0.1 ··· 30 65 4 0 0.1 0.1 ··· 45 50 5 0 0.1 0.1 ··· 60 54 6 0.2 0.3 0.2 ··· 0 84 0.9 7 0.2 0.3 0.2 ··· 15 73 8 0.2 0.3 0.2 ··· 30 37 9 0.2 0.3 0.2 ··· 45 40 10 0.2 0.3 0.2 ··· 60 35 11 0.4 0.2 0.3 0 24 ··· ··· ··· ··· ··· ··· ··· ··· Average 0.92
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表 4 優化出的高強高導銅合金成分及預測性能
Table 4. Compositions and predicted properties of the optimized high-strength and high-conductivity copper alloys
No. Composition（mass fraction）/% Predicted properties UTS/MPa EC/%IACS 1# Cu–0.5Cr–0.2Zr–0.1Mg–0.05Ti–0.06Fe 613 78.1 2# Cu–0.6Cr–0.15Zr–0.1Mg–0.02Ti–0.05Sn 635 77.2 3# Cu–0.6Cr–0.1Zr–0.1Mg–0.02Ti 607 83.4
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