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Accurate prediction of higher-level electronic structure energies for large databases using neural networks, Hartree–Fock energies, and small subsets of the database
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10.1063/1.3231686
/content/aip/journal/jcp/131/12/10.1063/1.3231686
http://aip.metastore.ingenta.com/content/aip/journal/jcp/131/12/10.1063/1.3231686

Figures

Image of FIG. 1.
FIG. 1.

Variation in the MP4(SDQ) energies as a function of the HF energy in eV for the 68 308 configurations in the vinyl bromide database.

Image of FIG. 2.
FIG. 2.

Comparison of the spread of the differences between the predicted MP4(SDQ) energies obtained from the median network with and the computed ab initio MP4(SDQ) energies in eV. The spread increases significantly at larger energies due to the much wider spread of vinyl bromide configurations that have a given HF energy, each with differing amounts of correlation energy.

Image of FIG. 3.
FIG. 3.

Distribution of the predicted MP4(SDQ) energies obtained from the median NN with and the computed ab initio MP4(SDQ) energies in eV using GAUSSIAN-03. The MAE of the distribution is 0.0549 eV.

Image of FIG. 4.
FIG. 4.

Comparison of the spread of the differences between the predicted MP4(SDQ) energies obtained from the median network with and the computed ab initio MP4(SDQ) energies in eV. The spread increases significantly at larger energies due to the much wider spread of vinyl bromide configurations that have a given HF energy, each with differing amounts of correlation energy. It is smaller than that seen in Fig. 2 because of the much larger training set.

Image of FIG. 5.
FIG. 5.

Distribution of the predicted MP4(SDQ) energies obtained from the median NN with and a training set that comprises 60% of the database and the computed ab initio MP4(SDQ) energies in eV using GAUSSIAN-03. The MAE of the distribution is 0.0292 eV.

Image of FIG. 6.
FIG. 6.

Atom notation used to specify the NN input elements for vinyl bromide.

Image of FIG. 7.
FIG. 7.

Variation in the MAE on the testing set vs data with and without HF energy in the NN input vector.

Image of FIG. 8.
FIG. 8.

Flowchart for practical application of the method.

Image of FIG. 9.
FIG. 9.

Illustration of single-input neuron.

Image of FIG. 10.
FIG. 10.

Illustration of a NN layer with neurons.

Image of FIG. 11.
FIG. 11.

Matrix illustration of a three-layer MLP network.

Tables

Generic image for table
Table I.

Typical fitting accuracy of MSI methods for three- and four-body systems undergoing a single, two-center bond dissociation reaction.

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Table II.

Typical fitting accuracy of IMLS methods for three- and four-body systems undergoing a single, two-center bond dissociation reaction.

Generic image for table
Table III.

Typical fitting accuracy of NN methods for three-, four-, and six-body systems undergoing multiple two-, three-, and/or four-center reactions.

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Table IV.

Database sizes required for convergence of some systems whose dynamics has been investigated using ab initio methods.

Generic image for table
Table V.

Median error values, RMSE and MAE, using nine different random samplings for eight different values of for the 68 308 point database for vinyl bromide. All errors are given in units of eV. In each case, . In each case, the input vector for the (16-80-1) NN comprises the 15 bond distances and the HF energy for vinyl bromide.

Generic image for table
Table VI.

Typical MAE results obtained for each of nine different random samplings of the vinyl bromide database for , 0.40, and 0.70. All errors are given in units of eV. The standard deviations of the nine results from the median results reported in Table V are given at the bottom of each column. These deviations are a measure of the expected sampling error of the method.

Generic image for table
Table VII.

Sensitivity of the computed MSE of the NN to each of the 16 input elements for the median NNs trained with and 10% of the database used for training and with and 60% of the database used for training. The results are all normalized with the largest sensitivity being set equal to unity. See text for the definition of sensitivity. The notation for the input elements follows the atom numbering given in Fig. 6.

Generic image for table
Table VIII.

Median error values, RMSE and MAE, using nine different random samplings for four different values of for the 68 308 point database for vinyl bromide with a NN that does not contain the HF energy of the configuration as one of the input elements. Therefore, the (15-80-1) NN has only the 15 configuration coordinates in the input vector. All errors are given in units of eV. In each case, .

Generic image for table
Table IX.

Testing set errors for the NNs and as described in the text. All errors are given in eV. In each case, the architecture of the networks is (15-140-1).

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/content/aip/journal/jcp/131/12/10.1063/1.3231686
2009-09-30
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Accurate prediction of higher-level electronic structure energies for large databases using neural networks, Hartree–Fock energies, and small subsets of the database
http://aip.metastore.ingenta.com/content/aip/journal/jcp/131/12/10.1063/1.3231686
10.1063/1.3231686
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