cond-mat.dis-nnApr 6, 2026
SaveInterpretation of Crystal Energy Landscapes with Kolmogorov-Arnold Networks
Organizations: Max Planck Institute for Chemical Physics of Solids, 01187, Dresden, Germany · Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA · Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
Abstract
Characterizing crystalline energy landscapes is essential to predicting thermodynamic stability, electronic structure, and functional behavior. While machine learning (ML) enables rapid property predictions, the "black-box" nature of most models limits their utility for generating new scientific insights. Here, we introduce Kolmogorov-Arnold Networks (KANs) as an interpretable framework to bridge this gap. Unlike conventional neural networks with fixed activation functions, KANs employ learnable functions that reveal underlying physical relationships. We developed the Element-Weighted KAN, a composition-only model that achieves state-of-the-art accuracy in predicting formation energy, band gap, and work function across large-scale datasets. Crucially, without any explicit physical constraints, KANs uncover interpretable chemical trends aligned with the periodic table and quantum mechanical principles through embedding analysis, correlation studies, and principal component analysis. These results demonstrate that KANs provide a powerful framework with high predictive performance and scientific interpretability, establishing a new paradigm for transparent, chemistry-based materials informatics.
Figures & tables
Figure 1: Workflow schematic of the Element-Weighted Kolmogorov–Arnold Network (EWKAN). Elemental compositions are transformed into weighted embeddings that capture each element’s contribution to the target property. These embeddings are passed into a two-layer KAN with trainable B-spline activation functions, which provide interpretable nonlinear mappings. The output layer yields property predictions such as band gap or formation energy.
Figure 2: Bandgap prediction under EWKAN model. (a) Bandgap schematic diagram. (b) Histogram of data distribution of matbench_mp_gap. (c) MAE results for bandgap prediction under different KAN structure.
Figure 3: Work function prediction under EWKAN model. (a) Work function schematic diagram. (b) Histogram of data distribution of JARVIS-C2DB containing work function entries. (c) MAE results for work function prediction under different KAN structure.
Figure 4: Formation energy prediction under EWKAN model. (a) Formation energy schematic diagram(take sodium chloride as an example). (b) Histogram of data distribution of JARVIS-DFT containing formation energy entries. (c) MAE results for formation energy prediction under different KAN structure.
Figure 5: Comparison of (a) the original PC1 values obtained from the KAN model and (b) normalized Pauling electronegativity across the periodic table layout. The close resemblance between the two maps indicates that the dominant latent axis learned by the model captures an effective “electronegativity–metallicity” dimension, highlighting the emergence of chemically interpretable features from a purely data-driven representation. According to the PCA variance analysis (Fig. S11 ), the PC1 achieves 32.69% variance while the first four components account for over 80% of the total variance
Figure 6: Comparison of model efficiency and performance across different materials property prediction tasks. (a) Mean absolute error (MAE) vs. number of parameters for band gap prediction models. (b) MAE vs. number of parameters for formation energy prediction. Dashed lines represent iso-efficiency contours, with higher curves indicating better accuracy–efficiency trade-offs. These contours are derived following the NetScore methodology [ 62 ] , which jointly considers prediction accuracy and model complexity; see Supplementary Materials for details. EWKAN (this work) achieves competitive MAEs with significantly fewer parameters, demonstrating strong predictive efficiency in both tasks
Figure S1: Bandgap dataset analysis (matbench_mp_gap). (a) Frequency distribution of bandgap values showing a heavily right-skewed distribution with mean 1.21 eV and median 0.27 eV, indicating a predominance of small-bandgap materials. (b) Box plot summary revealing the statistical spread of bandgap values across the materials dataset, with outliers extending to approximately 10 eV. (c) Cumulative distribution function demonstrating that approximately 25%, 50%, and 75% of materials exhibit bandgaps below 0.27 eV, 1.21 eV, and 2.5 eV, respectively. (d) Scatter plot of bandgap versus number of elements per compound, color-coded by bandgap magnitude, showing that binary and ternary compounds span the full range of bandgap values, while higher-order compounds tend toward smaller bandgaps. The dataset comprises diverse inorganic compounds with bandgaps ranging from 0 to 9.8 eV, providing a comprehensive benchmark for evaluating machine learning models on semiconductor property prediction
Figure S2: Work function dataset analysis (C2DB). (a) Frequency distribution of work function values exhibiting a near-normal distribution with mean 4.90 eV and median 4.91 eV, centered around typical values for two-dimensional materials. (b) Box plot summary illustrating the statistical spread of work function values across the C2DB dataset, with interquartile range spanning approximately 4.5–5.7 eV and outliers extending to both lower and higher extremes. (c) Cumulative distribution function showing that 25%, 50%, and 75% of materials exhibit work functions below approximately 4.5 eV, 4.9 eV, and 5.7 eV, respectively, with vertical dashed lines marking key quartiles. (d) Scatter plot of work function versus number of elements per compound, color-coded by work function magnitude, revealing that the dataset is dominated by binary compounds with relatively uniform work function distribution, while ternary and quaternary systems show similar ranges but with reduced sample density. The C2DB dataset provides a comprehensive collection of two-dimensional materials with work functions ranging from approximately 1.5 to 9.0 eV, representing diverse electronic properties relevant for surface science and device applications
Figure S3: Formation energy dataset analysis (JARVIS-DFT). (a) Frequency distribution of formation energies showing a slightly right-skewed distribution with mean eV/atom and median eV/atom, indicating that most compounds in the dataset are thermodynamically stable relative to their constituent elements. (b) Box plot summary revealing the statistical spread of formation energies, with the majority of compounds exhibiting negative formation energies (thermodynamically favorable) and outliers extending to both highly stable ( eV/atom) and metastable (5.0 eV/atom) regimes. (c) Cumulative distribution function demonstrating that approximately 25%, 50%, and 75% of materials exhibit formation energies below eV/atom, eV/atom, and 0 eV/atom, respectively. (d) Scatter plot of formation energy versus number of elements per compound, color-coded by formation energy magnitude, showing that binary and ternary compounds dominate the dataset and span the full stability range, while higher-order compounds tend toward intermediate formation energies. The JARVIS-DFT dataset encompasses diverse inorganic compounds with formation energies ranging from to 5.0 eV/atom, providing a comprehensive benchmark for thermodynamic stability prediction across different chemical compositions and structural complexities
Figure S4: Bandgap prediction for the ElementWeighted Model. (a) use the as the activation function to strict the final output. (b) use the linear combination of the weighted contribution for each element as the output directly
| Model | Mean Absolute Error (eV) | |||
| Linear Model, | 0.8835±0.0071 | |||
| ReLU Model, | 0.6344±0.0071 |
Table S1: MAE results comparison for and linear combination method under 10-fold cross-validation approach
Figure S5: Periodic table visualization of the learned parameters
Figure S6: Different pre-defined activation function test for ElementWeighted model
Figure S7: Simple efficiency score under different KAN structure for three different properties
Figure S8: Evolution of KAN activation functions along PC1 across different architectures for bandgap. The explained variance percentage indicates the information concentration in the primary component, revealing the learning convergence and functional complexity evolution from simple structures to the optimal configuration
Figure S9: Evolution of KAN activation functions along PC1 across different architectures for work function. The explained variance percentage indicates the information concentration in the primary component, revealing the learning convergence and functional complexity evolution from simple structures to the optimal configuration
Figure S10: Evolution of KAN activation functions along PC1 across different architectures for formation energy. The explained variance percentage indicates the information concentration in the primary component, revealing the learning convergence and functional complexity evolution from simple structures to the optimal configuration
Figure S11: Explained variance ratio of the seven principal components. The bar plot indicates the variance explained by each individual component, while the red line shows the cumulative explained variance. PC1 alone accounts for the largest fraction of the variance, and the first three PCs together explain over 70% of the total variance, demonstrating that a small number of latent dimensions capture the dominant chemical trends in the input representation
Figure S12: PCA projection of elements colored by chemical category. The first two principal components (PC1 and PC2) capture 32.69% and 22.19% of the variance, respectively. Clear grouping of elements according to their chemical families (e.g., alkali metals, transition metals, halogens, noble gases) indicates that the latent space preserves chemically meaningful structure, with separation driven largely by electronegativity and atomic radius trends
Figure S13: (a) Learned activation functions of the seven input dimensions in the KAN model, compared with the standard ReLU (red dashed line). The learned nonlinearities deviate substantially from the fixed ReLU form, indicating that the model adapts its internal representation to capture chemically relevant transformations of the input features. (b) Correlation matrix between PCA principal components (PC1–PC7) and tabulated elemental properties. PC1 shows the strongest correlations with electronegativity (Pauling and Gordy scales) and covalent radius, confirming that the dominant latent axis learned by the model encodes a chemically meaningful “electronegativity–radius” dimension
| Model Name | MAE (eV) | Parameters (K) | Efficiency |
| CGCNN | 0.2280 | 128.0 | 0.499772 |
| MEGNet | 0.2350 | 168.0 | 0.405707 |
| ALIGNN | 0.2500 | 4030.0 | 0.054942 |
| EWKAN (This Work) | 0.3582 | 4.5 | 1.891839 |
Table S2: Bandgap Prediction Model Performance (MatBench MP Gap Dataset)
| Model Name | MAE (eV/atom) | Parameters (K) | Efficiency |
| eSEN-30M-OAM | 0.0180 | 30200.0 | 0.849351 |
| SevenNet-MF-ompa | 0.0210 | 25700.0 | 0.742525 |
| GRACE-2L-OAM | 0.0230 | 12600.0 | 0.993541 |
| DPA-3.1-3M-FT | 0.0230 | 3270.0 | 2.231925 |
| ORB v3 | 0.0240 | 25500.0 | 0.610603 |
| AlphaNet-v1-OMA | 0.0240 | 4650.0 | 1.695163 |
Table S3: Formation Energy Prediction Model Performance (MatBench Discovery Dataset)
Figure S14: Examples of chemical formula degeneracy in bandgap prediction. Bandgap differences between polymorphs of hydrogen and carbon highlight the EW-KAN model’s current limitation in distinguishing structures with identical compositions. Red dashed lines indicate the model’s composition-based predictions
Figure S15: Percentage of samples associated with polymorphic compositions in the three benchmark datasets. Numbers inside the bars indicate the number of polymorphic entries over the total dataset size. Although the fraction is relatively small (2.1% for band gap, 1.4% for formation energy, and 0.3% for work function), these cases highlight scenarios where composition-only models cannot distinguish structural variations.
Figure S16: Effect of incorporating space group information on band gap prediction. Top panels: Predicted versus DFT band gaps for the baseline (formula-only) and enhanced (+ space group) models. Bottom panels: Representative polymorphic cases showing band gap variations across different space groups. The baseline model assigns identical predictions to all phases of the same composition, whereas the enhanced model differentiates structures and improves overall accuracy (lower MAE and higher ).
Figure S17: Effect of incorporating space group information on work function prediction. Top panels: Predicted versus DFT work functions for the baseline and enhanced models. Bottom panels: Representative polymorphic materials illustrating structural splitting. Including space group information enables the model to partially resolve composition-induced degeneracy and improves predictive performance.
Figure S18: Effect of incorporating space group information on formation energy prediction. Top panels: Predicted versus DFT formation energies for the baseline and enhanced models. Bottom panels: Selected polymorphic compounds demonstrating that the baseline model collapses all phases to a single value, while the enhanced model differentiates space groups and better captures structure-dependent variations.
| Formula | Predicted (eV) | Database Value (eV) |
| Fe 3 Sn 2 | 0.0270 | 0.0288 |
| Fe 3 Sn | 0.0749 | 0.0706, 0.0706, 0.1336, 0.0586, 0.0586 |
| FeO | -1.0145 | -1.0038, -1.0979, -1.0041, -1.0141, -1.0039, -0.9914, -1.0036, -1.1110, -1.0139 |
| Fe 2 O 3 | -1.3497 | -1.4099, -0.8025, -1.4496, -1.3610, -1.3297, -1.2657 |
| Fe 3 O 4 | -1.2822 | -1.3434, -1.3431, -1.2450, -1.2103, -1.2207, -1.3553, -1.2232 |
| FeS | -0.3833 | -0.3806, -0.3819, -0.3736, -0.3502, -0.3809, -0.3499, -0.3510, |
Table S4: Predicted and database formation energies of representative Fe-based compounds.