Appendix A
See illustration. 20, 21, 22, 23, 24
This figure compares the training loss (blue) and validation loss (red) curves for the baseline cGAN model.
Pearson correlation matrix of elemental concentrations for real alloy data (left panel), cWGAN-generated alloy (middle panel), and cGAN-generated alloy (right panel).
Element-wise KDE comparing the actual alloy composition (dotted blue line) to the alloy composition generated by cWGAN (black dot-dashed line) and cGAN model (solid red line).
Element-wise KDE comparing the actual alloy composition (dotted blue line) to the alloy composition generated by cWGAN (black dot-dashed line) and cGAN model (solid red line).
Element-wise KDE comparing the actual alloy composition (dotted blue line) to the alloy composition generated by cWGAN (black dot-dashed line) and cGAN model (solid red line).
Appendix B
adversarial loss function
For conditional Wasserstein GAN (cWGAN), the purpose of the identifier20 is given by
$$\begin{aligned} \mathcal {L}_{\text {identifier}} = \mathbb {E}_{\textbf{x} \sim \mathbb {P}_r} [D(\textbf{x}, \textbf{y})] – \mathbb {E}_{\hat{\textbf{x}} \sim \mathbb {P}_g} [D(\hat{\textbf{x}}, \textbf{y})] + \lambda _{\text {GP}} \mathbb {E}_{\tilde{\textbf{x}} \sim \mathbb {P}_{\tilde{x}}} \left[ \left( \Vert \nabla _{\tilde{\textbf{x}}} D(\tilde{\textbf{x}}, \textbf{y}) \Vert _2 – 1 \right) ^2 \right] ,\end{align}$$
(1)
where \(\textbf{y}\) is the conditioning vector (H, E), \(\mathbb {P}_r\) is the actual data distribution, \(\mathbb {P}_g\) is the model distribution, \(\mathbb {P}_{\tilde{x}}\) is the distribution of the interpolated samples. The cWGAN generator loss is calculated as follows:
$$\begin{aligned} \mathcal {L}_{\text {gen}} = -\mathbb {E}_{\hat{\textbf{x}} \sim \mathbb {P}_g} [D(\hat{\textbf{x}}, \textbf{y})]. \end{Align}$$
(2)
For cGAN38the discriminator (D) and generator (G) losses are written as:
$$\begin{aligned} \mathcal {L}_D = – \mathbb {E}_{\textbf{x} \sim \mathbb {P}_r} [\log D(\textbf{x}, \textbf{y})] – \mathbb {E}_{\hat{\textbf{x}} \sim \mathbb {P}_g} [\log (1 – D(\hat{\textbf{x}}, \textbf{y}))],\end{align}$$
(3)
$$\begin{aligned} \mathcal {L}_G = – \mathbb {E}_{\hat{\textbf{x}} \sim \mathbb {P}_g} [\log D(\hat{\textbf{x}}, \textbf{y})]. \end{Align}$$
(4)
here \(\textbf{x}\) Shows the actual alloy composition vector extracted from the true data distribution \(\mathbb {P}_r\)and \(\hat{\textbf{x}}\) Here is the generated configuration extracted from the model distribution \(\mathbb {P}_g\). conditioning vector \(\textbf{y} = (H, E)\) represent the target H and E values. identifier \(D(\cdot )\) Outputs the probability that the specified input composition matches the actual distribution under the conditions \(\textbf{y}\). expectation value operator \(\mathbb {E}[\cdot ]\) indicates the average across the mini-batches of training samples.
Evaluation index
Let me \(\textbf{c} = (c_1, \dots , c_n)\) and \(\hat{\textbf{c}} = (\hat{c}_1, \dots , \hat{c}_n)\) denote the actual resultant vector and the generated resultant vector, respectively. The root mean square error (RMSE) is defined as:
$$\begin{aligned} \text {RMSE} = \sqrt{ \frac{1}{n} \sum _{i=1}^{n} (c_i – \hat{c}_i)^2 }. \end{Align}$$
(5)
The mean absolute deviation (MAD) is defined as:
$$\begin{aligned} \text {MAD} = \frac{1}{n} \sum _{i=1}^{n} |c_i – \hat{c}_i |. \end{Align}$$
(6)
Cosine similarity (CosSim) is defined as:
$$\begin{aligned} \text {CosSim} = \frac{\textbf{c} \cdot \hat{\textbf{c}}}{\Vert \textbf{c} \Vert _2 \, \Vert \hat{\textbf{c}} \Vert _2}. \end{Align}$$
(7)
RMSE measures the average magnitude of the reconstruction error, with lower values indicating a better match between the produced alloy and the actual alloy. MAD captures the average deviation for each element. Smaller values reflect a more consistent reproduction of elemental fractions. CosSim evaluates the directional similarity between the actual and generated resultant vectors. Values closer to 1 indicate stronger alignment in composite space.
Frobenius norm for composition
The Frobenius norm was used to measure the overall deviation between the actual and generated composition matrices. Let me \(\textbf{X}_{\text {real}}, \textbf{X}_{\text {gen}} \in \mathbb {R}^{N \times 15}\) Represents the actual dataset and the generated dataset. N Alloy and number of each entry \(x^{\text {real}}_{ij}\) or \(x^{\text {gen}}_{ij}\) Indicates the atomic fraction of an element j with alloy I. The Frobenius norm difference is defined as:
$$\begin{aligned} \Vert \textbf{X}_{\text {gen}} – \textbf{X}_{\text {real}} \Vert _F = \sqrt{\sum _{i=1}^{N}\sum _{j=1}^{15} \left( x^{\text {gen}}_{ij} – x^{\text {real}}_{ij}\right) ^2 }. \end{Align}$$
(8)
This metric captures the overall compositional deviation between the two datasets, with lower values indicating a better match between the produced alloy and the actual alloy.
Mean Absolute Spearman Correlation (MASC)
Let me \(\rho _s(j, p)\) will be the Spearman correlation between the proportions of the elements j (where \(j = 1, \dots , 15\) Al, Co, Cr, Fe, Ni, Cu, Mn, V, Mo, Ti, Nb, Hf, W, Ta, Zr) and properties \(p \in \{H, E\}\). after that
$$\begin{aligned} \text {MASC} = \frac{1}{15 \times 2} \sum _{j=1}^{15} \sum _{p \in \{H,E\}} \left| \rho _s(j, p) \right| . \end{Align}$$
(9)
This index measures the extent to which the produced alloy maintains the statistical relationship between individual elemental fractions and target properties ( H and E). Higher values indicate better retention of these compositional and property trends.
Preservation of correlation structure
For Pearson correlation matrix \(\textbf{R}_{\text {real}}\) and \(\textbf{R}_{\text {gen}}\) The difference in the Frobenius norm for the elemental fractions is:
$$\begin{aligned} \Vert \textbf{R}_{\text {gen}} – \textbf{R}_{\text {real}} \Vert _F = \sqrt{\sum _{i=1}^{15}\sum _{j=1}^{15} (r^{\text {gen}}_{ij} – r^{\text {real}}_{ij})^2 }. \end{Align}$$
(10)
This metric quantifies the extent to which the resulting alloy retains the interelement correlation patterns found in the real dataset. A lower value indicates a more accurate reproduction of the underlying chemical relationships.
Kernel Density Estimation (KDE)
Kernel density estimation was used in the section “Inversely generated compositions” to sample the composite property pair (H, E). Given a set of observations, \(\{x_i\}_{i=1}^n\)KDE is defined as:
$$\begin{aligned} \hat{f}_h(x) = \frac{1}{nh} \sum _{i=1}^{n} K \left( \frac{x – x_i}{h} \right) , \end{aligned}$$
(11)
where \(K(\cdot )\) is a kernel function, h Bandwidth parameter. This method provides a smooth approximation of the probability distribution of the data. Applying a Gaussian kernel enabled a realistic and diverse sampling of new property pairs (H, E).
activation function
Two activation functions were used in the generative model. The first is a sigmoid function, which maps real-valued inputs to the range (0, 1).
$$\begin{aligned} \sigma (x) = \frac{1}{1 + e^{-x}}. \end{Align}$$
(12)
This activation function makes it suitable for outputs that need to be interpreted as probabilities. The softmax activation function normalizes a vector of values into a probability distribution where all components are positive and sum to 1. The softmax function is given as:
$$\begin{aligned} \text {softmax}(x_i) = \frac{e^{x_i}}{\sum _{j=1}^{n} e^{x_j}}, \quad i = 1, \dots , n. \end{Align}$$
(13)
In this study, a softmax function was applied to the output of the generator to ensure that the alloy composition vector is physically valid and the elemental fractions sum to unity. All evaluation metrics presented in this appendix were calculated over 50 independent runs, and both mean and standard deviation are reported to capture variation due to stochastic generation.
