12.1. How to tune Fitting/embedding-net size ?

Here are some test forms on fitting-net size tuning or embedding-net size tuning performed on several different systems.

12.1.1. Al2O3

12.1.1.1. Fitting net size tuning form on Al2O3: (embedding-net size: [25,50,100])

Fitting-net size

Energy L2err(eV)

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[240,240,240]

1.742252e-02

7.259383e-05

4.014115e-02

[80,80,80]

1.799349e-02

7.497287e-05

4.042977e-02

[40,40,40]

1.799036e-02

7.495984e-05

4.068806e-02

[20,20,20]

1.834032e-02

7.641801e-05

4.094784e-02

[10,10,10]

1.913058e-02

7.971073e-05

4.154775e-02

[5,5,5]

1.932914e-02

8.053808e-05

4.188052e-02

[4,4,4]

1.944832e-02

8.103467e-05

4.217826e-02

[3,3,3]

2.068631e-02

8.619296e-05

4.300497e-02

[2,2,2]

2.267962e-02

9.449840e-05

4.413609e-02

[1,1,1]

2.813596e-02

1.172332e-04

4.781115e-02

[]

3.135002e-02

1.306251e-04

5.373120e-02

[] means no hidden layer, but there is still a linear output layer. This situation is equal to the linear regression.

12.1.1.2. Embedding net size tuning form on Al2O3: (Fitting-net size: [240,240,240])

Embedding-net size

Energy L2err(eV)

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[25,50,100]

1.742252e-02

7.259383e-05

4.014115e-02

[10,20,40]

2.909990e-02

1.212496e-04

4.734667e-02

[5,10,20]

3.357767e-02

1.399070e-04

5.706385e-02

[4,8,16]

6.060367e-02

2.525153e-04

7.333304e-02

[3,6,12]

5.656043e-02

2.356685e-04

7.793539e-02

[2,4,8]

5.277023e-02

2.198759e-04

7.459995e-02

[1,2,4]

1.302282e-01

5.426174e-04

9.672238e-02

12.1.2. Cu

12.1.2.1. Fitting net size tuning form on Cu: (embedding-net size: [25,50,100])

Fitting-net size

Energy L2err(eV)

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[240,240,240]

4.135548e-02

1.615449e-04

8.940946e-02

[20,20,20]

4.323858e-02

1.689007e-04

8.955762e-02

[10,10,10]

4.399364e-02

1.718502e-04

8.962891e-02

[5,5,5]

4.468404e-02

1.745470e-04

8.970111e-02

[4,4,4]

4.463580e-02

1.743586e-04

8.972011e-02

[3,3,3]

4.493758e-02

1.755374e-04

8.971303e-02

[2,2,2]

4.500736e-02

1.758100e-04

8.973878e-02

[1,1,1]

4.542073e-02

1.774247e-04

8.964761e-02

[]

4.545168e-02

1.775456e-04

8.983201e-02

12.1.2.2. Embedding net size tuning form on Cu: (Fitting-net size: [240,240,240])

Embedding-net size

Energy L2err(eV)

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[25,50,100]

4.135548e-02

1.615449e-04

8.940946e-02

[20,40,80]

4.203562e-02

1.642016e-04

8.925881e-02

[15,30,60]

4.146672e-02

1.619794e-04

8.936911e-02

[10,20,40]

4.263060e-02

1.665258e-04

8.955818e-02

[5,10,20]

4.994913e-02

1.951138e-04

9.007786e-02

[4,8,16]

1.022157e-01

3.992802e-04

9.532119e-02

[3,6,12]

1.362098e-01

5.320695e-04

1.073860e-01

[2,4,8]

7.061800e-02

2.758515e-04

9.126418e-02

[1,2,4] && seed = 1

9.843161e-02

3.844985e-04

9.348505e-02

[1,2,4] && seed = 2

9.404335e-02

3.673568e-04

9.304089e-02

[1,2,4] && seed = 3

1.508016e-01

5.890688e-04

1.382356e-01

[1,2,4] && seed = 4

9.686949e-02

3.783965e-04

9.294820e-02

12.1.3. Water

12.1.3.1. Fitting net size tuning form on water: (embedding-net size: [25,50,100])

Fitting-net size

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[240,240,240]

9.1589E-04

5.1540E-02

[200,200,200]

9.3221E-04

5.2366E-02

[160,160,160]

9.4274E-04

5.3403E-02

[120,120,120]

9.5407E-04

5.3093E-02

[80,80,80]

9.4605E-04

5.3402E-02

[40,40,40]

9.8533E-04

5.5790E-02

[20,20,20]

1.0057E-03

5.8232E-02

[10,10,10]

1.0466E-03

6.2279E-02

[5,5,5]

1.1154E-03

6.7994E-02

[4,4,4]

1.1289E-03

6.9613E-02

[3,3,3]

1.2368E-03

7.9786E-02

[2,2,2]

1.3558E-03

9.7042E-02

[1,1,1]

1.4633E-03

1.1265E-01

[]

1.5193E-03

1.2136E-01

12.1.3.2. Embedding net size tuning form on water: (Fitting-net size: [240,240,240])

Embedding-net size

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[25,50,100]

9.1589E-04

5.1540E-02

[20,40,80]

9.5080E-04

5.3593E-02

[15,30,60]

9.7996E-04

5.6338E-02

[10,20,40]

1.0353E-03

6.2776E-02

[5,10,20]

1.1254E-03

7.3195E-02

[4,8,16]

1.2495E-03

8.0371E-02

[3,6,12]

1.3604E-03

9.9883E-02

[2,4,8]

1.4358E-03

9.7389E-02

[1,2,4]

2.1765E-03

1.7276E-01

12.1.4. Mg-Al

12.1.4.1. Fitting net size tuning form on Mg-Al: (embedding-net size: [25,50,100])

Fitting-net size

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[240,240,240]

3.9606e-03

1.6289e-02

[200,200,200]

3.9449e-03

1.6471e-02

[160,160,160]

4.0947e-03

1.6413e-02

[120,120,120]

3.9234e-03

1.6283e-02

[80,80,80]

3.9758e-03

1.6506e-02

[40,40,40]

3.9142e-03

1.6348e-02

[20,20,20]

4.1302e-03

1.7006e-02

[10,10,10]

4.3433e-03

1.7524e-02

[5,5,5]

5.3154e-03

1.9716e-02

[4,4,4]

5.4210e-03

1.9710e-02

[2,2,2]

6.2667e-03

2.2568e-02

[1,1,1]

7.3676e-03

2.6375e-02

[]

7.3999e-03

2.6097e-02

12.1.4.2. Embedding net size tuning form on Mg-Al: (Fitting-net size: [240,240,240])

Embedding-net size

Energy L2err/Natoms(eV)

Force L2err(eV/Angstrom)

[25,50,100]

3.9606e-03

1.6289e-02

[20,40,80]

4.0292e-03

1.6555e-02

[15,30,60]

4.1743e-03

1.7026e-02

[10,20,40]

4.8138e-03

1.8516e-02

[5,10,20]

5.6052e-03

2.0709e-02

[4,8,16]

6.1335e-03

2.1450e-02

[3,6,12]

6.6469e-03

2.3003e-02

[2,4,8]

6.8222e-03

2.6318e-02

[1,2,4]

1.0678e-02

3.9559e-02