Molecular Potential Energy Computation via Graph Edge Aggregate Attention Neural Network
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Abstract: Accurate potential energy surface (PES) calculation is the basis of molecular dynamics research. Using deep learning (DL) methods can improve the speed of PES calculation while achieving competitive accuracy to ab initio methods. However, the performance of DL model is extremely sensitive to the distribution of training data. Without sufficient training data, the DL model suffers from overfitting issues that lead to catastrophic performance degradation on unseen samples. To solve this problem, based on the message passing paradigm of graph neural networks, we innovatively propose an edge-aggregate-attention mechanism, which specifies the weight based on node and edge information. Experiments on MD17 and QM9 datasets show that our model not only achieves higher PES calculation accuracy but also has better generalization ability, compare with Schnet, which demonstrates that edge-aggregate-attention can better capture the inherent features of equilibrium and non-equilibrium molecular conformations.
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Table I. Mean absolute errors for energy and force prediction ( in kcal/mol and kcal·mol−1· Å−1 ) for different training samples on MD17 dataset. GM-sNN [18], EANN [16], SchNet [33], PhysNet [35] results are compared. EANN does not provide results on Benzene molecule.The lowest error is emphasized in bold.
Mean absolute error N = 1000 N = 50000 Schnet EANN GM-sNN EAA Schnet PhysNet GM-sNN EAA Benzene Energy 0.08 0.08 ${\bf{0.06}}$ ${\bf{0.07}}$ ${\bf{0.07}}$ ${\bf{0.07}}$ 0.10 Force 0.31 ${\bf{0.21}}$ 0.34 0.17 0.15 ${\bf{0.14}}$ 0.17 Toluene Energy 0.12 ${\bf{0.11}}$ 0.15 ${\bf{0.11}}$ 0.09 0.10 0.14 ${\bf{0.07}}$ Force 0.57 0.38 ${\bf{0.34}}$ 0.42 0.09 ${\bf{0.03}}$ 0.10 0.05 Malonaldehyde Energy 0.13 0.14 0.12 ${\bf{0.10}}$ 0.08 ${\bf{0.07}}$ 0.12 ${\bf{0.07}}$ Force 0.66 0.62 0.45 ${\bf{0.40}}$ 0.08 ${\bf{0.04}}$ 0.08 ${\bf{0.04}}$ Salicylic acid Energy 0.20 ${\bf{0.14}}$ 0.19 0.16 0.10 0.11 0.19 ${\bf{ 0.09}}$ Force 0.85 0.51 ${\bf{0.49}}$ 0.52 0.19 ${\bf{0.04}}$ 0.14 0.09 Aspirin Energy 0.37 0.33 0.38 ${\bf{0.32}}$ 0.12 0.12 0.19 ${\bf{0.11}}$ Force 1.35 0.99 ${\bf{0.69}}$ 0.99 0.33 ${\bf{0.06}}$ 0.26 0.14 Ethanol Energy 0.08 0.10 0.10 ${\bf{0.07}}$ 0.05 0.05 0.05 ${\bf{0.04}}$ Force 0.39 0.47 0.33 ${\bf{0.22}}$ 0.05 0.03 0.06 ${\bf{0.02}}$ Uracil Energy 0.14 ${\bf{0.11}}$ 0.12 ${\bf{0.11}}$ ${\bf{0.10}}$ ${\bf{0.10}}$ ${\bf{0.10}}$ ${\bf{0.10}}$ Force 0.56 0.35 ${\bf{0.33}}$ 0.36 0.11 ${\bf{0.03}}$ 0.07 0.04 Naphthalene Energy 0.16 ${\bf{0.12}}$ 0.17 0.15 0.11 0.12 0.13 ${\bf{0.08}}$ Force 0.58 ${\bf{0.27}}$ 0.36 0.36 0.11 ${\bf{0.04}}$ 0.13 0.05 Table II. Results with different training samples N. Scores are given by mean absolute errors of energy (kcal/mol) and force (kcal/(mol· Å)) prediction.
Dataset N Mean absolute error Schnet EAA Energy Force Energy Force Malonaldehyde 200 0.48 1.25 0.28 1.44 400 0.27 0.92 0.18 0.89 600 0.21 0.78 0.14 0.66 800 0.17 0.71 0.12 0.52 1000 0.13 0.66 0.10 0.40 Uracil 200 0.40 1.45 0.20 1.12 400 0.36 1.09 0.15 0.58 600 0.28 0.89 0.13 0.50 800 0.20 0.74 0.12 0.42 1000 0.14 0.56 0.11 0.36 Table III. MAE results with different attention mechanisms. Scores are given by mean absolute errors of energy (kcal/mol) and force (kcal/mol/Å) prediction.
Attention
on nodeAttention
on edgeMalonaldehyde Uracil Energy Force Energy Force W/O W/O 0.13 0.66 0.14 0.56 W W/O 0.14 0.49 0.12 0.44 W W 0.10 0.44 0.11 0.36 Table IV. Mean absolute error per molecule of predictions for different target properties of the QM9 dataset using 110k training examples. The lowest error is emphasized in bold. Provably powerful graph networks (PPGN) [19], SchNet [33] and enn-s2s [24] results are compared.
Parameters Description Unit PPGN SchNet enn-s2s EAA ${\epsilon_{\rm{homo}} }$ Energy of highest occupied molecular orbital ${\rm{meV}}$ 40 41 43 ${\bf{33}}$ ${\epsilon_{\rm{lumo}} }$ Energy of lowest unoccupied molecular orbital ${\rm{meV}}$ 33 34 37 ${\bf{27}}$ ${\Delta\epsilon }$ Difference between LUMO and HOMO ${\rm{meV}}$ 60 63 69 ${\bf{54}}$ ${\rm{ZPVE}}$ Dipole moment ${\rm{meV} }$ 3.12 1.7 ${\bf{1.5}}$ 1.6 ${\mu}$ Dipole moment ${\rm{Debye}}$ 0.047 0.033 ${\bf{0.030}}$ 0.042 ${\alpha}$ Isotropic polarizability ${\rm{Bohr}^3}$ 0.131 0.235 0.092 ${\bf{0.086}}$ $\langle R^2 \rangle$ Electronic spatial extent ${\rm{Bohr}^2}$ 0.592 ${\bf{0.073}}$ 0.180 0.241 ${U_0 }$ Internal energy at 0 K ${\rm{meV}}$ 37 14 19 ${\bf{12}}$ ${U}$ Internal energy at 298.15 K ${\rm{meV}}$ 37 19 19 ${\bf{15}}$ ${H}$ Enthalpy at 298.15 K ${\rm{meV}}$ 36 ${\bf{14}}$ 17 ${\bf{14}}$ ${G}$ Free energy at 298.15 K ${\rm{meV}}$ 36 ${\bf{14}}$ 19 ${\bf{14}}$ ${C_v}$ Heat capacity at 298.15 K ${\rm{cal\cdot mol^{-1}\cdot K^{-1}} }$ 0.055 0.033 0.040 ${\bf{0.030}}$ -
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