Table 3 Benchmarking MoE on 19 data-scarce regression tasks.

From: Towards overcoming data scarcity in materials science: unifying models and datasets with a mixture of experts framework

Downstream task (dataset size)

STL

TL from MP Ef

MoE-(18)

Expt. \({{E}_{{{{\rm{f}}}}}}^{{\mathrm{a}}}\) (1709)

0.194 ± 0.027

\(\underline{0.117\pm 0.007}\)

0.0946 ± 0.0054

\({{E}_{{{{\rm{exfol}}}}}}^{{\mathrm{b}}}\) (636)

62.0 ± 13.6

52.7 ± 9.0

\(\underline{52.8\pm 10.5}\)

\({{d}_{33}}^{{\mathrm{c}}}\) (941)

\(\underline{0.228\pm 0.033}\)

0.276 ± 0.091

0.206 ± 0.026

PhonDOS peakd (1265)

0.126 ± 0.014

0.0768 ± 0.0042

\(\underline{0.103\pm 0.009}\)

2D \({{E}_{{{{\rm{f}}}}}}^{{\mathrm{e}}}\) (633)

0.165 ± 0.024

\(\underline{0.139\pm 0.011}\)

0.105 ± 0.011

2D Eg, Optf (522)

0.693 ± 0.148

\(\underline{0.679\pm 0.100}\)

0.543 ± 0.101

2D Eg, Tbmbjg (120)

1.31 ± 0.29

0.942 ± 0.159

\(\underline{1.06\pm 0.12}\)

\({{A}^{{{{\rm{U}}}}}}^{{\mathrm{h}}}\) (1181)

3.69 ± 2.82

2.44 ± 1.49

\(\underline{3.08\pm 2.37}\)

Log\({({\epsilon }_{\infty })}^{{\mathrm{i}}}\) (1296)

0.170 ± 0.039

0.146 ± 0.032

\(\underline{0.147\pm 0.037}\)

Log\({({\epsilon }_{{{{\rm{total}}}}})}^{{\mathrm{j}}}\) (1296)

0.254 ± 0.038

\(\underline{0.238\pm 0.031}\)

0.231 ± 0.029

Poisson ratiok (1181)

\(\underline{0.0325\pm 0.0003}\)

0.0340 ± 0.0019

0.0292 ± 0.0017

\({{\epsilon }_{{{{\rm{poly}}}}}^{\infty }}^{{\mathrm{l}}}\) (1056)

2.94 ± 0.89

\(\underline{2.93\pm 0.508}\)

2.70 ± 0.667

\({{\epsilon }_{{{{\rm{poly}}}}}}^{{\mathrm{m}}}\) (1056)

\(\underline{6.40\pm 1.54}\)

7.02 ± 0.45

5.58 ± 1.37

2D n, Optn (522)

\(\underline{2.71\pm 0.55}\)

2.98 ± 0.56

2.27 ± 0.40

2D n, Tbmbjo (120)

\(\underline{6.89\pm 2.24}\)

9.47 ± 6.72

6.42 ± 1.48

3D n, PBEp (4764)

0.0860 ± 0.0131

\(\underline{0.0820\pm 0.0126}\)

0.0779 ± 0.0105

Expt. \({{E}_{{{{\rm{g}}}}}}^{{\mathrm{q}}}\) (2481)

0.460 ± 0.046

\(\underline{0.446\pm 0.074}\)

0.376 ± 0.051

ϵavg, Tbmbjr (8.043)

\(\underline{32.7\pm 2.6}\)

182. ± 139. 

28.3 ± 3.5

Eg, Tbmbjs (7348)

0.503 ± 0.018

\(\underline{0.448\pm 0.067}\)

0.353 ± 0.015

  1. aExperimental formation enthalpies (eV/atom) from Matminer’s expt_formation_enthalpy37 and expt_formation_enthalpy_kingsbury datasets36. The former was preferred when duplicates arose.
  2. bExfoliation energies (meV/atom) from Matminer’s jarvis_dft_2d dataset35.
  3. cPiezoelectric modulus from Matminer’s piezoelectric_tensor dataset34.
  4. dHighest frequency optical phonon mode peak (cm−1) from Matminer’s matbench_phonons dataset47.
  5. eFormation energies (eV/atom) from Matminer’s jarvis_dft_2d dataset35.
  6. fBand gap of 2D materials (eV) from Matminer’s jarvis_dft_2d dataset35, calculated with the OptB88vDW DFT functional.
  7. gBand gap of 2D materials (eV) from Matminer’s jarvis_dft_2d dataset35, calculated with the TBMBJ DFT functional.
  8. hElastic anisotropy index from Matminer’s elastic_tensor_2015 dataset1.
  9. iElectronic contribution to dielectric constant from Matminer’s phonon_dielectric_mp dataset47.
  10. jDielectric constant from Matminer’s phonon_dielectric_mp dataset47.
  11. kPoisson ratio from Matminer’s elastic_tensor_2015 dataset44.
  12. lAverage eigenvalue of the dielectric tensor’s electronic component from Matminer’s dielectric_constant dataset48.
  13. mAverage dielectric tensor eigenvalue from Matminer’s dielectric_constant dataset48.
  14. nDielectric constant of 2D materials, computed by the OptB88vDW functional, from Matminer’s jarvis_dft_2d dataset35.
  15. oDielectric constant of 2D materials, computed by the TBMBJ functional, from Matminer’s jarvis_dft_2d dataset35.
  16. pRefractive index from Matminer’s dielectric_constant dataset6,48.
  17. qExperimental bandgaps (eV) from Matminer’s expt_gap_kingsbury dataset38.
  18. rAverage eigenvalue of dielectric tensor, calculated with the TBMBJ DFT functional, from the jarvis_dft_3d database in Matminer45.
  19. sBand gap (eV), calculated with the TBMBJ DFT functional, from Matminer’s jarvis_dft_3d dataset45.
  20. Average test MAEs and standard deviations over five random seeds on 19 data-scarce regression tasks for single-task learning, transfer learning, and mixture of experts. The first and second-best MAEs per task are bolded and underlined, respectively. MoE source tasks are listed in Supplementary Table 2.
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