Molecular Potential Energy Computation via Graph Edge Aggregate Attention Neural Network

Jian Chang¹,
Yiming Kuai^{1, 2},
Xian Wei²,
Hui Yu^{2, 3},
Hai Lan^{2, 3
, ,}

1.
College of Software, Liaoning Technical University, Huludao 125105, China
2.
Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences, Fuzhou 362200, China
c. Fujian Science & Technology Innovation Laboratory for Optoelectronic Information of China, Fuzhou, 350108, China

More Information

Corresponding author: E-mail: lanhai09@fjirsm.ac.cn

摘要

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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.
- Potential energy surface /
- Message passing paradigm /
- Attention mechanism /
- Graph deep learning

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Figure 1. Framework of our model. MPNN is used as backbone (middle). The framework consists of feature embedding block (orange dotted box), EAA interaction block (blue dotted box), output block (green dotted box).

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Figure 2. Schematic diagram of EAA.

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Figure 3. Potential energy distribution of eight organic molecules in MD17 dataset.

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Figure 4. The fitting errors for the malonaldehyde (upper), uracil (bottom) testing dataset with Schnet (left) and EAA (right). The horizontal axis denotes the distribution of ground truth energies of a given molecule dataset.

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Figure 5. (a) Performance comparison of models with different head numbers. (b) Performance comparison of models with different layer numbers.

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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

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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

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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 node	Attention on edge	Malonaldehyde		Uracil
Attention on node	Attention on edge	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

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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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