RBF神经元网络和遗传算法的研究及其在化工中的应用
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摘要
建立准确的模型对化工过程的理论研究和实际应用都具有重要的意义。但是，化工过程往往比较复杂，而且机理也不甚明了，很难直接通过机理建立准确的模型。神经元网络建模不需要考虑机理，根据样本数据即可建立起过程模型。本文重点研究了径向基函数网络(Radial Basis Function Networks，RBF网络)学习方法的改进和网络结构的优化；系统论述了遗传算法(Genetic Algorithms，GA)，并对它做了有效的改进，将遗传算法应用于参数的估计和优化，取得了较好的效果。全文主要内容如下：
(1)将主成分回归(PCR)和偏最小二乘回归(PLSR)引入RBF网络中，建立了RBF—PCR模型和RBF—PLSR模型。由于PCR和PLSR从原有自变量数据矩阵中提取相互正交的成分，并略去方差非常小的成分，从而消除了原有自变量间所存在的复共线性，使回归过程更为稳健。
(2)将循环子空间回归(CSR)引入到RBF网络中，建立了RBF—CSR模型，并将它应用于实际化工过程。与RBF—PCR、RBF—PLSR相比，RBF—CSR运用了更为泛化的方法求解回归问题，在更广的范围内寻求最优的网络权值，因此，它所建立的模型是一种更优的网络回归模型。
(3)针对常规遗传算法(SGA)寻优效率偏低、易陷入局部最优的弱点，通过增加单纯形寻优算子、改进交叉算子、自适应地调整交叉率和变异率等措施，设计了一种优进遗传算法(Eugenic Evolution Genetic Algorithms，EGA)，并将它应用于动力学模型的参数估计，取得了良好的效果。
(4)利用混沌变量的遍历性，将其引入到遗传操作，设计了一种混沌遗传算法(Chaos Genetic Algorithms，CGA)。将CGA应用于RBF网络隐含层结构的优化，并设计了一种适应度函数，提出了CGA—RBFN模型。该模型应用于烃类热裂解丙烯产率的预测，取得了满意的效果。
文章最后对所做的工作进行了总结，并在此基础上，提出了今后的研究发展方向。
Accurate models are important to the research and application of chemical engineering process. However, most problems in chemical engineering process are complex and we know little about their principles. So it is difficult to build accurate models directly by the principles. Neural networks build models without the principles, it modeling chemical engineering process by sample data. The main focus of this thesis is on improving the learning algorithms of the radial basis function networks (RBF networks), and optimizing the structure of the RBF networks, and reviewing the genetic algorithms (GA) and improving it effectively, and applying the GA to estimating and optimizing parameters and good effect is obtained. This article includes the following parts mainly:
(1) The principal component regression (PCR) and partial least square regression (PLSR) methods are applied to determine the weight of the RBF networks, the RBF-PCR and RBF-PLSR model is built. The PCR and PLSR picks-up the orthogonal components form the primary independent variable data matrix, and ignores the components of very little variance. So they eliminate the multicollinearity between the primary independent variables and ensure the regression process being steady.
(2) Through the cyclic subspace regression (CSR) being applied to determine the weight of the RBF networks, the RBF-CSR model is produced. The model is applied to the actual chemical engineering process. Comparing with the RBF-PCR and RBF-PLSR, the RBF-CSR uses its high generalization ability to solve the regressive problem, and finds the optimal coefficient in winder space. So the RBF-CSR model is a better networks regression model.
(3) An eugenic evolution genetic algorithm (EGA) is proposed to improve the efficiency of simple genetic algorithm (SGA) searching and the performance of global optimization through introducing deterministic simplex searching operation, and improving the crossover operator, and modifying adaptive crossover probability and
adaptive mutation probability, and others. The EGA is applied to estimate the kinetic model parameter and perfect results are obtained.
(4) A chaos genetic algorithms (CGA) is designed through introducing chaos variable to genetic algorithm by the full use of its ergodic property. The CGA is applied to optimize the hidden layer structure of the RBF networks, at the same time a fitness function is designed, then the CGA-RBFN model is proposed. The model is applied to predict the propylene yield in the course of the thermal cracking of hydrocarbon, satisfied result is obtained.
In the end of this paper, we make a summary and describe the further works.
(1)将主成分回归(PCR)和偏最小二乘回归(PLSR)引入RBF网络中，建立了RBF—PCR模型和RBF—PLSR模型。由于PCR和PLSR从原有自变量数据矩阵中提取相互正交的成分，并略去方差非常小的成分，从而消除了原有自变量间所存在的复共线性，使回归过程更为稳健。
(2)将循环子空间回归(CSR)引入到RBF网络中，建立了RBF—CSR模型，并将它应用于实际化工过程。与RBF—PCR、RBF—PLSR相比，RBF—CSR运用了更为泛化的方法求解回归问题，在更广的范围内寻求最优的网络权值，因此，它所建立的模型是一种更优的网络回归模型。
(3)针对常规遗传算法(SGA)寻优效率偏低、易陷入局部最优的弱点，通过增加单纯形寻优算子、改进交叉算子、自适应地调整交叉率和变异率等措施，设计了一种优进遗传算法(Eugenic Evolution Genetic Algorithms，EGA)，并将它应用于动力学模型的参数估计，取得了良好的效果。
(4)利用混沌变量的遍历性，将其引入到遗传操作，设计了一种混沌遗传算法(Chaos Genetic Algorithms，CGA)。将CGA应用于RBF网络隐含层结构的优化，并设计了一种适应度函数，提出了CGA—RBFN模型。该模型应用于烃类热裂解丙烯产率的预测，取得了满意的效果。
文章最后对所做的工作进行了总结，并在此基础上，提出了今后的研究发展方向。
Accurate models are important to the research and application of chemical engineering process. However, most problems in chemical engineering process are complex and we know little about their principles. So it is difficult to build accurate models directly by the principles. Neural networks build models without the principles, it modeling chemical engineering process by sample data. The main focus of this thesis is on improving the learning algorithms of the radial basis function networks (RBF networks), and optimizing the structure of the RBF networks, and reviewing the genetic algorithms (GA) and improving it effectively, and applying the GA to estimating and optimizing parameters and good effect is obtained. This article includes the following parts mainly:
(1) The principal component regression (PCR) and partial least square regression (PLSR) methods are applied to determine the weight of the RBF networks, the RBF-PCR and RBF-PLSR model is built. The PCR and PLSR picks-up the orthogonal components form the primary independent variable data matrix, and ignores the components of very little variance. So they eliminate the multicollinearity between the primary independent variables and ensure the regression process being steady.
(2) Through the cyclic subspace regression (CSR) being applied to determine the weight of the RBF networks, the RBF-CSR model is produced. The model is applied to the actual chemical engineering process. Comparing with the RBF-PCR and RBF-PLSR, the RBF-CSR uses its high generalization ability to solve the regressive problem, and finds the optimal coefficient in winder space. So the RBF-CSR model is a better networks regression model.
(3) An eugenic evolution genetic algorithm (EGA) is proposed to improve the efficiency of simple genetic algorithm (SGA) searching and the performance of global optimization through introducing deterministic simplex searching operation, and improving the crossover operator, and modifying adaptive crossover probability and
adaptive mutation probability, and others. The EGA is applied to estimate the kinetic model parameter and perfect results are obtained.
(4) A chaos genetic algorithms (CGA) is designed through introducing chaos variable to genetic algorithm by the full use of its ergodic property. The CGA is applied to optimize the hidden layer structure of the RBF networks, at the same time a fitness function is designed, then the CGA-RBFN model is proposed. The model is applied to predict the propylene yield in the course of the thermal cracking of hydrocarbon, satisfied result is obtained.
In the end of this paper, we make a summary and describe the further works.
引文
1. W.S. McCulloch, W. Pitts. A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophysics, 1943, 5(1): 115
2． 王永骥，涂健．神经元网络控制．北京：机械工业出版社，1998
3. D.O. Hebb. The Organization of Behavior. New York: John wiley, 1949
4. F. Rosenblatt. The perceptron: a Probabilistic Model for Information Storage and Organization in the Brain, Psychological Review, 1958, 65:386-408
5. M. Minsky, S. Papert. Perceptrons. Cambridge: MIT Press, 1969
6. Hopfield J J. Neural Network and Physical Systems with Emergent Collective Computational Abilities, Proceedings of the National Academy of Science of the U S A 79, 1982:2554-2558
7. Rumelhart D E, Hinton G E, Willams R J. Learning representation by back propagating errors. Nature, 1986, 323 (9): 553
8. Willamson R C, Helmeke U, Existence and uniquences results for neural network approximation. IEEE Trans. Neural Networks, 1995, 6(1): 2-13
9. Chen T, Chen, M, Universal approximation to nonlinear operators by neural networks with arbitary activation functions and its application to dynamical systems. IEEE Trans. Neural Networks, 1995, 6(4): 991
10. Hummels D M, Ahemed W, Musavi M T. Adaptive Detection of small Sinusodial Signals in Non-gaussian Noise using an RBF Neural Network. IEEE Trans on Neural Networks, 1995,6(1): 214-219
11. M.J.D. Powell, "radial basis function approximations to polynomials" in proc. 12th Biennial Numerical Analysis Conf. (Dundee), 1987, pp. 223-241
12. M.J.D. Powell, "radial basis function for multivariable interpolation: A review," in Algorithms for Approximation, J,C. Mason and M.G. Cox, Eds. Oxford, 1987, pp. 143-167
13. Moody, Darken C. Fast Learning in Networks of Locally-tuned Processing Units. Neural Computation, 1989,1(1): 281-294
14. Chen S, Cowan C F N, Grant P M. Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks. IEEE Trans on Neural Networks, 1991, 2(2): 302-309
15. Chen S, Billings S A, Grant P M. Recursive Hybrid Algorithm for Nonlinear Identification Using Radial Basis Function Networks. International Journal of Control, 1995, 55(5): 1051-I070
16. Holcomb, Tyler, Morari Manfred. Local Training for radial basis function networks: toward solving the hidden unit problem. American Control Conference, 1993,2331-2335
17．孙毅刚，战强．一种用于径向基函数(RBF)神经网络训练的有效方法．哈尔滨工业大学学报，1997，29(4)：103-106
18．汪小帆，王执铨，宋文忠．径向基函数神经网络的新型混合递推学习算法．控制理论与应用，1998，15(2)：272-276
19．王凌，郑大钟．径向基函数神经网络结构的混合优化策略，清华大学学报(自然科学版)，]999，39(7)：50-53
20. B. Walezak, D.L. Massart. The Radial Basis Functions-Partial Least Squares approach as a flexible non-linear regression technique. Analytic Chimica Acta 331 (1996): 177-185
21．王旭东，邵惠鹤，范懋基．改进的RBF神经元网络及其应用．上海交通大学学报，1996，30(4)：132-136
22．赵伟祥，陈德钊，胡上序．势RBF网络及其在水质诊断中的应用．浙江大学学报(自然科学版)，1999，33(2)：147-151
23. E.S.Chng, S.Chen, and B. Mulgrew. Gradient Radial Basis Function Networks for Nonlinear and Nonstationary Time Series Prediction. IEEE Trans on Neural Networks, 1996, 7(1): 190-194
24. Steve A. Billings, X. Hong. Dual-orthogonal radial basis function networks for nonlinear time series prediction. Neural Networks, 1998(11): 479-493
25. Cover, T.M. Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Trans. Electron. Computer, EC-14, 1965: 326-334
26．张尧庭，方开泰．多元统计分析引论．科学出版社，1982
27．胡上序，陈德钊．观测数据的分析与处理．浙江大学出版社，1996
28．陈德钊．多元数据处理．化学工业出版社，1998
29．陈亚秋．基于多层前传神经网络的优化组合建模及应用．浙江大学博士学位论文，2000
30．罗洪波，钟本和．磷铵及其不溶物的组成研究．四川大学学报，1994，(6)：1-6
31．唐盛伟，刘代俊，应建康，梁斌．酸性磷铵料浆在浓缩中氟逸出率的测定．化肥工业，2002，28 (6)：23-25
32．陈滨．乙烯工学．北京：化学工业出版社，1997
33．王松汉．乙烯装置技术．北京：中国石化出版社，1994
34．郝红，熊国华，张少锋，陈波．烃类热裂解的丙烯产率计算．化学工程，1999，27(2)：54-59
35. S. Wold, N. Kettaneh-Wold and B. Skagerberg, Nonlinear PLS Modeling, Chemometrics and Intelligent Laboratory Systems, 7, pp. 53-65, 1989
36．葛显良．应用泛函分析．杭州：浙江大学出版社，1996
37. Kalivas J H. Interrelationships Between PCR And PLS by Cyclic Subspace Regression with Applications, New Trends In Chemometrics. Changsha: Hunan University Press, 1997:87
38. Lang P M, Brencheley J M, Nieves R G, Kalivas J H. Cyclic Subspace Regression. Journal of Multivariate Analysis, 1998, 65:58-70
39. John H. Kalivas. New Trends in Chemometrics, Changsha: Hunan University Press, 1997:87
40．徐树方．矩阵计算的理论与方法．北京：北京大学出版社，1995
41. Patrick M. L., Jason M. B., Reinaldo G. N. et al.. Journal of Multivariable Analysis, 1998,65:58
42．庄凌，陈德钊，陈亚秋，胡上序．循环子空间回归的一种快速算法及其在定量构效关系中的应用．分析化学，1999，27 (12)：1386-1390
43．Markerlong E W，Gerlase T M，编．最新碱法制浆技术．曹邦威，译．北京：中国轻工出版社，1998
44．罗琪．硫酸盐法蒸煮过程纸浆Kappa值软测量技术的研究．华南理工大学博士论文，1998
45．李向阳，朱学峰，刘焕彬．硫酸盐蒸煮过程Kappa值预测模型的研究与应用．广东造纸，2000，5：23-27
46．李向阳，朱学峰，刘焕彬．间歇制浆蒸煮过程的混合建模方法研究．中国造纸学报，2001，16：24-28
47. Holland J H. Adaptation in Nature and Artificial Systems. MIT Press, 1975
48. De Jong K A. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Ph. D
Dissertation, University of Michigan, No.76-9381, 1975
49. Goldberg D E. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989
50．周明，孙树栋．遗传算法原理及应用．北京：国防工业出版社，1999
51. Wang Xu-Fa, Mini-Micro Systems, 1995, 16(2): 59
52．孙增圻，张再兴，邓志东．智能控制理论与技术．北京：清华大学出版社，1997
53．邵军力，张景，魏长华．人工智能基础．北京：电子工业出版社，2000
54．王小平，曹立明．遗传算法．西安：西安交通大学出版社，2002
55．孙卫国．有机药物的定量构效关系．浙江大学博士学位论文，1999
56. Yanagiya M. IEEE Int. Conf on Evolutionary Computation-Troceedings (vol. 1), 1994:456
57. DAVIS L D. Handbook of Genetic Algorithms. New York: Van Nostrand Reinhold, 1991
58. Steve A. Billings and Guang L. Zheng, Radial Basis Function Networks Configuration Using Genetic Algorithms, Neural Networks, 1995, 6:877-890
59．席裕庚，柴天佑，恽为民．遗传算法综述．控制理论与应用，1996，13(6)：697-708
60．宋晓峰，陈德钊，胡上序，等．基于优进策略的遗传算法对重油热解模型参数的估计．高校化学工程学报(已录用待发表)
61. Michalewicz Z. Genetic Algorithms + Data Structures = Evolution Programs. Berlin: Springer-Verlag Berlin Heidelberg, 1996
62．黄晓峰，潘立登，陈标华，等．用改进的实数编码遗传算法估计反应动力学参数．高校化学工程学报，1999，13(1)：50-55
63．陈振华，叶华，刘今．铯—铷—钒系低温硫酸催化剂上SO_2氧化反应速率的机理解释．化学反应工程与工艺，2001，17(2)：119-123
64．王旭东，邵惠鹤．RBF神经网络在非线性系统建模中的应用．控制理论与应用，1997,14(1)：59-66
65．颜学峰，陈德钊，胡上序，丁军委．混沌遗传算法估计反应动力学参数．化工学报，2002，53(8)：810-814
66．李兵，蒋慰孙．混沌优化方法及其应用．控制理论与应用，1997,14(4)：613-615
67．钱富才，费楚红，万百五．利用混沌搜索全局最优的一种混合算法．信息与控制，
1998,27(3)：232-235
68．李一波，黄小原．基于改进排序遗传算法的径向基函数神经网络色谱峰解析．分析化学，2001，29(3)：253-257
69．王梦松．神经元网络的研究及其在复杂化学信息处理中的应用．浙江大学硕士学位论文，2002
2． 王永骥，涂健．神经元网络控制．北京：机械工业出版社，1998
3. D.O. Hebb. The Organization of Behavior. New York: John wiley, 1949
4. F. Rosenblatt. The perceptron: a Probabilistic Model for Information Storage and Organization in the Brain, Psychological Review, 1958, 65:386-408
5. M. Minsky, S. Papert. Perceptrons. Cambridge: MIT Press, 1969
6. Hopfield J J. Neural Network and Physical Systems with Emergent Collective Computational Abilities, Proceedings of the National Academy of Science of the U S A 79, 1982:2554-2558
7. Rumelhart D E, Hinton G E, Willams R J. Learning representation by back propagating errors. Nature, 1986, 323 (9): 553
8. Willamson R C, Helmeke U, Existence and uniquences results for neural network approximation. IEEE Trans. Neural Networks, 1995, 6(1): 2-13
9. Chen T, Chen, M, Universal approximation to nonlinear operators by neural networks with arbitary activation functions and its application to dynamical systems. IEEE Trans. Neural Networks, 1995, 6(4): 991
10. Hummels D M, Ahemed W, Musavi M T. Adaptive Detection of small Sinusodial Signals in Non-gaussian Noise using an RBF Neural Network. IEEE Trans on Neural Networks, 1995,6(1): 214-219
11. M.J.D. Powell, "radial basis function approximations to polynomials" in proc. 12th Biennial Numerical Analysis Conf. (Dundee), 1987, pp. 223-241
12. M.J.D. Powell, "radial basis function for multivariable interpolation: A review," in Algorithms for Approximation, J,C. Mason and M.G. Cox, Eds. Oxford, 1987, pp. 143-167
13. Moody, Darken C. Fast Learning in Networks of Locally-tuned Processing Units. Neural Computation, 1989,1(1): 281-294
14. Chen S, Cowan C F N, Grant P M. Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks. IEEE Trans on Neural Networks, 1991, 2(2): 302-309
15. Chen S, Billings S A, Grant P M. Recursive Hybrid Algorithm for Nonlinear Identification Using Radial Basis Function Networks. International Journal of Control, 1995, 55(5): 1051-I070
16. Holcomb, Tyler, Morari Manfred. Local Training for radial basis function networks: toward solving the hidden unit problem. American Control Conference, 1993,2331-2335
17．孙毅刚，战强．一种用于径向基函数(RBF)神经网络训练的有效方法．哈尔滨工业大学学报，1997，29(4)：103-106
18．汪小帆，王执铨，宋文忠．径向基函数神经网络的新型混合递推学习算法．控制理论与应用，1998，15(2)：272-276
19．王凌，郑大钟．径向基函数神经网络结构的混合优化策略，清华大学学报(自然科学版)，]999，39(7)：50-53
20. B. Walezak, D.L. Massart. The Radial Basis Functions-Partial Least Squares approach as a flexible non-linear regression technique. Analytic Chimica Acta 331 (1996): 177-185
21．王旭东，邵惠鹤，范懋基．改进的RBF神经元网络及其应用．上海交通大学学报，1996，30(4)：132-136
22．赵伟祥，陈德钊，胡上序．势RBF网络及其在水质诊断中的应用．浙江大学学报(自然科学版)，1999，33(2)：147-151
23. E.S.Chng, S.Chen, and B. Mulgrew. Gradient Radial Basis Function Networks for Nonlinear and Nonstationary Time Series Prediction. IEEE Trans on Neural Networks, 1996, 7(1): 190-194
24. Steve A. Billings, X. Hong. Dual-orthogonal radial basis function networks for nonlinear time series prediction. Neural Networks, 1998(11): 479-493
25. Cover, T.M. Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Trans. Electron. Computer, EC-14, 1965: 326-334
26．张尧庭，方开泰．多元统计分析引论．科学出版社，1982
27．胡上序，陈德钊．观测数据的分析与处理．浙江大学出版社，1996
28．陈德钊．多元数据处理．化学工业出版社，1998
29．陈亚秋．基于多层前传神经网络的优化组合建模及应用．浙江大学博士学位论文，2000
30．罗洪波，钟本和．磷铵及其不溶物的组成研究．四川大学学报，1994，(6)：1-6
31．唐盛伟，刘代俊，应建康，梁斌．酸性磷铵料浆在浓缩中氟逸出率的测定．化肥工业，2002，28 (6)：23-25
32．陈滨．乙烯工学．北京：化学工业出版社，1997
33．王松汉．乙烯装置技术．北京：中国石化出版社，1994
34．郝红，熊国华，张少锋，陈波．烃类热裂解的丙烯产率计算．化学工程，1999，27(2)：54-59
35. S. Wold, N. Kettaneh-Wold and B. Skagerberg, Nonlinear PLS Modeling, Chemometrics and Intelligent Laboratory Systems, 7, pp. 53-65, 1989
36．葛显良．应用泛函分析．杭州：浙江大学出版社，1996
37. Kalivas J H. Interrelationships Between PCR And PLS by Cyclic Subspace Regression with Applications, New Trends In Chemometrics. Changsha: Hunan University Press, 1997:87
38. Lang P M, Brencheley J M, Nieves R G, Kalivas J H. Cyclic Subspace Regression. Journal of Multivariate Analysis, 1998, 65:58-70
39. John H. Kalivas. New Trends in Chemometrics, Changsha: Hunan University Press, 1997:87
40．徐树方．矩阵计算的理论与方法．北京：北京大学出版社，1995
41. Patrick M. L., Jason M. B., Reinaldo G. N. et al.. Journal of Multivariable Analysis, 1998,65:58
42．庄凌，陈德钊，陈亚秋，胡上序．循环子空间回归的一种快速算法及其在定量构效关系中的应用．分析化学，1999，27 (12)：1386-1390
43．Markerlong E W，Gerlase T M，编．最新碱法制浆技术．曹邦威，译．北京：中国轻工出版社，1998
44．罗琪．硫酸盐法蒸煮过程纸浆Kappa值软测量技术的研究．华南理工大学博士论文，1998
45．李向阳，朱学峰，刘焕彬．硫酸盐蒸煮过程Kappa值预测模型的研究与应用．广东造纸，2000，5：23-27
46．李向阳，朱学峰，刘焕彬．间歇制浆蒸煮过程的混合建模方法研究．中国造纸学报，2001，16：24-28
47. Holland J H. Adaptation in Nature and Artificial Systems. MIT Press, 1975
48. De Jong K A. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Ph. D
Dissertation, University of Michigan, No.76-9381, 1975
49. Goldberg D E. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989
50．周明，孙树栋．遗传算法原理及应用．北京：国防工业出版社，1999
51. Wang Xu-Fa, Mini-Micro Systems, 1995, 16(2): 59
52．孙增圻，张再兴，邓志东．智能控制理论与技术．北京：清华大学出版社，1997
53．邵军力，张景，魏长华．人工智能基础．北京：电子工业出版社，2000
54．王小平，曹立明．遗传算法．西安：西安交通大学出版社，2002
55．孙卫国．有机药物的定量构效关系．浙江大学博士学位论文，1999
56. Yanagiya M. IEEE Int. Conf on Evolutionary Computation-Troceedings (vol. 1), 1994:456
57. DAVIS L D. Handbook of Genetic Algorithms. New York: Van Nostrand Reinhold, 1991
58. Steve A. Billings and Guang L. Zheng, Radial Basis Function Networks Configuration Using Genetic Algorithms, Neural Networks, 1995, 6:877-890
59．席裕庚，柴天佑，恽为民．遗传算法综述．控制理论与应用，1996，13(6)：697-708
60．宋晓峰，陈德钊，胡上序，等．基于优进策略的遗传算法对重油热解模型参数的估计．高校化学工程学报(已录用待发表)
61. Michalewicz Z. Genetic Algorithms + Data Structures = Evolution Programs. Berlin: Springer-Verlag Berlin Heidelberg, 1996
62．黄晓峰，潘立登，陈标华，等．用改进的实数编码遗传算法估计反应动力学参数．高校化学工程学报，1999，13(1)：50-55
63．陈振华，叶华，刘今．铯—铷—钒系低温硫酸催化剂上SO_2氧化反应速率的机理解释．化学反应工程与工艺，2001，17(2)：119-123
64．王旭东，邵惠鹤．RBF神经网络在非线性系统建模中的应用．控制理论与应用，1997,14(1)：59-66
65．颜学峰，陈德钊，胡上序，丁军委．混沌遗传算法估计反应动力学参数．化工学报，2002，53(8)：810-814
66．李兵，蒋慰孙．混沌优化方法及其应用．控制理论与应用，1997,14(4)：613-615
67．钱富才，费楚红，万百五．利用混沌搜索全局最优的一种混合算法．信息与控制，
1998,27(3)：232-235
68．李一波，黄小原．基于改进排序遗传算法的径向基函数神经网络色谱峰解析．分析化学，2001，29(3)：253-257
69．王梦松．神经元网络的研究及其在复杂化学信息处理中的应用．浙江大学硕士学位论文，2002