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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.

Al2O3

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.

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

Cu

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

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
[3,9,12] 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

Water

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

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

Mg-Al

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

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