Table 2 UQ statistics for the crystal formation volume of different crystal phases relative to the volume of the BCC phase (c.f., Fig. 3)

From: Uncertainty quantification for misspecified machine learned interatomic potentials

Structure

MLE Error

Std

Low Error

High Error

bcc

0.00 × 100

0.00 × 100

0.00 × 100

0.00 × 100

A15

− 8.41 × 10−3

1.77 × 10−2

− 5.19 × 10−2

5.84 × 10−2

σ

− 2.36 × 10−3

1.35 × 10−2

− 3.28 × 10−2

4.57 × 10−2

Al4W

− 3.48 × 10−3

1.35 × 10−2

− 3.52 × 10−2

4.62 × 10−2

χ

− 1.08 × 10−2

1.19 × 10−2

− 4.26 × 10−2

2.72 × 10−2

μ

− 2.78 × 10−3

1.41 × 10−2

− 4.23 × 10−2

4.64 × 10−2

C15

− 1.78 × 10−3

2.08 × 10−2

− 5.88 × 10−2

5.46 × 10−2

C36

− 3.78 × 10−3

1.78 × 10−2

− 5.46 × 10−2

4.24 × 10−2

C14

− 1.27 × 10−2

1.53 × 10−2

− 5.57 × 10−2

3.13 × 10−2

Al5W

1.10 × 10−2

7.93 × 10−3

− 1.62 × 10−2

3.52 × 10−2

fcc

1.86 × 10−3

9.07 × 10−3

− 2.54 × 10−2

2.52 × 10−2

hcp

1.35 × 10−2

9.44 × 10−3

− 1.93 × 10−2

4.33 × 10−2

Al − 12W

1.34 × 10−3

1.05 × 10−2

− 3.85 × 10−2

2.90 × 10−2

\({\rm{Mean}}\,\frac{{\rm{Error}}}{{\rm{Std}}}\)

0.52

\({\rm{Mean}}\,\frac{{\rm{Error}}}{{\rm{High}}-{\rm{Low}}}\)

0.085

  1. MLE error corresponds to the errors of the MLE relative to the DFT reference, Std is the standard deviation of the predictions of \({\pi }_{{\mathcal{H}}}^{* }\), Low Error is the difference between the smallest prediction in \({\pi }_{{\mathcal{H}}}^{* }\) and the DFT reference, and High Error is the difference between the largest prediction in \({\pi }_{{\mathcal{H}}}^{* }\) and the DFT reference. A negative Low Error and a positive High Error indicate that the predictions from the UQ ensemble bracket the reference value.