L. Campos, A.Dias, J. Dutra and W. da Silva,

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

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

Exploring Digital Twins

of Nonlinear Systems

through Meta-Modeling

with Echo State

Networks

ARTICLE HISTORY

Received 08 March 2024

Accepted 08 May 2024

Published 8 July 2024

Laisa Cristina Juo Campos

Universidade Federal do Espírito Santo

Alegre, Brasil

laisacampos01@gmail.com

ORCID: 0009-0003-4427-0395

Wellington Betencurte da Silva

Universidade Federal do Espírito Santo

Alegre, Brasil

wellinton.betencurte@ufes.br

ORCID: 0000-0003-2242-7825

Ana Carolina Spindola Rangel Dias

Serviço Nacional de Aprendizagem Industrial (SENAI)

Rio de Janeiro, Brasil

acspdias@gmail.com

ORCID: 0000-0001-7376-0703

Julio Cesar Sampaio Dutra

Universidade Federal do Espírito Santo

Alegre, Brasil

julio.dutra@ufes.br

ORCID: 0000-0001-6784-4150

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

This work is licensed under a Creative Commons

Attribution-NonCommercial-ShareAlike 4.0 International License.

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July - December 2024

Exploring Digital Twins of Nonlinear Systems

through Meta-Modeling with Echo State Networks

Laisa Cristina Juffo Campos

Universidade Federal do Espírito Santo

Departamento de Engenharia Rural

Alegre, Brasil

laisacampos01@gmail.com

Ana Carolina Spindola Rangel Dias

Serviço Nacional de Aprendizagem Industrial

(SENAI)

Rio de Janeiro, Brasil

acspdias@gmail.com

Wellington Betencurte da Silva

Universidade Federal do Espírito Santo

Departamento de Engenharia Rural

Alegre, Brasil

wellinton.betencurte@ufes.br

Julio Cesar Sampaio Dutra

Universidade Federal do Espírito Santo

Departamento de Engenharia Rural

Alegre, Brasil

julio.dutra@ufes.br

Abstract— Effective process 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.

Keywords— Echo State Networks, Dynamic systems, Digital

twins

I. I

NTRODUCTION

In recent years, rapid technological progress has resulted

in substantial enhancements across diverse sectors, notably

in enhancing quality and safety within chemical processes.

The ubiquitous incorporation of computers into process

management has empowered control over various variables,

that include temperature, pressure, and chemical

composition, thereby generating extensive and diverse data

archives [1]. Design challenges necessitating intensive

computational resources are increasingly prevalent in

manufacturing industries [2]. Moreover, creating tools

capable of analyzing data and constructing predictive

mathematical models has become imperative for real-time

process monitoring and control.

Creating rigorous models that accurately capture the

dynamics and nonlinearity of real systems may be

impractical at plant sites, where rapid responses are crucial.

One practical approach is to utilize metamodeling strategies

[2][3] to tackle the challenges inherent in process systems.

Widely utilized across engineering, computer science, and

optimization, these strategies involve developing simplified

models that approximate the behavior of complex systems

or processes [4]. These simplified representations, named

meta-models or surrogate models, aim to balance accuracy

and computational efficiency.

In this context, digital twins emerge as virtual

representations capable of reflecting the behavior of

physical systems in real-time, this shows potential for online

monitoring and process optimization [5]. By generating

simplified yet computationally efficient models, digital

twins enable dynamic data analytics and rapid decision-

making to optimize industrial plant control and

performance.

Expanding on recent data science research,

metamodeling can draw upon various machine learning

techniques [2][6]. Artificial Neural Networks (ANNs) are

widely recognized for their ability to approximate complex

functions [7]. Modeled after the functioning mechanism of

biological neurons, ANNs comprise an input layer, a hidden

layer housing artificial neurons in quantities necessary to

represent the data, and an output layer. Additionally, ANNs

possess memory storage and learning capabilities, making

them particularly suitable for dynamic and nonlinear

systems. This work precisely investigates this characteristic

regarding applying neural meta-models for generating

digital twins of complex chemical processes [8][9]. The aim

is to develop computationally efficient representations that

approximately capture the underlying dynamics of these

systems.

Depending on the network architecture, various types of

neural networks exist, including Feedforward Neural

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI: https://doi.org/10.33333/lajc.vol11n2.01

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

https://doi.org/10.33333/lajc.vol11n2.01

L. Campos, A.Dias, J. Dutra and W. da Silva,

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

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

Networks (FNNs) and Recurrent Neural Networks (RNNs).

RNNs offer computational advantages for dynamic process

systems owing to their inherent feedback loops. However,

training traditional RNNs can be complicated due to issues

like the "vanishing gradient" problem [10]. To address this,

[11] introduced the Echo State Network (ESN). Unlike

traditional RNNs that adjust all synaptic weights, ESNs

maintain fixed input and recurrent connections, focusing

solely on training output connections through a relatively

simple linear regression process. This approach circumvents

the complexities of training recurrent connections and

mitigates gradient-related challenges. Consequently, ESNs

present an effective solution for harnessing the power of

RNNs while mitigating training complexities, particularly

in scenarios where efficient learning is essential.

This article proposes using an Echo State Network as a

meta-model to approximate dynamic nonlinear models and

evaluate the performance in a closed-loop application. This

work assesses the potential of this approach for this purpose,

analyzing the performance of different methodologies in

modeling a CSTR reactor through the construction of a

digital twin. Section 2 presents a brief background on the

metamodeling problem. Section 3 elaborates a case study

based on a simulated bioreactor and details the data

acquisition procedure. The theory, rationale, and

construction of the Echo State Network are described in

Section 4, followed by the discussion of simulation results.

The contribution of this article lies in presenting initial

steps towards developing digital twins of chemical

processes using ESN-driven meta-modeling. By

demonstrating the efficacy of ESNs in generating

computationally efficient surrogate representations of a

classical nonlinear dynamical system, this work opens space

for online process monitoring and optimized control of

industrial plants.

II. T

ETAMODELING

ROBLEM

A meta-model (or surrogate model) can be conceived as

a "model of a model" [6], functioning as a simplified

representation of a high-fidelity simulation model [12]. It

emulates the response by delineating the relationship

between inputs () and outputs () based on data acquired

with known precision or uncertainty [13]. The importance

of metamodeling lies in its ability to balance accuracy and

computational efficiency. Hence, metamodeling emerges as

an essential approach to navigating real-world system

intricacies, especially those characterized by nonlinear

relationships, numerous variables, and complex behaviors.

In industrial settings, meta-models are employed for

tasks which necessitate the establishment of a (complex)

relationship between the inputs and outputs of a process

system. This relationship can be encapsulated by an

extended meta-model equation that incorporates the

feedback signal (1):





= 

(





, 



)

+ 

(1)

Where 



represents the current output, 



denotes the

current inputs, 



is the previous output (feedback

signal), () is the relationship that incorporates inputs and

feedback, and  represents error or uncertainty in the meta-

model prediction.

By offering a simplified representation of burdensome

simulations, meta-models facilitate quicker evaluations and

decision-making - crucial aspects in industries that demand

real-time solutions. This approach enables approaching

complex systems without the need of resource-intensive

full-scale simulations, which can be computationally

demanding and time-consuming. Some commonly used

metamodeling techniques encompass polynomial surface

response models, Kriging, Radial Basis Functions, Support

Vector Regression, and Artificial Neural Networks

[13][14]. These techniques generate approximated

mappings from inputs to outputs. The choice depends on

problem characteristics, available data, and required

predictions.

Metamodeling using neural networks adopts a data-

driven approach that harnesses the principles of ANNs to

construct efficient approximations of complex systems.

This methodology entails training the neural network on a

dataset that reflects the system behavior under scrutiny. This

dataset consists of input variables paired with corresponding

output values, that facilitates the network identification of

underlying patterns and correlations. Following training, the

neural network can provide predictions for new input data,

substantially which alleviates computational burdens

compared to resource-intensive full-scale simulations.

The increased processing speed has dramatically

expanded the applicability of neural network-based

metamodeling. For example, [15] employed a neural

network as a meta-model to approximate a copper porphyry

mine comminution circuit, which leads to a significant

acceleration of simulations compared to traditional

phenomenological models. Additionally, [16] utilized

neural networks in the metamodeling of reactive transport,

and this reduces computational time for scenarios requiring

multiple realizations. These studies highlight the versatility

of neural network-based metamodeling in improving

efficiency, accuracy, and computational performance across

various domains.

Modeling and Data Generation

The mathematical model employed to generate the data

was adapted from [17], outlining the dynamic behavior of a

bioreactor. The equations that govern substrate balance, S,

and cell balance, , are expressed by (2) and (3),

respectively, while the reaction rate, (), is defined by (4),

where  is defined as the dilution rate, that represents the

ratio between the volumetric feed flow rate and the reactor

volume, and 



stands for the substrate feed concentration.





= 





(



)





/

(2)





= 

(



)



(3)



(



)











+ 

(4)

All code implementations were developed in Python,

with the free Spyder development environment (version

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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July - December 2024

3.9.16). The code was compiled and executed on a computer

system featuring 128 GB of DDR4 RAM, and an Intel®

Core I7-12700k processor operating at 5.00 GHz.

This specific case study adopted a supervised training

strategy to construct the neural model. This approach

required the generation of input and output data. The input

data was synthesized by a Random Gaussian Signal (RGS)

algorithm [18]. The RGS technique is widely utilized for

dynamic systems identification, which enables a thorough

exploration of the input space. Consequently, it effectively

stimulates the process response across diverse conditions.

The input variables were the dilution rate and substrate

feed concentration, with mean values of 0.1 h⁻¹ and 10.0 g

L⁻¹, respectively. Each variable displayed variations of ± 0.1

h⁻¹ and ± 2.5 g L⁻¹. A total of 2500 samples were generated

and collected at intervals of 0.25 h. The sampling interval

was modified to 8 h to generate the second dataset, while

the other parameters were kept constant. As for the output

data, represented by  and , these were derived by solving

the system of ordinary differential equations outlined in (2)

and (3), using the solve_ivp function from the

scipy.integrate library for this purpose. Gaussian random

noise was added to the simulated result to make output data

more complex and realistic, with a standard deviation of

5%. This makes the resulting data more complex while

pushing the meta-model to discover the underlying patterns

in a way that enhances its robustness against noise and

variability when it transfers to actual operation.

Subsequently, all datasets were organized and stored within

a spreadsheet.

The generated data is showcased in Figs. 1-4 which

illustrate the obtained data with higher (Figs. 1-2) and lower

frequency (Figs. 3-4). The red data points indicate outputs

with the addition of measurement noise, which was

introduced to a better approximate reality and attenuate

potential overfitting.

Fig. 1. Input data for the first dataset

Fig. 2. Output data for the first dataset

Fig. 3. Input data for the second dataset

Fig. 4. Output data for the second dataset

III. E

CHO

TATE

ETWORK

Acknowledging the potential of RNNs, [8] introduced a

groundbreaking neural network architecture called the Echo

State Network (ESN). The primary aim of this architecture

is to harness the capabilities of effectively addressing

complex problems while it simplifes the learning process.

In the conventional training of ANNs, with the adjustment

of synaptic weights across input, output, and feedback

layers can impose substantial computational demands, often

requiring significant computational resources. However,

Jaeger's innovative network design focuses solely on

training output weights, accomplished through a relatively

straightforward linear regression process. This approach

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

https://doi.org/10.33333/lajc.vol11n2.01

L. Campos, A.Dias, J. Dutra and W. da Silva,

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

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

offers significant advantages in terms of computational

efficiency and streamlining the intricate task of fine-tuning

complex feedback loops.

The ESN remarkably simplifies the training process by

compartmentalizing the learning process into distinct stages

- initially training output weights while keeping other

weights fixed. This streamlined approach enhances

computational efficiency and facilitates faster convergence

during the training phase. Furthermore, the methodology

unlocks potential applications in scenarios where efficient

learning is paramount. The innovative design of the ESN

offers a promising pathway to address challenges related to

training complexity, which makes it well-suited for

scenarios demanding both computational efficiency and

enhanced learning performance.

In this implementation, the ESN network algorithm was

coded following the equations outlined by [8], with specific

hyperparameters maintained at fixed values (Table I). These

predetermined values were determined empirically. An

optimization method was utilized and implemented through

Python programming to identify the optimal

hyperparameters - neuron count, sparsity, and leaking rate.

Following this, the resulting network was validated using

the fine-tuned hyperparameters.

TABLE I. N

ETWORK

YPERPARAMETERS

Hyperparameter Value

Reservoir size 1222

Leaking rate 0.6964

Sparsity 0.3536

Spectral radius 0.70

Train fraction 0.35

Ridge 4E-4

Noise level 1E-5

Random seed 13042023

IV. C

ONTROLLER TUNING AND CLOSED

LOOP

Another test was applied to evaluate the performance in

a closed-loop simulation, allowing for the assessment of the

feasibility of applying the trained network as a meta-model

(that is, the digital twin). The control objective was to

maintain cell concentration (X) around desired values, and

it considers the substrate concentration in the feed (Sf) as

the disturbance and the dilution rate (D) as the manipulated

variable. For this purpose, we used a PI controller with the

velocity algorithm.

A transfer function of the reactor dynamics was obtained

to tune the controller, with a step test of -5% on D,

performed on the differential model from its initial

conditions. The steady-state response obtained was Xs = 4.5

g L⁻¹ and Ss = 1.0 g L⁻¹. With the approach of [19], it was

possible to approximate the process with a first-order plus

dead time (FOPDT) system. Fig. 5 comparatively illustrates

the original process (differential model), represented by red

points, and the approximated process. The parameters

obtained through such an approach are shown in Table II.

Fig. 5. Process simulation and obtained model

TABLE II. P

ROCESS PARAMETERS

Parameter Value

(L g

-1

) -6.6642

θ (h)

0.0700

 (h) 1.0050

After conducting tests on different controllers, three

tuning techniques were applied: Internal Model Control

(IMC), Integral of Time multiplied by Absolute Error for

servo test (ITAE), and manual fine-tuning [17]. The

parameters for each tuning technique are described in Table

III. It was concluded that the manually tuned controller was

the best choice for this study, even though it was a more

conservative option. The manually tuned controller yielded

a favorable result of less oscillation in the manipulated

variable during closed-loop tests. Additionally, it

demonstrated a slight difference in response time compared

to the other controllers examined. The gain margin of the

manually fine-tuned controller was 56.8437, which is

significantly higher than the gain margins of the IMC

(22.9541) and ITAE-servo test (3.0869) methods. This

result suggests that the manually fine-tuned controller is

more robust than the other methods. As a result, the

manually fine-tuned controller was chosen due to its quick,

highly stable, and oscillation-free response.

The results of the closed-loop simulation using the selected

controller are presented in Figs. 6-7. Fig. 6 illustrates the

behavior of the manipulated and disturbance variables,

while Fig. 7 depicts the controlled variable with its setpoint,

along with the other output.

TABLE III. T

UNING

ETHODS AND

ONTROLLER

ARAMETERS

Parameter

Tuning method

IMC

ITAE

(servo test)

Manual

(L g

-1

) -0.15164 -1.12761 -0.06123

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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July - December 2024



(h)

1.00500

1.00752 6.70000

Fig. 6. Inputs of the closed-loop

To assess the neural network efficacy in accurately

representing the behavior of the simulated system, as

required for a digital twin, its response was evaluated within

a closed-loop control framework. Within this framework,

the control actions computed for the original process (based

on the differential model) with the tuned proportional-

integral (PI) controller were integrated as one of the

network's inputs. Moreover, these inputs encompassed

process disturbance information and a feedback signal

generated by the network predictions rather than simulated

measurements from the differential model simulation.

Consequently, the neural network can autonomously adapt

over time, dynamically responding to the evolving process

inputs.

Fig. 7. Outputs of the closed-loop

V. R

ESULTS

After fine-tuning the hyperparameters, the network

performance was evaluated on both datasets. The higher-

frequency dataset was used to assess the network predictive

capacity. The neural network demonstrated exceptional

training performance, accurately predicting the test data and

effectively capturing the underlying dataset patterns and

relationships (Fig. 8). This success highlights the robust

ability of the model to generalize from complex training

examples to unseen data, this showcases its deep

understanding of system dynamics.

An autocorrelation analysis of the training modeling

errors (residual) indicated significant autocorrelation only at

lag = 0, resembling a Dirac delta function (Fig. 9), which

confirms that the residual distribution follows a white noise

correlogram pattern. We can see this result as an indication

of the absence of systematic errors or patterns in the model

predictions. Additionally, a white noise correlogram pattern

suggests that the model has effectively captured all relevant

information from the data, and the predictions are based on

genuine signals rather than noise.

The following run evaluates the pre-trained network

adaptability to a distinct scenario (second dataset), as

illustrated in Fig. 10-11. As can be seen, the successful

prediction of the second test dataset resulted in a residual

distribution that also adheres to a white noise correlogram

pattern. Remarkably, despite being trained with higher-

frequency data, the model ability to accurately represent

lower-frequency data underscores its robustness and

versatility in capturing the system dynamics across different

temporal scales.

In the closed-loop control scenario, the neural network

functioned autonomously, providing its feedback signal

based on the predicted outputs. However, Fig. 9 reveals a

systematic deviation between the predicted and actual

responses, likely stemming from the absence of feedback

control dynamical effects in the training data. This

discrepancy highlights the challenge of accurately capturing

real-time system behavior under closed-loop control

conditions.

Fig. 8. Network performance for the first dataset

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

https://doi.org/10.33333/lajc.vol11n2.01

L. Campos, A.Dias, J. Dutra and W. da Silva,

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

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

Fig. 9. Network residual analysis for the training of the first run

A bias b(k)was introduced to mitigate this issue, and

this represents the disparity between the simulated process

measurements, y_m(k), and the predicted outputs,

\hat{y}(k). This adjustment on the predicted outputs, being

\hat{y}\left(k\right)+b\left(k-1\right) with b(0)=0, yielded

a maximum relative error of just 1.1%, compared to the

2.7% observed without bias. The graphical representations

that depict the predictions in the absence and presence of

bias correction are presented in Figs. 12 and 13,

correspondingly.

Detailed performance metrics for the training, testing,

and closed-loop application phases are provided in Table

IV. The findings demonstrate the exceptional predictive

capabilities of the network, which achieves outstanding

performance in forecasting output data despite being

trained on a comparatively small dataset — and contrasts

with the higher training percentages commonly used in the

literature. Notably, the network accurately captured the

output dynamics in the first dataset with remarkable

precision. Furthermore, the successful modeling of a

scenario with lower variability in the second dataset

suggests its versatility and robustness. Thus, inferring that

the acquired meta-model fits both scenarios is reasonable.

Moreover, the closed-loop results showcase the neural

network potential as a virtual representation that reflects

real-time process responses, thereby mimicking real-world

scenarios with fidelity.

TABLE I. N

ETWORK PERFORMANCE METRICS

Metrics

Dataset 1

Dataset 2

Closed loop

Training Test Test

Without

bias

With

bias

0.9790 0.9490 0.9812 0.9930 0.9996

MSE 2.6676E-02 3.14347E-02 4.6535E-02 0.0007 0.0001

ExpVar 0.9790 0.9491 0.9812 0.9979 0.9998

Fig. 10. Network performance for the second dataset

Fig. 11. Network residual analysis for the training of the second run

Fig. 12. Network performance for the closed-loop, without bias

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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July - December 2024

Fig. 13. Network performance for the closed-loop, with bias

VI. C

ONCLUSION

This study employed an Echo State Network (ESN) as a

meta-model to tackle the complexities of a classical

nonlinear bioreactor. Unlike traditional Recurrent Neural

Networks, ESNs simplify learning by maintaining fixed

input and recurrent connections, while training only output

connections through linear regression. This approach

mitigates the challenges associated with training recurrent

connections.

The outcomes of our study showcase the robust

predictive capabilities of the ESN, adeptly handling noisy

data and limited samples across a broad spectrum of

oscillations. These results underscore the ESN adaptability

to the diverse scenarios commonly encountered in industrial

contexts. The results of the closed-loop test validate the

efficacy of ESNs, with maximum errors limited to just 3%.

This underscores the potential for further exploration of

ESN applications in constructing digital twins, which

represents a paradigm shift from traditional models towards

real-time control and monitoring contexts.

Moreover, the findings confirm the practical and

effective utility of the ESN for metamodeling in industrial

processes. The versatility and potential integration of ESNs

into Process Control and Monitoring practices facilitate

precise simulations and streamline optimization procedures,

thereby enhancing the efficiency and effectiveness of

industrial processes. However, it is essential to

acknowledge the ongoing need for evaluating and

discussing alternative strategies to enhance the network

predictive accuracy, given the inherent complexity and

challenges inherent in industrial process control. Continued

research in this area promises to unlock further

advancements in ESN applications, driving innovation and

optimization within industrial processes.

CKNOWLEDGMENT

This study was funded in part by the Fundação de

Amparo à Pesquisa e Inovação do Espírito Santo – FAPES.

The authors also acknowledge the financial support

from the CNPq and FAPERJ funding agencies.

EFERENCES

[1] A. J. Silva Neto and J. C. Becceneri, “Técnicas de inteligência

computacional inspiradas na natureza: Aplicação em problemas

inversos em transferência radiativa,” 2009.

[2] 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.

[3] 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.

[4] F. P. Incropera et al., Fundamentals of Heat and Mass Transfer, vol.

6, New York, Wiley, 1996.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[10] 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.

[11] M. N. Özışık and H. R. Orlande, Inverse Heat Transfer:

Fundamentals and Applications, 2021.

[12] 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.

[13] 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.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

AUTHORS

Laisa Cristina Juo Campos is Brazilian, born in the state of Espírito

Santo. She completed a technical course in Informatics at the Federal

Institute of Espírito Santo (IFES), where she gained experience with

various programming languages. Her dedication led her to participate

in an extension project during her second year, involving Arduino

programming and the development of mobile device software.

Currently, she is pursuing a degree in Chemical Engineering at the

Federal University of Espírito Santo (UFES). During her studies, she

has applied her programming knowledge to various course-related

problems, particularly in Numerical Methods and Process Control.

Her research focuses on exploring the applications of artiﬁcial neural

networks, with a speciﬁc emphasis on the Echo State Network

(ESN) architecture. Recently, she began studying physics-informed

neural networks (PINNs) and plans to combine PINNs and ESN in her

future research. With a strong interest in the intersection of artiﬁcial

intelligence and chemical engineering, she aims to develop innovative

methodologies that can contribute to signiﬁcant advancements in

the ﬁeld. In her free time, she enjoys drawing, playing the piano, and

birdwatching.

Wellington Betencurte da Silva is a Brazilian professor and researcher

with a strong academic background and extensive experience

in Mathematics and Mechanical Engineering. He graduated in

Mathematics from the Federal Fluminense University in 2006, followed

by a master's degree in Mechanical Engineering from the Military

Institute of Engineering in 2008 and a Ph.D. in Mechanical Engineering

from the Federal University of Rio de Janeiro in 2012. Currently, he

serves as an associate professor at the Federal University of Espírito

Santo, where he conducts research in the areas of Inverse Problems,

Bayesian Filters, State and Parameter Estimation, and Heat Transfer.

His expertise and academic contributions have been recognized in

various master's dissertations and undergraduate thesis projects,

where he has served as a supervisor and participated in examining

boards. With a solid academic background and a commitment

to excellence in research and teaching, Wellington Betencurte da

Silva is a prominent ﬁgure in the Brazilian academic scene, making

signiﬁcant contributions to the advancement of knowledge in his ﬁeld

of expertise.

Laisa Cristina Juo Campos

Wellington Betencurte da Silva

L. Campos, A.Dias, J. Dutra and W. da Silva,

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

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

AUTHORS

Ana Carolina Spindola Rangel Dias, a Brazilian born in Minas Gerais

state, holds a Bachelor's degree in Chemical Engineering from the

Federal University of Espírito Santo (2015) and a Master's degree in

Chemical Engineering from the same institution (2017). Her master's

dissertation focused on the control of a propylene polymerization

reactor using particle ﬁlters and neural networks. A research internship

was completed at the Norwegian University of Science and Technology

(NTNU) from April 2019 to October 2020. She completed a Ph.D. in

Chemical and Biochemical Process Engineering from the School of

Chemistry at the Federal University of Rio de Janeiro in February

2023, with a thesis on developing predictive and self-optimizing

controllers based on operational data. Previously, she worked as a

temporary professor in the Department of Rural Engineering at the

Federal University of Espírito Santo from March 2015 to December

2016. Currently, she is the lead researcher for the intelligent systems

team on the Chemical Processes platform at the Senai Institute of

Innovation in Biosynthetic and Fibers. Research interests include

modeling, simulation, control, and optimization of processes. In her

free time, she enjoys kpop music, movies and traveling.

Julio Cesar Sampaio Dutra, born in Rio de Janeiro, Brazil, obtained his

Bachelor's in Chemical Engineering from the Federal Rural University

of Rio de Janeiro (UFRRJ) in 2007. He completed a direct-entry PhD

in Chemical Engineering at the Federal University of Rio de Janeiro

(UFRJ) in 2012, which included a research exchange at Norges Teknisk-

Naturvitenskapelige Universitet (NTNU). Julio has been a faculty

member since 2013 at the Federal University of Espírito Santo (UFES)

as an Associate Professor. His research focuses on Mathematical

Modeling, Process Simulation, and Process Control. He is particularly

interested in machine learning and estimation schemes for monitoring,

combined with control structure design using PID controllers, advanced

process control strategies, and estimation algorithms, like Kalman

ﬁlters and Sequential Monte Carlo methods. Considering such topics,

he has extensive experience teaching and advising undergraduate

and graduate students in Chemical Engineering. In addition to these

activities, Julio is involved in administrative activities and committed

to serving on deliberative committees. He enjoys cooking, drinking

red wine, and traveling in his free time. In the future, Julio plans to

continue exploring new techniques to address emerging challenges

in Chemical Engineering.

Ana Carolina Spindola Rangel Dias

Julio Cesar Sampaio Dutra

L. Campos, A.Dias, J. Dutra and W. da Silva,

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

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.