Abstract:
A recent method for solving differential equations using feedforward neural networks was applied to a non-steady fixed bed non-catalytic solid-gas reactor. As neural networks have universal approximation capabilities, it is possible to postulate them as solutions for a given DE problem that defines an unsupervised error. The training was performed using genetic algorithms and the gradient descent method. The solution was found with uniform accuracy (MSE ∼ 10-9) and the trained neural network provides a compact expression for the analytical solution over the entire finite domain. The problem was also solved with a traditional numerical method. In this case, solution is known only over a discrete grid of points and its computational complexity grows rapidly with the size of the grid. Although solutions in both cases are identical, the neural networks approach to the DE problem is qualitatively better since, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. © 2003 Elsevier Science B.V. All rights reserved.
Registro:
| Documento: |
Artículo
|
| Título: | Solving differential equations with unsupervised neural networks |
| Autor: | Parisi, D.R.; Mariani, M.C.; Laborde, M.A. |
| Filiación: | Depto de Ing. Química, Facultad de Ingeniería, Universidad de Buenos Aires, Buenos Aires, Argentina
Departamento de Matemática, Facultad de Cie Exact y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina
|
| Palabras clave: | Neural networks differential equations; Non-catalytic solid-gas reactor simulations; Approximation theory; Computational complexity; Error analysis; Genetic algorithms; Problem solving; Solid-gas reactors; Feedforward neural networks; computer modeling; differential equation; neural network; reactor |
| Año: | 2003
|
| Volumen: | 42
|
| Número: | 8-9
|
| Página de inicio: | 715
|
| Página de fin: | 721
|
| DOI: |
http://dx.doi.org/10.1016/S0255-2701(02)00207-6 |
| Título revista: | Chemical Engineering and Processing: Process Intensification
|
| Título revista abreviado: | Chem. Eng. Process.: Process Intensif.
|
| ISSN: | 02552701
|
| CODEN: | CENPE
|
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02552701_v42_n8-9_p715_Parisi |
Referencias:
- Ph van Milligen, B., Tribaldos V, Jiménez, J.A., Neural network differential equation and plasma equilibrium solver (1995) Physical Review Letters, 75, pp. 3594-3597
- Monterola, C., Saloma C, Characterizing the dynamics of constrain physical systems with unsupervised neural network (1998) Physical Review E, 57, pp. 1247R-1250R
- Monterola, C., Saloma, C., Solving the nonlinear Schrodinger equation with an unsupervised neural network (2001) Optics Express, 9, pp. 72-84
- Quito, M., Monterola, C., Saloma, C., Solving N-body problems with neural networks (2001) Physical Review Letters, 86, pp. 4741-4744
- Parisi, D.R., Laborde, M.A., Modeling steady state heterogeneous gas-solid reactors using feedforward neural networks (2001) Computer and Chemical Engineering, 25, pp. 1241-1250
- Spretcher, D.A., On the structure of continuous function of several variables (1965) Transactions of the American Mathematical Society, 115, p. 340
- Hetch-Nielsen, R., Kolmogorov's mapping neural networks existence theorem (1987) 1st IEEE Inter. Conf. on Neural Networks, 3, p. 11. , San Diego CA
- Hetch-Nielsen, R., Theory of backpropagation neural network (1989) Proc. Int. Joint Conf. on Neural Networks, pp. I.593-I.605. , Washington D.C. New York, IEEE Press
- Cybenko, G., Approximation by superposition of a sigmoidal function (1989) Mathematics of Control, Signals and Systems, 2, p. 303
- Funahashi, K.I., On the approximate realization of continuous mappings by neural networks (1989) Neural Networks, 2, p. 183
- Hornik, K., Stichcombe, M., White, H., Multilayer feedforward networks are universal approximators (1989) Neural Networks, 2, p. 359
- Hornik, K., Stichcombe, M., White, H., Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks (1990) Neural Networks, 3, p. 551
- Hornik, K., Approximation capabilities of multilayer feedforward networks (1991) Neural Networks, 4, p. 251
- Hetch-Nielsen, R., (1990) Neurocomputing, , Addison Wesley Publishing Company
- Hertz, J., Krogh, A., Palmer, R.G., (1991) Introduction to the Theory of Neural Computation, , Addison-Wesley Publishing Company
- Freeman, J.A., Skapura, D.M., (1991) Neural Networks: Algorithms, Applications, and Programming Techniques, , Addison-Wesley Publishing Company
- Hagan, M.T., Demuth, H.B., Beale, M.H., (1996) Neural Network Design, , PWS Publishing, Boston, MA
- Haykin, S., (1999) Neural Networks: A Comprehensive Foundation, , Prentice-Hall Inc
- Schaffer, J.D., Whitley, D., Eshelman L.J, Combinations of genetic algorithms and neural networks: A survey of the state of the art (1992) COGANN-92: International Workshop on Combinations of Genetic Algorithms and Neural Networks, , Whitley, L.D. Schaffer J.D. (Eds.), IEEE Computer Society Press
- Whitley, D., Hanson, T., Optimizing Neural Networks Using Faster, More Accurate Genetic Search (1989) Proceedings of the Third International Conference on Genetic Algorithms, pp. 391-396. , J.D. Schaffer (Ed.), (Arlington, 1989), San Mateo, Morgan Kauffmann
- Montana, D.J., Davis, L., Training Feedforward Networks Using Genetics Algorithm (1989) Eleventh International Joint Conference on Artificial Intelligence, pp. 762-767. , N.S. Sridharan (Ed.), (Detroit, 1989), San Mateo, Morgan Kauffmann
- Martín del Brio, B., Sanz Molina, A., (1997) Redes Neuronales Y Sistemas Borrosos, , Ra-Ma
- Goldberg, D.E., (1989) Genetic Algorithms in Search, Optimization, and Machine Learning, , Addison Wesley
- Mitchell, M., (1999) an Introduction to Genetic Algorithms, , MIT Press
- Froment, G.F., Bischoff, K.B., (1979) Chemical Reactor Analysis and Design, , John Willey - Sons Inc
- Dormand, J.R., Prince, P.J., A family of embedded Runge-Kutta formulae (1980) J. Comp. Appl. Math., 6, p. 19
Citas:
---------- APA ----------
Parisi, D.R., Mariani, M.C. & Laborde, M.A.
(2003)
. Solving differential equations with unsupervised neural networks. Chemical Engineering and Processing: Process Intensification, 42(8-9), 715-721.
http://dx.doi.org/10.1016/S0255-2701(02)00207-6---------- CHICAGO ----------
Parisi, D.R., Mariani, M.C., Laborde, M.A.
"Solving differential equations with unsupervised neural networks"
. Chemical Engineering and Processing: Process Intensification 42, no. 8-9
(2003) : 715-721.
http://dx.doi.org/10.1016/S0255-2701(02)00207-6---------- MLA ----------
Parisi, D.R., Mariani, M.C., Laborde, M.A.
"Solving differential equations with unsupervised neural networks"
. Chemical Engineering and Processing: Process Intensification, vol. 42, no. 8-9, 2003, pp. 715-721.
http://dx.doi.org/10.1016/S0255-2701(02)00207-6---------- VANCOUVER ----------
Parisi, D.R., Mariani, M.C., Laborde, M.A. Solving differential equations with unsupervised neural networks. Chem. Eng. Process.: Process Intensif. 2003;42(8-9):715-721.
http://dx.doi.org/10.1016/S0255-2701(02)00207-6
