Table 1 UQ statistics for the crystal formation energy of different crystal phases relative to 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

− 1.30 × 10−2

4.60 × 10−2

− 1.53 × 10−1

1.07 × 10−1

σ

− 5.33 × 10−2

3.49 × 10−2

− 1.45 × 10−1

8.25 × 10−2

Al4W

− 3.00 × 10−2

3.42 × 10−2

− 1.10 × 10−1

9.30 × 10−2

χ

− 5.81 × 10−2

3.58 × 10−2

− 1.61 × 10−1

3.76 × 10−2

μ

− 5.92 × 10−2

4.07 × 10−2

− 1.80 × 10−1

6.03 × 10−2

C15

− 7.25 × 10−2

5.03 × 10−2

− 2.09 × 10−1

8.43 × 10−2

C36

− 6.43 × 10−2

4.25 × 10−2

− 1.86 × 10−1

1.06 × 10−1

C14

− 8.01 × 10−2

4.46 × 10−2

− 2.29 × 10−1

8.08 × 10−2

Al5W

− 1.74 × 10−2

2.08 × 10−2

− 7.41 × 10−2

4.44 × 10−2

fcc

− 2.98 × 10−2

2.15 × 10−2

− 9.83 × 10−2

3.71 × 10−2

hcp

− 4.67 × 10−2

2.84 × 10−2

− 1.31 × 10−1

3.24 × 10−2

Al − 12W

6.69 × 10−3

3.87 × 10−2

− 1.27 × 10−1

1.09 × 10−1

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

1.21

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

0.19

  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.