From individual elements to macroscopic materials: in search of new superconductors via machine learning

The study addresses the challenge of discovering new superconductors, a process traditionally guided by expert intuition and exhibiting a low success rate (~3%). The authors propose leveraging machine learning, specifically a Deep Sets architecture, to handle materials as unordered sets of constituent elements. The research aims to: (i) classify whether a compound is superconducting or not and (ii) regress its critical temperature (Tc) directly from elemental composition, avoiding arbitrary feature engineering or biases from ordered representations (e.g., periodic table layouts). The work underscores the importance of robust generalization to unseen materials and the potential to accelerate superconductor discovery beyond conventional methods.

Prior work includes random forest models applied to large superconducting databases (e.g., Stanev et al., predicting Tc>10 K) that relied on hand-crafted chemical features, which may introduce bias. Convolutional neural networks mapping stoichiometries to a 2D periodic table representation (e.g., Konno et al.) bypass explicit feature engineering but still depend on chosen spatial encodings (rows/columns), which may impose unintended structure. Reported metrics can be misleading when test sets differ (e.g., high r² but large RMSE≈80 K in Li et al., 2020). The present approach adopts Deep Sets to ensure permutation invariance and automated feature learning directly from element-level descriptors, aiming to improve both regression accuracy and classification performance over these earlier methods.

Model: A Deep Sets architecture is used, with two neural-network components: φ (element-wise encoder) and ρ (set-level decoder). Inputs are sets of elements forming a compound; permutation invariance is enforced by summing φ-transformed element embeddings before passing to ρ. Input features: For each element (Z<96), a 22-feature vector from the Mendeleev Python API is used: atomic number, atomic volume, periodic-table block, density (295 K), dipole polarizability, electron affinity, evaporation heat, fusion heat, group, lattice constant, lattice structure symbol, melting temperature, period, specific heat (293.15 K), thermal conductivity (293.15 K), van der Waals radius, covalent radius, Pauling electronegativity, atomic weight, atomic radius, ionization energies (eV), and valence. Missing values (~4%) were supplemented from Mathematica datasets. The stoichiometric integer for each element is appended. Embeddings are linearly combined with stoichiometric weights. Latent space: Main results use latent dimension d=300; interpretability studies in SI consider d=1. Datasets: SuperCon (after cleaning: 16,395 entries across oxides/metallic and organic superconductors). For regression, SuperCon is split 80/20 into train/test, repeated 50 times (random splits). For classification, a "garbage in" strategy augments non-superconductors using the Crystallography Open Database (COD): randomly sampled entries (up to ~50,000) are labeled non-superconducting under the assumption that superconductors are a small fraction. The Hosono database (distilled set: 207 materials, 39 superconductors and 168 non-superconductors) is used as an external test. Training: Implemented in TensorFlow with Adam (lr=0.001), max 400 epochs, early stopping (patience=40), batch size 64, hyperparameters tuned via Hyperband. - Regression model: φ is a 7-layer MLP (96–992 nodes, ReLU; final layer linear), ρ is a 13-layer MLP (128–960 nodes, ReLU). Loss: MSE. - Classification model: φ is a 4-layer MLP (300 nodes/layer, ReLU); ρ is a 3-layer MLP (300, 300, 100 nodes + sigmoid output). Loss: binary cross-entropy. Outputs p∈[0,1]; classify SC if p>ε_th. A majority-vote ensemble over 60 independent runs improves robustness; ensemble threshold Eth (e.g., 0.85) is used to declare SC if the majority predict SC at that ε_th. Experimental validation: Synthetic analogues of three mineral candidates predicted as superconductors (temagamite Pd₃HgTe₃, michenerite PdBiTe, monchetundraite Pd₂NiTe₂) were synthesized in sealed silica tubes. Composition was verified via EPMA (CAMECA SX-100) and powder XRD (Bruker D8; Rietveld refinements with Topas 5). AC and DC magnetic measurements (Quantum Design MPMS, He-3 insert) probed superconducting transitions and critical fields under zero-field-cooled conditions, with H_ac≈3 Oe and 10 Hz for AC measurements. Impurity phases and space groups were quantified via Rietveld analysis.

- Regression performance: On SuperCon test sets (50 random splits), predicted vs observed Tc aligns near the diagonal. Overall RMSE≈9.5 K with r²≈0.92; ensemble averaging reduces RMSE to ≈9 K with r²≈0.93. On the Hosono set, RMSE≈7 K and r²≈0.84. - Classification performance: Precision–recall on SuperCon+COD can exceed 90% for both precision and recall with suitable ε_th. For the Hosono set, using an ensemble majority rule with Eth=0.85 identified 29/39 superconductors with 19 false positives; baseline precision for Hosono is 20% (random), and the method substantially exceeds this. Compared with prior methods on similar datasets, the Deep Sets approach exhibits superior precision/recall and F1. - Mineral screening: Applying the trained classifier (Eth=0.85, 60 runs) to the International Mineralogical Association list yielded candidates of which ~44% had prior reports of superconductivity (29% when excluding high-pressure and thin-film cases). None of the minerals were in training. - Experimental validation: Two of three selected mineral analogues showed superconductivity with Tc close to predictions: • PdBiTe (michenerite): predicted Tc=1.6±0.8 K; measured Tc=2.10±0.05 K; impurity α-PdBi₂ at 0.9% (Tc≈1.7 K), but the observed susceptibility suggests intrinsic SC. • Pd₂NiTe₂ (monchetundraite): predicted Tc=1.18±0.7 K; measured Tc=1.06±0.05 K; impurity Pd₀.₇Ni₀.₃Te (PdTe-type) at 4.5% but unlikely to account for the observed sharp transition; AC/DC field dependence at 0.50 K shows first critical field ≈20 Oe and suppression of diamagnetism by ≈160 Oe (AC) and ≈70 Oe (DC), indicating type-II behavior. • Pd₃HgTe₃ (temagamite): predicted Tc=1.8±1.8 K; no SC observed down to 0.45 K, despite minor PdTe impurity. - Interpretability (d=1 latent space): An element-wise scalar contribution x₁ highlights elemental trends (e.g., Ca associated with higher Tc materials); a simple threshold on summed x₁ achieves ≈84% accuracy for classifying above/below Tc=10 K. Median Tc for Ca-containing superconductors is ≈73 K (IQR ≈41–88 K).

Treating compositions as unordered sets via Deep Sets eliminates biases from arbitrary ordering or 2D periodic-table mappings and allows data-driven feature learning from elemental descriptors. The strong regression accuracy (RMSE≈9–9.5 K on SuperCon; 7 K on Hosono) demonstrates that elemental composition contains sufficient signal to estimate Tc across diverse families. The classifier, trained with explicit non-superconductors via the garbage-in strategy, robustly distinguishes SC vs non-SC and generalizes to a challenging, expert-curated Hosono set, outperforming prior DL approaches. The compact d=1 latent space offers interpretability by assigning per-element contributions that linearly compose to a material score, enabling chemical insight (e.g., the prominence of Ca in higher-Tc materials) while sacrificing only modest performance. Experimentally, two predicted superconductors (PdBiTe and Pd₂NiTe₂) were confirmed with Tc in close agreement to predictions, validating the pipeline end-to-end; Pd₂NiTe₂ represents the first certified superconductor identified by an AI methodology. The magnetic-field dependence in Pd₂NiTe₂ suggests type-II behavior, consistent with complex vortex dynamics in granular powders. The absence of SC in Pd₃HgTe₃ down to 0.45 K underscores prediction uncertainty and the role of sample quality and impurities.

This work introduces a Deep Sets-based ML framework for superconductor discovery that directly ingests elemental compositions, achieving state-of-the-art Tc regression accuracy and robust SC classification. Applied to a comprehensive mineral list, the model prioritized candidates, two of which (PdBiTe and Pd₂NiTe₂) were experimentally confirmed as superconductors with Tc matching predictions; notably, Pd₂NiTe₂ is the first superconductor certified via AI-guided discovery. The approach offers interpretable insights when compressing the latent space and demonstrates superiority to prior feature-engineered and CNN periodic-table methods. Future directions include enriching databases with reliable negatives, annotating contextual factors (pressure, film thickness), improving sample purity in validation experiments, exploring ensemble strategies, and integrating structural or electronic descriptors to further refine predictions and interpretability.

- Database assumptions: The garbage-in strategy labels COD entries as non-superconductors, potentially introducing mislabeled positives; however, this is expected to be a small fraction. - Annotation gaps: Current models do not account for pressure, dimensionality (e.g., thin films), or synthesis details that can critically affect superconductivity. - Experimental constraints: Validation samples contained impurities and were measured on loose powders, complicating precise volume-fraction estimates and Meissner fraction due to demagnetization and grain-size effects; SC shielding was incomplete at lowest temperatures. - Model trade-offs: Reducing latent dimension (d=1) improves interpretability at the expense of some predictive performance. - Feature coverage: Despite efforts, ~4% of elemental properties were imputed from alternative sources; residual inconsistencies may affect learning. - Generalizability: Predictions were based solely on elemental composition without explicit structural/electronic features, which may limit accuracy for specific families.