提高密度泛函理论方法计算吸收能的精度：神经网络和遗传算法
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摘要
电子跃迁吸收能是分子的一个重要的物理属性，它包含分子的内在结构信息和电子性质，所以精确地预测吸收能是计算化学领域的一个重要问题。量子化学方法已经超过了仅仅验证实验值的水平，它能够在实验值不知道或不确定的时候来精确地预测吸收能，然而并不是所有的计算结果都是十分精确地，特别是对于复杂分子或者较大的系统，导致这种局限性的主要原因是计算方法本身采用固有的近似引起的。要解决这个问题，期望找到一些简单而有效的方法来校正理论计算的误差。
本论文针对150个有机小分子体系，用神经网络、遗传算法、神经网络集成以及K近邻等方法来校正量子化学方法计算的结果，提高量子化学计算电子光谱吸收能的精度。这些方法为准确地预测分子的各种性质提供了一种新的研究手段，拓展了理论方法的可靠性和适用性。
研究工作主要包括如下几个部分：
1．基于量子化学TDDFT/B3LYP方法计算有机小分子的紫外可见吸收光谱的吸收能，利用遗传算法和BP神经网络（GANN）来提高有机小分子吸收能的计算精度。在GANN方法中，GA被用来搜索神经网络的最优初始权值，BP被用来进一步训练神经网络来获得最优的最终连接权值。该方法被用来校正150个有机分子的光谱吸收能的理论计算误差。通过BPN的校正，均方根误差由B3LYP/6-31G(d)计算得到的0.47降到了0.22 eV，而对于GANN校正方法，误差则降到了0.16 eV。GANN方法避免了传统BP算法易陷入局部极小的缺陷，同时在提高DFT方法计算精度时优于BP神经网络校正方法。
2．利用神经网络集成（NNE）的方法来提高单一神经网络的泛化能力，其中NNE采用了bagging技术来生成集成中的6个个体神经网络，在集成时使用基于简单平均的结果合成（NNEA）和加权平均的结果合成（NNEW）方法。包含150个分子的实验数据被随机分成两个数据集，训练集包含120个分子，测试集包含30个分子。对于BPN、NNEA和NNEW校正方法将训练集中120个分子的误差分别由B3LYP/6-31G(d)计算得到的0.48降到0.20，0.22，0.22 eV，对于测试集中的30个分子，误差则分别由原来的0.41降到0.26，0.20和0.18 eV。从测试集的数据仿真结果表明神经网络集成能够降低单一神经网络的泛化误差。
3．提出了用神经网络集成和K近邻方法（NNEKNN）来精确预测150个有机分子的电子跃迁吸收能。传统的前向神经网络是一个无记忆的方法，这意味着当神经网络训练结束后，所有有关输入的相关信息都被存储在网络的连接权值中，这时不再需要输入数据了。相反，K近邻方法代表的是一种基于记忆的方法，该方法在记忆中存储了输入数据的整个数据库，然后它的预测结果是基于这些已存储数据的局部近似值。在近邻选择上，NNEKNN方法使用集成输出的结果与位于训练集中的近邻之间的欧几里德距离作为衡量方法。对于NNEKNNA和NNEKNNW校正方法，对于训练集的120个分子而言，误差均由原来的0.48降到0.16 eV，而对于测试集中的30有机分子来说，误差分别由原来的0.41降到0.14和0.10 eV。结果表明NNEKNN方法能够有效地降低密度泛函理论的计算误差，可以以比BPN和NNE方法提供更高的精度来预测光谱吸收能。
Absorption energy is a significant physical property for a molecule, which impliesinherent structure information and electronic properties. In this regard, the accurate predictionof the absorption energy is of great significance in computational chemistry. Quantumchemical methods have been developed beyond the level of just reproducing experimentaldata and can now accurately predict the absorption energies which are unknown or uncertainexperimentally. However, the calculation results are not accurate enough for all systems,especially for large systems. This is caused by the inherent approximations adopted infirst-principles methods. To resolve this, simple yet efficient way to correct such errors isdesired.
In the present work, genetic algorithm, neural network, neural network ensemble andk-nearest-neighbors approaches have been applied to improve the calculation accuracy ofquantum chemical methods for absorption energies of 150 small organic molecules. Thesecombined methods can greatly eliminate the systemic errors of theoretical calculation andimprove the calculation accuracy of density functional theory (DFT) for absorption energies,which provide a novel tool for predicting the properties of the molecules.
Our work has been focus on following aspects:
1. The combination of genetic algorithm and neural network correction approach(GANN) has successfully improved the calculation accuracy of the absorption energies afterquantum chemical methods calculated UV-visible absorption spectra. The raw calculatedabsorption energies are evaluated by TDDFT/B3LYP method. In this GANN approach, GA isadopted in searching the optimal initial synaptic weights for neural networks of pre-specifiedtopology, while BP is employed in further training the neural networks to find the optimalfinal synaptic weights. It is employed to reduce the errors of calculated absorption energy of150 molecules. Upon the traditional BP neural networks correction approach, the RMSdeviation of the calculated absorption energies of 150 organic molecules is reduced from 0.47eV to 0.22 eV for the TDDFT/B3LYP/6-31G(d) calculation. With the GANN correction, theRMS deviation is reduced from 0.47 eV to 0.16 eV. This combined GANN correctionapproach avoids being trapped at local minima of the traditional BPN approach, thus leads toimproved DFT calculation results as compared to those of BPN.
2. The generalization ability of NN can be substantially improved by using an averagingtechnique with a neural network ensemble (NNE). We use bagging on the training set togenerate six individual base BP networks. The NNE with simple averaging method (NNEA)and weighted averaging method (NNEW) is adopted as for combining the predictions ofcomponent NNs. The experimental data of 150 organic molecules are randomly divided into atraining set with 120 molecules and a testing set with 30 molecules. The BPN, NNEA andNNEW approaches reduce the RMS deviations from 0.48 to 0.20 and both 0.22 eV for the120 absorption energies, respectively. For the 30 absorption energies, they are from 0.41 to0.26, 0.20 and 0.18 eV. Statistical tests show that generalization errors of the NNE approach is significantly lower than that of the TDDFT/B3LYP method, and they attain still lower errors than BPN.
3. In this paper, we propose an ensemble of NNs and the KNN approach (NNEKNN) to improve the calculation accuracy of DFT. The traditional artificial feed-forward NN is a memoryless approach. This means that after training is complete all information about the input patterns is stored in the NN weights and the input data is no longer needed. Contrary to that, the k-nearest-neighbors (KNN) approach represents the memory-based approach. The approach keep in memory the entire database of examples, and their predictions are based on some local approximation of the stored examples. The approach is applied to predict the optical absorption energies of 150 organic molecules. This method uses the distance between ensemble responses and the neighbors in the training set as a measure amid the analyzed cases for the nearest neighbor technique. The NNEKNNA and NNEKNNW approach improved DFT calculation results and reduced the RMS deviations from 0.48 to both 0.16 eV for the training set of 120 organic molecules, respectively, while for the testing set of 30 organic molecules, they are from 0.41 to 0.14 and 0.10 eV. Simulation results and comparison of the BPN and NNE corrected values demonstrates the feasibility and effectiveness of the NNEKNN approach to reduce the calculation errors of DFT and it could indeed predict the absorption energies with higher accuracy.
本论文针对150个有机小分子体系，用神经网络、遗传算法、神经网络集成以及K近邻等方法来校正量子化学方法计算的结果，提高量子化学计算电子光谱吸收能的精度。这些方法为准确地预测分子的各种性质提供了一种新的研究手段，拓展了理论方法的可靠性和适用性。
研究工作主要包括如下几个部分：
1．基于量子化学TDDFT/B3LYP方法计算有机小分子的紫外可见吸收光谱的吸收能，利用遗传算法和BP神经网络（GANN）来提高有机小分子吸收能的计算精度。在GANN方法中，GA被用来搜索神经网络的最优初始权值，BP被用来进一步训练神经网络来获得最优的最终连接权值。该方法被用来校正150个有机分子的光谱吸收能的理论计算误差。通过BPN的校正，均方根误差由B3LYP/6-31G(d)计算得到的0.47降到了0.22 eV，而对于GANN校正方法，误差则降到了0.16 eV。GANN方法避免了传统BP算法易陷入局部极小的缺陷，同时在提高DFT方法计算精度时优于BP神经网络校正方法。
2．利用神经网络集成（NNE）的方法来提高单一神经网络的泛化能力，其中NNE采用了bagging技术来生成集成中的6个个体神经网络，在集成时使用基于简单平均的结果合成（NNEA）和加权平均的结果合成（NNEW）方法。包含150个分子的实验数据被随机分成两个数据集，训练集包含120个分子，测试集包含30个分子。对于BPN、NNEA和NNEW校正方法将训练集中120个分子的误差分别由B3LYP/6-31G(d)计算得到的0.48降到0.20，0.22，0.22 eV，对于测试集中的30个分子，误差则分别由原来的0.41降到0.26，0.20和0.18 eV。从测试集的数据仿真结果表明神经网络集成能够降低单一神经网络的泛化误差。
3．提出了用神经网络集成和K近邻方法（NNEKNN）来精确预测150个有机分子的电子跃迁吸收能。传统的前向神经网络是一个无记忆的方法，这意味着当神经网络训练结束后，所有有关输入的相关信息都被存储在网络的连接权值中，这时不再需要输入数据了。相反，K近邻方法代表的是一种基于记忆的方法，该方法在记忆中存储了输入数据的整个数据库，然后它的预测结果是基于这些已存储数据的局部近似值。在近邻选择上，NNEKNN方法使用集成输出的结果与位于训练集中的近邻之间的欧几里德距离作为衡量方法。对于NNEKNNA和NNEKNNW校正方法，对于训练集的120个分子而言，误差均由原来的0.48降到0.16 eV，而对于测试集中的30有机分子来说，误差分别由原来的0.41降到0.14和0.10 eV。结果表明NNEKNN方法能够有效地降低密度泛函理论的计算误差，可以以比BPN和NNE方法提供更高的精度来预测光谱吸收能。
Absorption energy is a significant physical property for a molecule, which impliesinherent structure information and electronic properties. In this regard, the accurate predictionof the absorption energy is of great significance in computational chemistry. Quantumchemical methods have been developed beyond the level of just reproducing experimentaldata and can now accurately predict the absorption energies which are unknown or uncertainexperimentally. However, the calculation results are not accurate enough for all systems,especially for large systems. This is caused by the inherent approximations adopted infirst-principles methods. To resolve this, simple yet efficient way to correct such errors isdesired.
In the present work, genetic algorithm, neural network, neural network ensemble andk-nearest-neighbors approaches have been applied to improve the calculation accuracy ofquantum chemical methods for absorption energies of 150 small organic molecules. Thesecombined methods can greatly eliminate the systemic errors of theoretical calculation andimprove the calculation accuracy of density functional theory (DFT) for absorption energies,which provide a novel tool for predicting the properties of the molecules.
Our work has been focus on following aspects:
1. The combination of genetic algorithm and neural network correction approach(GANN) has successfully improved the calculation accuracy of the absorption energies afterquantum chemical methods calculated UV-visible absorption spectra. The raw calculatedabsorption energies are evaluated by TDDFT/B3LYP method. In this GANN approach, GA isadopted in searching the optimal initial synaptic weights for neural networks of pre-specifiedtopology, while BP is employed in further training the neural networks to find the optimalfinal synaptic weights. It is employed to reduce the errors of calculated absorption energy of150 molecules. Upon the traditional BP neural networks correction approach, the RMSdeviation of the calculated absorption energies of 150 organic molecules is reduced from 0.47eV to 0.22 eV for the TDDFT/B3LYP/6-31G(d) calculation. With the GANN correction, theRMS deviation is reduced from 0.47 eV to 0.16 eV. This combined GANN correctionapproach avoids being trapped at local minima of the traditional BPN approach, thus leads toimproved DFT calculation results as compared to those of BPN.
2. The generalization ability of NN can be substantially improved by using an averagingtechnique with a neural network ensemble (NNE). We use bagging on the training set togenerate six individual base BP networks. The NNE with simple averaging method (NNEA)and weighted averaging method (NNEW) is adopted as for combining the predictions ofcomponent NNs. The experimental data of 150 organic molecules are randomly divided into atraining set with 120 molecules and a testing set with 30 molecules. The BPN, NNEA andNNEW approaches reduce the RMS deviations from 0.48 to 0.20 and both 0.22 eV for the120 absorption energies, respectively. For the 30 absorption energies, they are from 0.41 to0.26, 0.20 and 0.18 eV. Statistical tests show that generalization errors of the NNE approach is significantly lower than that of the TDDFT/B3LYP method, and they attain still lower errors than BPN.
3. In this paper, we propose an ensemble of NNs and the KNN approach (NNEKNN) to improve the calculation accuracy of DFT. The traditional artificial feed-forward NN is a memoryless approach. This means that after training is complete all information about the input patterns is stored in the NN weights and the input data is no longer needed. Contrary to that, the k-nearest-neighbors (KNN) approach represents the memory-based approach. The approach keep in memory the entire database of examples, and their predictions are based on some local approximation of the stored examples. The approach is applied to predict the optical absorption energies of 150 organic molecules. This method uses the distance between ensemble responses and the neighbors in the training set as a measure amid the analyzed cases for the nearest neighbor technique. The NNEKNNA and NNEKNNW approach improved DFT calculation results and reduced the RMS deviations from 0.48 to both 0.16 eV for the training set of 120 organic molecules, respectively, while for the testing set of 30 organic molecules, they are from 0.41 to 0.14 and 0.10 eV. Simulation results and comparison of the BPN and NNE corrected values demonstrates the feasibility and effectiveness of the NNEKNN approach to reduce the calculation errors of DFT and it could indeed predict the absorption energies with higher accuracy.
引文
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