Physics-guided auxiliary multi-task learning for temperature-dependent thermoelectric 𝑧𝑇 curve prediction

Technical Summary: Physics-guided Auxiliary Multi-task Learning for Temperature-Dependent Thermoelectric $zT$ Curve Prediction

Problem Statement
Thermoelectric performance is governed by the dimensionless figure of merit $zT = S^2\sigma T / \kappa$, where $S$ is the Seebeck coefficient, $\sigma$ is electrical conductivity, and $\kappa$ is thermal conductivity. These properties are strongly coupled and temperature-dependent; the magnitude, position, and width of the $zT(T)$ maximum determine the useful operating window for materials. However, comprehensive experimental characterization is laborious due to the need for dense temperature measurements across different instruments. While machine learning (ML) offers a route to accelerate discovery, existing approaches often rely on pointwise predictions or fail to account for the physical consistency between transport coefficients ($S, \sigma, \kappa$) and derived metrics ($PF, zT$). Furthermore, standard random splitting of data can lead to data leakage where temperatures from the same material appear in both training and test sets, failing to evaluate true generalization to unseen materials.

Methodology
The author proposes a $zT$-prioritized, physics-guided auxiliary multi-task learning framework designed to predict full temperature-dependent curves from chemical composition and temperature.

• Data Curation and Representation:

  • A curated dataset of 4,222 observations from 755 formula-defined materials and 270 elemental systems (275–1275 K) was assembled from literature and databases.
  • Rigorous consistency checks were applied: reported $PF$ and $zT$ values were verified against recalculated values from $S, \sigma, \kappa$, and $T$. Discrepancies >1% ($PF$) or >5% ($zT$) were investigated, and uniformly recalculated labels were used.
  • Features: 132 Magpie composition descriptors were reduced via variance filtering, correlation filtering ($|r| > 0.95$), and Principal Component Analysis (PCA) to 96 principal components.
  • Temperature Encoding: Temperature was normalized and expanded using Fourier encoding (4 frequencies) to capture nonlinear dependencies.

• Network Architecture:

  • A multi-task neural network with a shared trunk and five task-specific heads.
  • Inputs: Concatenated 96-dimensional composition vector and 9-dimensional Fourier temperature vector.
  • Outputs: Simultaneous prediction of $S, \sigma, \kappa, PF$, and $zT$.
  • Auxiliary vs. Principal: While all properties are predicted, the $zT$ head is the principal task, with the other properties serving as auxiliary supervision.

• Training Strategy and Physics Guidance:

  • Two-Stage Optimization:
    • Stage 1: Promotes shared multi-property learning with balanced task weights.
    • Stage 2: Reduces auxiliary weights and concentrates optimization on $zT$.
  • High-$zT$ Reweighting: A smooth loss weight function emphasizes high-performance regions (70th percentile and above) to address imbalanced continuous regression, increasing weights from ~1 to ~3 for high-$zT$ observations.
  • Dynamic Physical Consistency: The model reconstructs $PF$ and $zT$ from the predicted $S, \sigma, \kappa$ using thermoelectric relations ($\hat{PF}_{phys} = \hat{S}^2\hat{\sigma}$, $\hat{zT}_{phys} = \hat{PF}_{phys}T/\hat{\kappa}$). A consistency penalty (Smooth L1 loss in log space) is applied between the direct predictions and these reconstructed values. This penalty is introduced gradually (30-epoch cosine ramp) to allow auxiliary heads to stabilize before enforcing strong physical constraints.
  • Curve Smoothing: A weak regularizer penalizes large second differences in the predicted $zT$ curve to suppress unsupported local oscillations.

• Evaluation Protocol:

  • Material-Grouped Splitting: All temperature points for a specific material are kept in the same fold to prevent data leakage and ensure evaluation on unseen materials.
  • Ensemble: A 10-seed ensemble (seeds 42–51) is used to average predictions and stabilize the learned response.
  • Metrics: Evaluation includes pointwise metrics (MAE, RMSE, $R^2$) and curve-level metrics (peak $zT$ error, peak temperature error).

Key Contributions

1. $zT$-Prioritized Framework: A novel multi-task architecture that explicitly prioritizes the screening-relevant $zT$ metric while leveraging coupled transport properties ($S, \sigma, \kappa$) as auxiliary tasks.
2. Progressive Physics-Guided Learning: The introduction of a dynamic consistency penalty that couples the direct $zT$ prediction with physics-reconstructed values, ensuring the model respects thermoelectric relations without destabilizing early training.
3. Rigorous Data Curation: A workflow that integrates literature-derived data with explicit unit checks and consistency verification against physical laws, creating a high-quality dataset for curve prediction.
4. Material-Level Generalization: The use of material-disjoint cross-validation and external validation on literature-derived curves to demonstrate true generalization capabilities beyond interpolation.

Results

• Training Performance: The 10-seed ensemble achieved $R^2 = 0.9563$ for $PF$ and $R^2 = 0.9405$ for $zT$ on the training set. The direct and physics-reconstructed outputs remained closely aligned (log-space MAE of 0.0399 for $PF$ and 0.0441 for $zT$).
• Generalization (Out-of-Fold): In 10-fold material-grouped cross-validation, the model achieved $R^2 = 0.8927$ for $PF$ and $R^2 = 0.8765$ for $zT$. The $zT$ MAE was 0.0764, and RMSE was 0.1166. The peak $zT$ MAE was 0.1141, and peak temperature MAE was 14.07 K.
• External Validation: On four literature-derived materials (BaSnS$_3$, Na$_2$InCuF$_6$, LaOBiS$_2$, MgI$_2$Bi$_2$S$_3$) comprising 24 data points, the model achieved an overall $R^2 = 0.9755$, MAE = 0.0128, and RMSE = 0.0198.
• Curve Prediction: The model successfully reproduced the temperature-dependent trends, peak locations, and magnitudes for unseen materials across the 275–1275 K range.

Significance
The paper claims that this workflow provides a reliable, composition-based surrogate for screening thermoelectric materials. By prioritizing $zT$ and enforcing physical consistency, the model supports candidate ranking and operating-window selection (identifying the temperature range of maximum performance) without requiring detailed crystal structure information. The study demonstrates that integrating physical constraints and multi-task learning can effectively bridge the gap between composition and complex, temperature-dependent transport properties, offering a practical tool for materials informatics in thermoelectrics.