A generative deep learning framework for inverse design of compositionally complex bulk metallic glasses

Glass-forming metallic alloys (metallic glasses, MGs) possess exceptional properties but relatively low glass-forming ability (GFA), making discovery of bulk metallic glass (BMG) compositions (>1 mm castability) challenging. Traditional empirical, trial-and-error discovery is slow and inefficient. While supervised machine learning has accelerated alloy design, it struggles to efficiently explore the astronomically large compositional spaces of compositionally complex MGs (e.g., high-entropy systems with >5 principal elements). This work addresses the need for efficient, scalable exploration by proposing a generative deep-learning framework capable of directly generating candidate BMG compositions and enabling inverse design guided by targeted properties.

Prior ML approaches in BMG design have largely used supervised learning with descriptors derived from composition, empirical rules, or physics-based models to classify/regress properties such as GFA and thermal metrics. These methods require human-guided sampling of composition space and face bottlenecks as compositional complexity grows (e.g., high-entropy MGs). Empirical rules (eutectic rule, confusion principle) and empirical descriptors (atomic size difference, mixing enthalpy, etc.) have been effective in supervised contexts, but their direct use in unsupervised generative models can introduce large data scattering and hinder convergence. Recent advances suggest theory-guided descriptors can improve ML performance on small datasets and may better support generative modeling. Existing literature also highlights the scarcity and diversity of BMG datasets (~300 compositions across several systems), creating challenges for model training and generalization.

Data: Four primary BMG datasets (Zr-, La-, Fe-, Ti-based), totaling ~300 literature compositions across affordable, non-toxic elements; additional expanded dataset included Cu-, ZrCu-, Ni-, and Ti-based BMGs to enrich diversity. Hierarchical clustering assessed compositional diversity and subtype structure of datasets. Descriptors: Three descriptor sets were evaluated: (D1) compositional fractions of constituent elements (normalized to [0,1]); (D2) empirical parameters (13 features) including mean atomic radius, atomic size difference, average and standard deviation of melting temperature, average and standard deviation of mixing enthalpy, ideal mixing entropy, average and standard deviation of electronegativity, valence electron concentration and its standard deviation, and mean and standard deviation of bulk modulus; (D3) theory-guided descriptors (8 features) based on models of atomic size effects (residual strain and derived scalar measures), local chemical affinity (cohesive and interaction energy statistics), and nonideal mixing entropy. Compound descriptors tested: CD1 = D1 + D3; CD2 = D1 + D2. Properties (Tg, Tx, Tl) were appended to CD1 for inverse design experiments. Generative model (GAN): Implemented in MATLAB R2020a. Generator: fully connected network with layers (100 input neurons, 1000 hidden neurons with Leaky ReLU, output sized to descriptor dimension, Sigmoid activation). Discriminator: fully connected (input matching descriptor size, 1000 hidden neurons with Leaky ReLU, single-neuron output with Sigmoid). Optimizer: Adam, learning rate 0.001. Training used small BMG datasets with random sampling to generate compositions and associated attributes (when properties included in descriptors). Convergence and performance monitored via generator/discriminator losses. Surrogate evaluator (MG classifier): Boosted Trees classifier trained on ~7000 labeled data points (~5000 MGs, ~2000 non-MGs; non-MGs oversampled by 150% using SMOTE), using 8 theory-guided descriptors (D3) and 10-fold cross-validation. This classifier served to label GAN-generated compositions as glass-forming or not. Visualization and metrics: PCA (on normalized descriptor matrices) and kernel smoothing function estimate (KSFE) for 2D density visualization; Wasserstein distance computed via Sinkhorn iterations (PyTorch) to quantify distributional differences between generated and literature data. Inverse design: Procedure involved (1) augmenting CD1 with thermal properties (Tg, Tx, Tl); (2) training GAN to output composition–property pairs (composition–property mapping); (3) selecting compositions that meet targeted property criteria (e.g., high Tl, reduced Tg/Tl). Candidate compositions were validated experimentally. Experimental validation: Alloys prepared via arc-melting and Cu-mold casting into 2 mm diameter rods under Ar with Ti-gettered vacuum; XRD (Rigaku MiniFlex 600, Cu Kα) for amorphous confirmation; FWHM of main amorphous peak computed; DSC (METTLER TOLEDO DSC3/700 and TGA-DSC3+HT/1600) at 20 K/min, Ar 50 mL/min, to measure Tg, Tx, Tl; EDX mapping for compositional homogeneity.

- Descriptor effects: GAN training converged and performed well with D1 and CD1, but poorly with CD2 (D1+D2), indicating empirical descriptors often caused non-convergence and low-quality generation. Most generated compositions using D1 and CD1 were classified as MGs by the surrogate model, while only a fraction from CD2 passed. - Distributional analysis (Zr-based): Generated data using D1 more closely matched literature distributions than CD1 per PCA/KSFE. Wasserstein distances: 0.263 (original vs generated via D1) and 0.343 (original vs generated via CD1), confirming D1-generated distributions were closer to literature, while CD1 promoted greater diversity beyond the training distribution. - Data expansion and hybrid generation: Training on an expanded dataset (adding 44 Cu-, 21 ZrCu-, 17 Ni-, and 4 Ti-based BMGs) yielded 100 generated BMGs including compositionally complex candidates resembling high-entropy MGs. A specific example, Zr27.3Cu26.1Hf16.9Al9.6Ti7.2Ni6.4Ag5.9Nb0.4 (at.%), was cast as a 2 mm rod and shown to be amorphous by XRD; FWHM of the main amorphous peak Ag ≈ 0.38 Å−1. EDX confirmed uniform elemental distributions and overall composition matching the generated target. - Inverse design demonstration (Zr-based): Augmented GAN with thermal properties (Tg, Tx, Tl) produced composition–property mappings with generated Tg ~550–720 K, Tx ~560–800 K, Tl ~1000–1250 K. Three candidates targeting high Tl and favorable Tg/Tl were selected: S1: Zr49.8Cu32.9Al7.4Ni3.9Ag2.4Ti1.7Nb1.4Fe0.5; S2: Zr50.5Cu36.3Ag4.7Al4.1Ni3.4Nb0.2Ti0.2; S3: Zr45.2Cu37.8Al9.1Ag4.9Ni3. All three formed 2 mm amorphous rods (XRD amorphous; Aq ≈ 0.32–0.37 Å−1). DSC-measured Tg, Tx, Tl agreed closely with the GAN-generated targets.

The study demonstrates that an unsupervised generative framework (GAN) can efficiently explore and generate candidate BMG compositions, addressing the limitations of supervised, human-guided search in high-dimensional, compositionally complex spaces. The success strongly depends on descriptor choice: theory-guided descriptors (combined with composition as CD1) stabilize GAN training with small, diverse datasets and expand the diversity of generated compositions beyond literature-like distributions. Conversely, empirical descriptors commonly used in supervised ML often introduce large data scattering that degrades generative training. By integrating properties into the descriptor, the generator yields composition–property mappings enabling inverse design—directly selecting compositions that meet targeted thermal property criteria. Experimental validations (one compositionally complex, high-entropy-like Zr–Cu–Hf–Al–Ti–Ni–Ag–Nb BMG and three inverse-designed Zr-based BMGs) confirm amorphous formation and targeted thermal characteristics, underscoring the framework’s relevance for accelerated BMG discovery and optimization.

This work introduces a generative deep-learning framework that couples a GAN-based generator with a boosted-tree surrogate classifier to design compositionally complex BMGs and perform inverse design via composition–property mappings. The framework: (1) generates novel BMG compositions, including hybrid/high-entropy-like systems, (2) leverages theory-guided descriptors to ensure convergence and diversity with limited data, and (3) enables targeted discovery by embedding properties into the generative process. Experimental validations confirm the feasibility and accuracy of the approach. Future research can extend the framework to broader alloy classes, incorporate additional properties (mechanical, corrosion, processing windows), integrate uncertainty quantification and active learning for data-efficient improvement, and expand datasets to further enhance generalization and robustness.

- Dataset size is small (~300 BMG compositions across four systems), posing challenges for training and generalization; performance is sensitive to descriptor design. - Empirical descriptor-based compound features (CD2) often led to GAN non-convergence or poor outputs, highlighting fragility to descriptor choices. - Experimental validation covered a limited number of generated compositions (one compositionally complex alloy and three inverse-designed Zr-based alloys), leaving broader validation to future work. - The surrogate classifier’s labeling influences evaluation; any biases or errors in the surrogate could affect assessment of generated candidates. - Property focus was on thermal parameters (Tg, Tx, Tl); other critical properties (mechanical behavior, corrosion resistance, processing robustness) were not explored within the generative loop.