Exploring Digital Twins of Nonlinear Systems through Meta-Modeling with Echo State Networks

Keywords: Echo State Networks, Dynamic systems, Digital twins


 monitoring, and control rely on precise dynamic models that can capture the inherent nonlinearities of chemical systems. However, rigorous modeling of complex industrial processes can be computationally demanding. Meta modeling using machine learning methodologies offers a viable approach to generate computationally efficient surrogate representations. Specifically, Echo State Networks (ESNs) are a promising neural network approach for meta-modeling nonlinear dynamical systems. ESNs simplify training through fixed input weights while they focus learning on output weights. This study explores the development of ESN-based digital twins for a nonlinear dynamic process. An ESN is employed to construct a meta-model of a simulated continuously stirred tank reactor with biochemical kinetic. The network was trained on input-output data obtained from the simulation of an ordinary differential equation system, and the performance was evaluated both in-sample and out-of-sample. The results indicate that the ESN meta-model can successfully approximate the underlying dynamics, accurately capturing temporal evolution. A closed-loop digital twin deployment using the ESN surrogate also showed reliable behavior. This work presents initial steps toward developing digital twins of chemical processes using ESN-driven meta-modeling. The findings suggest ESNs can effectively generate computationally efficient surrogate representations of nonlinear dynamical systems. Such digital twins hold promise for online process monitoring and optimized control of industrial plants.


Download data is not yet available.


A. J. Silva Neto and J. C. Becceneri, “Técnicas de inteligência computacional inspiradas na natureza: Aplicação em problemas inversos em transferência radiativa,” 2009.

C. P. Naveira-Cotta et al., “Eigenfunction expansions for transient diffusion in heterogeneous media,” International Journal of Heat and Mass Transfer, vol. 52, no. 21-22, pp. 5029–5039, 2009.

D. C. Knupp, “Integral transform technique for the direct identification of thermal conductivity and thermal capacity in heterogeneous media,” International Journal of Heat and Mass Transfer, 2021.

F. P. Incropera et al., Fundamentals of Heat and Mass Transfer, vol.6, New York, Wiley, 1996.

F. S. Mascouto et al., “Detection of contact failures employing combination of integral transforms with single-domain formulation, finite differences, and Bayesian inference,” Numerical Heat Transfer, Part A: Applications, 2020.

J. Beck and S.-K. Au, “Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation,” Journal of Engineering Mechanics, vol. 128, no. 4, pp. 380–391, 2002.

J. Ching and J. S. Wang, “Application of the transitional Markov chain Monte Carlo algorithm to probabilistic site characterization,” Engineering Geology, vol. 203, pp. 151–167, 2016.

J. Ching and Y. C. Chen, “Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging,” Journal of Engineering Mechanics, vol. 133, no.7, pp. 816–832, 2007.

J. P. Kaipio and C. Fox, “The Bayesian framework for inverse problems in heat transfer,” Heat Transfer Engineering, vol. 32, no.9, pp. 718–753, 2011.

L. A. Da Silva Abreu et al., “Estimativa do perfil de temperatura na entrada de dutos via Método de Monte Carlo com Cadeias de Markov,” Revista Cereus, vol. 14, no. 4, pp. 129–143, 2022.

M. N. Özışık and H. R. Orlande, Inverse Heat Transfer:Fundamentals and Applications, 2021.

P. Gardner, C. Lord, and R. J. Barthorpe, “A unifying framework for probabilistic validation metrics,” *Journal of Verification,Validation and Uncertainty Quantification*, vol. 4, no. 3, 031005,2019.

W. Betz, I. Papaioannou, and D. Straub, “Transitional Markov chain Monte Carlo: observations and improvements,” Journal of Engineering Mechanics, vol. 142, no. 5, 04016016, 2016.J. Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vol. 2.Oxford: Clarendon, pp.68–73, 1892

How to Cite
L. C. J. Campos, A. C. Dias, W. da Silva, and J. C. Dutra, “Exploring Digital Twins of Nonlinear Systems through Meta-Modeling with Echo State Networks”, LAJC, vol. 11, no. 2, pp. 13-22, Jul. 2024.
Research Articles for the Regular Issue