modelling and optimization of fed-batch fermentation processes using dynamic neural networks and...

Biochemical Engineering Journal 22 (2004) 51–61

Modelling and optimization of fed-batch fermentation processesusing dynamic neural networks and genetic algorithms

LeiZhi Chena, Sing Kiong Nguanga,∗, Xiao Dong Chenb, Xue Mei Lib

a Department of Electrical and Computer Engineering, The University of Auckland, New Zealandb Department of Chemical and Materials Engineering, The University of Auckland, New Zealand

Received 1 March 2004; accepted 26 July 2004

Abstract

Optimization of a fed-batch bioreactor using a cascade recurrent neural network (RNN) model and modified genetic algorithm (GA) isstudied in this paper. The complex nonlinear relationship between the manipulated feed rate and the biomass product is described by tworecurrent neural sub-models, in which outputs of one sub-model are fed into another sub-model to provide meaningful information for thebiomass prediction. The simulation results show that the error of prediction is less than 8%. Based on the neural network model, a modifiedG generate as te profileb perimentali©

K

1

cstbttimt

tmbar

plex-ngesses

dew

boveesition

ful ex-stentorob-et al.

lsim-lled

1d

A is employed to determine a smooth optimal feed rate. The evolution of feed rate profiles shows that the algorithm is able tomooth feed rate profile, where the optimality is still maintained. The final biomass quantity that yields from the optimal feed raased on the neural network model reaches 99.8% of the “real” optimal value obtained based on the mechanistic model. An ex

nvestigation has also been carried out to verify the feasibility of the proposed technique.2004 Elsevier B.V. All rights reserved.

eywords:Optimization; Fed-batch fermentation; Dynamic neural network; Genetic algorithm; Modelling; Artificial intelligence

. Introduction

The issue of sustainable development has recently drawnonsiderable attention worldwide, and has stimulated re-earchers and engineers to make greater effort to reducehe cost/benefit-ratio for development and manufacture ofio-industrial processes both economically and environmen-

ally. From the process engineering point of view, the wayo achieve this goal is through process optimization, whichs to control a bioprocess at its optimal state and to reach its

aximum productivity with lowest possible cost. In the meanime, the quality should be maintained.

Conventionally, optimizations of bioprocesses, such ashe fermentation process, are performed based on deter-inistic mathematical models, which are usually describedy a set of differential equations derived from mass bal-nces[1]. However, to obtain a sufficiently accurate andobust mathematical model for a bioprocess is a time-

∗ Corresponding author. Tel.: +64 9 3737 599; fax: +64 9 3737 461.E-mail address:[email protected] (S.K. Nguang).

consuming and costly task due to the exceeding comity in its physiology and performance. The major challeis the nonlinear and time-varying nature of such proce[2].

Artificial intelligence (AI), such as fuzzy logic (FL) anartificial neural network (ANN) approaches, provides nand powerful tools to handle the problem mentioned a[3]. Application of fuzzy control to industrial bioprocesshas been intensively reviewed by Honda and Kobayash[4].The usefulness of fuzzy control systems for the automaof bioprocesses has been demonstrated with successamples. Fuzzy control of vitamin B2 production, which ialso detailed by Horiuchi and Hiraga[5], has shown thafuzzy systems are able to operate a large-scale fermwith satisfactory results, provided that linguistic rulestained from experienced operators are available. Hondahave also reported a temperature control of theginjo sakemashing process using fuzzy neural networks[6]. The modecan be tuned automatically from observed data to give ailar control profile of temperature to that produced by skioperators.

369-703X/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.bej.2004.07.012

52 L. Chen et al. / Biochemical Engineering Journal 22 (2004) 51–61

In this paper, an extended recurrent neural network (RNN)is adopted to model fed-batch fermentation ofSaccharomycescerevisiae. The aim of building this model is to predict thebiomass concentration based purely on the information of thefeed rate, such that the model can be used to maximize thequantity of biomass at the end of the reaction by optimizingthe feed rate profiles.

The dynamic optimization problems of such complex sys-tems are difficult to solve. The conventional analytical meth-ods, such as Green’s theorem and the maximum (or mini-mum) principle of Pontryagin, are unable to provide a com-plete solution due to singular control problems[7]. Mean-while, conventional numerical methods, such as dynamicprogramming (DP), suffer from a large computational bur-den and may lead to sub-optimal solutions[8].

Genetic Algorithm (GA)[9,10], which is a member ofthe artificial intelligence family, is chosen to optimize thefeed rate profile. GA is a stochastic global search methodthat imitates principles of natural biological evolution. It si-multaneously evaluates many points in the parameter space,and the search is directed by the evolution principles. Hence,it is more likely to converge towards a global solution. Anexample of comparison between a GA and a DP is givenin [11]. Both methods are used for determining the opti-mal feed rate profile of a fed-batch culture of monoclonala uc-t %h n tot d tobf net-w ortedb o-c archf t ata witht mizet tionp

: inS astf to nted;i ont esti-g ni

2

tiale Thed ci n inF and

Fig. 1. Schematic illustration of the simulated fermentation model.

volume, were generated from a given feed rate by solving thesix differential equations.

Five different feed rate profiles, which are shown inFig. 2,are designated to excite the mathematical fermentation modelfor generating simulation data: (a) a square-wave feed flow,(b) a saw-wave feed flow, (c) a stair-shape feed flow, (d)an industrial feeding policy, and (e) a random-steps feedflow.

These feed patterns are used here to give excitation tothe simulated system as much as possible. Each of the firstfour feed rate profiles yielded 150 input–output (target) pairscorresponding to 6 min sampling time during a 15-h fermen-tation; the random-step feed rate yielded 450 data pairs dur-ing a 45-h fermentation with the same length of samplinginterval.

3. Development of dynamic neural network model

3.1. Cascade dynamic neural network model

A dynamic neural network model is proposed in this studyusing a cascade structure as shown inFig. 3. It contains twoextended recurrent neural blocks which model the dynamicsfrom inputs, feed rateF and volumeV, to the key variable,c noe r-m locka work[ , thee sec-o datam l pro-p e top e isa singE

athsa en-h ulti-p ddenl ers.T rela-t( deli

ntibodies (MAb). The result shows that the final prodion of MAb produced by using the GA is about 24igher than that produced by using the DP. In additio

he advantage of the global solution, GA can be applieoth “white box” and “black box” models[12,13]. This of-

ers a great opportunity to combine GAs with neuralorks. An example of such approaches has been repy Yoshikawa et al.[14] to control an activated sludge press. A GA is applied to a fuzzy neural network to seor proper control variables which can control the outpu

desired value. In the present work, GA is combinedhe extended recurrent neural network model to maxihe final quantity of biomass for the fed-batch fermentarocess.

The layout of the remainder of the paper is as followsection 2, the mechanistic model of industrial baker’s ye

ed-batch bioreaction is given; inSection 3, the developmenf the cascade recurrent neural network model is prese

n Section 4, the optimization of feed rate profile basedhe neural network model is described; experimental invation is presented inSection 5; and conclusions are draw

n Section 6.

. The mathematical simulation model

A mathematical model, which consists of six differenquations[15,16], is used to generate simulation data.etails of the model are given inAppendix A. Schemati

llustration of the simulated fermentation model is showig. 1. Three output variables, biomass, dissolved oxygen

oncentration of dissolved oxygenCo and the fermentatioutput (product), biomass concentrationX. The first blockstimates the trend ofCo which provides an important infoation to the second neural block. The second neural bcts exactly as a softsensor developed in the previous

17], except that instead of the measured value of DOstimated value of DO is used here as the input of thend neural block. The softsensor model requires DOeasured on line, whereas the cascade dynamic modeosed inFig. 3 basically needs only the data of feed ratredict the biomass concentration. Although the volumnother input for the model, it can be simply calculated uq. A.6.In each neural block, both feedforward and feedback p

re connected through tapped delay lines in order toance the dynamic behaviors. All connections could be mathes. Sigmoid activation functions are used for the hi

ayers and pure linear function is used for the output layhe structure of the neural blocks reflects the differential

ionships between inputs and outputs as given inEqs. (A.2)–A.6). A full mathematical description of the cascade mos given in the following equations. The output of theith neu-

L. Chen et al. / Biochemical Engineering Journal 22 (2004) 51–61 53

Fig. 2. Plots of simulation data for five different feed rates: (a) square-wave feed, (b) saw-wave feed, (c) industrial feed, (d) stair-shape feed, (e)randomfeed.


Fig. 3. Structure of the proposed recurrent neural model.

ron in the first hidden layer is of the form:

h1i(t) = f1

na∑j=0

W Iij u1(t − j) +

nb∑k=1

WRik Co(t − k)

+nc∑l=1

WH1il h1(t − l) + bH1

i

(1)

whereu1 andh1 are the vector values of the neural networkinput and the first hidden layer’s output, correspondingly;Co is the second hidden layer output;bH1

i is the bias ofithneuron in first hidden layer;na, nb, nc are the number ofinput delays, the number of the second hidden layer feed-back delays and the number of first hidden layer feedbackdelays, respectively;f1(·) is a sigmoidal function;W I

ij arethe weights connecting thejth delayed input toith neuron inthe first hidden layer,WR

ik are the weights connecting thekthdelayed second hidden layer output feedback to theith neu-ron in the first hidden layer,WH1

il are the weights connectingthe lth delayed activation feedback to theith neuron in thefirst hidden layer.

Note that one neuron is placed at the output of the secondhidden layer, so that:

C

( ng∑ )

w sc hes rsth

i d

hidden layer can be described as:

h3i(t) = f1

nd∑j=0

WPij u2(t − j) +

ne∑k=1

WOik X(t − k)

+nf∑l=1

WH3il h3(t − l) + bH3

i

(3)

whereu2 andh3 are the vector values of the input to thethird hidden layer and the third hidden layer’s output, cor-respondingly;X is the model’s output;bH3

i is the bias ofithneuron in the third hidden layer;nd , ne, nf are the num-ber of input delays to the third hidden layer, the number ofthe output layer feedback delays and the number of thirdhidden layer feedback delays, respectively;f1(·) is the sig-moidal function;WP

ij are the weights connecting thejth de-layed input of the third hidden layer to theith hidden neu-ron in the layer,WO

ik are the weights connecting thekthdelayed output feedback to theith neuron in the third hid-den layer,WH3

il are the weights connecting thelth delayedactivation feedback to theith neuron in the third hiddenlayer.

The model’s output, which is the estimated biomass con-centration can be expressed as:

X

w sc heo enl

ˆ o(t) = f2

m=1

WYm hm(t) + bY (2)

heref2(·) is a pure linear function;WYm are the weight

onnecting themth neuron in the first hidden layer to tecond hidden layer;ng is the number of neurons in the fiidden layer;bY is the bias of the second hidden layer.

The second neural block has an additional input,Co. Sim-lar to the first block, the output ofith neuron in the thir

ˆ (t) = f2

(nk∑m=1

WXm hm(t) + bX

)(4)

heref2(·) is a pure linear function;WXm are the weight

onnecting themth neuron in the third hidden layer to tutput layer;nk is the number of neurons in the third hidd

ayer;bX is the bias of the output layer.


Fig. 4. Schematic illustration of neural network model training and predic-tion.

3.2. Neural network training

A schematic illustration of the neural network model train-ing is shown inFig. 4. The output of the bioprocess is usedonly for training the network. The model predicts the processoutput using the same input as the process after training. Noadditional measurements are needed during the predictionphase.

The goal of network training is to minimize the MSE be-tween the measured value and the neural network’s outputby adjusting its weights and biases. The LMBP training al-gorithm is adopted to train the neural networks due to its fastconvergence and memory efficiency[18].

To prevent the neural network from being over-trained, anearly stopping method is used here. A set of data which isdifferent from the training data set (e.g., saw-wave) is usedas a validation data set. The error on the validation data set ismonitored during the training process. The validation errorwill normally decrease during the initial phase of training.However, when the network begins to over-fit the data, theerror on the validation set typically begins to rise. When thevalidation error increases for a specified number of iterations,the training is stopped, and the weights and biases of thenetwork at the minimum of the validation error are obtained.

The rest of the data sets, which are not seen by the neuraln ingt sedfw

E

wmi euralm

inedn rentt , ares net-w sed,b net-

works with different numbers of hidden neuron delays aretrained. For each network structure, 50 networks are trained;the one that produces the smallest RMSP error for the testingdata sets is retained. The number of hidden neurons for thefirst and the third hidden layers are 12 and 10, respectively.Errors for different training patterns and various combina-tions of input and feedback delays are shown inFig. 5. Asshown in this figure, the 6/4/4 structure (the feed rate delaysare six, the first block output delays and the second block out-put delays are four) has the smallest error and is chosen as theprocess model. The separated training method is more time-consuming but is not superior to the overall training. Thus,the overall training is chosen to train the network whenevernew data are available.

3.3. Simulation results of biomass prediction

The biomass concentrations predicted by the neural net-work model and the corresponding feed rates and predictionerror are plotted inFig. 6. As shown in the figure, the predic-tion error is quite big at the initial period of fermentation andgradually becomes smaller and smaller. The prediction erroris less than 8%.

4

canb tion.T ratep

s toa tog um-m itialp thefi se-l ormc pop-u arer

de-s antc mea ntrolv

F

T t ofb ctivef

m

w

etwork during the training period, are used in examinhe trained network. The performance function that is uor testing the neural networks is the RMSP error index[19],hich is defined as follows:

=√√√√∑N

t=1(Xmt − Xt)2∑N

t=1(Xmt )2

× 100 (5)

hereN is the number of sampling data pairs;Xmt is the

easured (actual) value of biomass at sampling timet; Xt

s the corresponding estimated value predicted by the nodels.A smaller error on the testing data set means the tra

etwork has achieved better generalization. Two differaining patterns, overall training and separated trainingtudied. When the overall training is used, the wholeork is trained together. When the separated training is ulocks one and two are trained separately. A number of

. Optimization of feed rate profile

Once the cascade recurrent neural model is built, ite used to perform the task of feed rate profile optimizahe GA is used in this work to search for the best feedrofiles.

GAs tend to seek for better and better approximationsolution of a problem when running from generation

eneration. A simple standard procedure of a GA is sarized here by the following five steps: (i) create an inopulation of a set of random individuals; (ii) evaluatetness of individuals using the objective function; (iii)ect individuals according to their fitness, and then perfrossover and mutation operations; (iv) generate a newlation; (v) repeat steps (ii)–(iv) until termination criteriaeached.

The feed flow rate, which is the input of the systemcribed inSection 2, is equally discretized into 150 constontrol actions. The total reaction time and the final volure fixed to be 15 h and 90 000 L, respectively. The coector of the feed rate sequence is:

= [F1, F2 · · · F150]T (6)

he optimization problem here is to maximize the amouniomass quantity at the end of the reaction. Thus, the obje

unction can be formulated as follows:

axF (t)

J = X(tf )V (tf ) (7)

heretf is the final reaction time.


Fig. 5. Biomass prediction error on testing data sets for neural models with different combinations of delays. ‘6/4/4’ indicates that the number of feed ratedelays is six, the number of the first block output feedback delays is four, and the number of the second block output feedback delays is four.

The optimization is subject to the constraints givenbelow:

0 ≤ F ≤ 3500 L/h

V (tf ) ≤ 90 000 L(8)

In this study, optimization based on the mathematical modelis first performed to find the best feed rate profile and the high-est possible final biomass productivity that can be obtained.Then, the optimization is performed again using the RNN

Fig. 6. Simulation result of model predic

model. The resulting optimal feed rate is applied to the math-ematical model to find the corresponding system responsesand the final biomass quantity. As mentioned above, the math-ematical model is considered here as the actual “plant”. Thus,the suitability of the proposed neural network model can beexamined by comparing these two simulation results.

The optimal profile that is obtained by using a standardGA is highly fluctuating. This makes the optimal feed rateprofile less attractive for practical use, because extra control

tion for an industrial feed rate profile.


Fig. 7. Evolution of feed rate profile using the modified GA based on the mathematical model.

Fig. 8. Evolution of feed rate profile using the modified GA based on the RNN model.


Fig. 9. Comparison of optimization results based on the mathematical model and RNN model.

costs are needed and unexpected disturbances may be addedinto the bioprocesses[5]. In order to eliminate the strong vari-ations on the optimal trajectory, the standard GA is modified.Instead of introducing new filter operators into the GA[20],a simple compensation method is integrated into the eval-uation function. The control sequenceF is amended insidethe evaluation function to produce a smoother curve of feed

Fig. 10. Optimization result yield based on the casc

trajectory while the evolutionary property of the GA is stillmaintained. This operation has no effect on the final volume.

The method includes three steps:

(1) Calculate the distance between two neighboring indi-vidualsFi andFi+1 usingd = |Fi − Fi+1|, wherei ∈(1,2, . . . ,150).

ade recurrent network model and genetic algorithm.


(2) If d is greater than a predefined value (e.g., 10 L/h), thenmoveFi andFi+1 by d/3 towards the middle ofFi andFi+1 to make them closer.

(3) Evaluate the performance indexJ for the new controlvariables.

(4) Repeat steps (1)–(3) until all individuals in the populationhave been checked.

The Matlab GAOT software is used to solve the prob-lem. The population size was chosen at 150. The develop-ment of the optimal feed rate profiles based on the mecha-nistic model and neural network model from the initial tra-jectory to the final shape is illustrated inFigs. 7 and 8. Asthe number of the generation increases, the feeding trajectorygradually becomes smoother and smoother, and the perfor-mance index,J, is also increased. The smoothing procedureworks in a more efficient way for the mathematical model; ittakes 2000 generations to obtain a smooth profile, while 2500generations are needed to smooth the profile for the neuralnetwork model. This is due to the disturbance rejection na-ture of the RNN. A small alteration in feed rate is treatedas a perturbation; thus, the network is rather insensitiveto it.

The simulation results of optimization using the modi-fied GA are plotted inFig. 9. The results based on the massbalance equations (MBEs) are shown from (a) to (e). As ac odela or tot o cal-c at thet nalb ont ieldf del.F arev g-u r tok end-i cen-t ordert closet

5

ratep lo3 Co.I asts -� ento d,N out.D then med

Table 1The measured and predicted final biomass concentrations and total reactiontimes for all experiments that have been carried out in this study

Run Total time (h) Final biomass (g/L)

Predicted Measured

1 12.5 – 8.452 12.5 – 7.63 12.5 – 9.24 12.5 – 9.655 12.5 – 8.56 12.5 – 9.57 12.5 – 6.5758 12.5 – 8.09 12.5 – 9.6510 8 10.67 11.02

based on the neural model. The optimal feed rate profile wasapplied to the last experiment run. The result of optimizationbased on the proposed technique in this work is given inFig. 10.

FromFig. 10, one can see a close prediction of biomassgrowth under optimal condition was also achieved. The fi-nal biomass predicted was 10.67 g/L, which was in a goodagreement with the experimental result (11.02 g/L). The pre-diction percentage error was less than 10% during the wholefermentation period.

The final biomass concentrations and total reaction timesof the experiments that were carried out are summarized inTable 1. Runs 1–9 are designed for experimental identifica-tion of the neural model. Run 10 was the optimization exper-iment.Table 1shows that the highest biomass concentrationand shortest reaction time was achieved in the run 10, whichis the optimization experiment based on the identified neuralnetwork model.

6. Conclusions

A cascade recurrent neural network model proposed inthis work has proven to be able to capture the dynamic non-linear underlying phenomena contained in the training datas or thep net-w stingm ntedf pa-b iza-t ntali ethodc Thep s ofh essm itedk herei cessm

omparison, the results based on the cascade RNN mre shown from (f) to (j). The responses of the bioreact

he optimal feed rate based on the neural model are alsulated using the mechanistic model. It can be seen thwo optimal trajectories are quite different. However, the fiiomass quantity yielded from the optimal profile based

he neural model is 281,956 C mol. This is 99.8% of the yrom the optimal profile based on the mathematical mourthermore, the reactions of glucose, ethanol and DOery similar for both optimal profiles. As shown in the fire, ethanol is first slowly formed and increased in ordeeep the biomass production rate at a high value. In theng stage of the fermentation, the residual glucose conration is reduced to zero, and ethanol is consumed ino make the overall substrate conversion into biomasso 100%.

. Experimental verification

Laboratory experiments controlled by different feedatterns, as shown inFig. 2, were carried out using BioF000 bench-top fermentors (New Brunswick Scientific

NC., USA). A pure culture of the prototrophic baker’s yetrain,S. cerevisiaeCM52(MATα his3-�200 ura3-52 leu21 lys2-�202 trp1-�63) was obtained from the Departmf Biologic Science, The University of Auckland (Aucklanew Zealand). Total 10 experimental runs were carriedata collected from nine of them were used for identifyingeural network model. The optimization was then perfor

et and can be used as the model of the bioprocess furpose of optimization. The structure of the neuralork model has been selected using validation and teethods. A modified genetic algorithm has been prese

or solving the optimization problem with a strong caility to produce smooth feed rate profiles. The optim

ion results obtained in both simulation and experimenvestigation have demonstrated that the proposed man be used for finding the optimal feed rate profiles.roposed approach can partly eliminate the difficultieaving to specify completely the structure of a bioprocodel. It is especially promising in situations where lim

nowledge is available about a complex process or wt is costly and infeasible to obtain a mechanistic pro

odel.


Acknowledgement

The research was financially supported in part by the Vice-Chancellor’s University Development Fund (VCUDF), TheUniversity of Auckland, New Zealand.

Appendix A. The industrial baker’s yeastfermentation model

A mathematical model of an industry fed-batch fermenta-tion process, which was given in[21], is used to describe thesystem. The kinetics of yeast metabolism that is consideredin the model is based on the bottleneck hypothesis[22]. Themodel is governed by a set of differential equations derivedfrom mass balances in the system. It comprises the followingequations:

Balance equations:

d(VCs)

dt= FS0 −

(µ

Yoxx/s

+ Qe,pr

Ye/s+m

)VX (A.1)

d(VCo)

dt= −QoVX + kLao(C∗

o − Co)V (A.2)

w otec xide,e ist fort ;k ;i

Q

Q

Q

Table A.1The parameter values of the industrial model

Parameters Values Parameters Values

m 0.00321 KLao 600Ke 0.0008 Yc/e 0.68Kl 0.0001 Yo/e 1028Ks 0.002 Yox

c/s 2.35Qe,max 0.70805 Y red

c/s 1.89Qs,max 0.06 Ye/s 1.9Qo,max 0.2 Yo/s 2.17µcr 0.15753 C∗

o 2.41× 10−4

Yx/e 2.0 C∗c 0.00001

Yoxx/s 4.57063 KLac 470.4

Y redx/s 0.1 Ko 3 × 10−6

Oxidative glucose metabolism:

Qs,ox = min

Qs

Qs,lim

Ys/oQo,lim

(A.10)

Reductive glucose metabolism:

Qs,red = Qs −Qs,ox (A.11)

Ethanol uptake rate:

Qe,up = Qe,maxCe

Ke + Ce

Kl

Kl + Cs(A.12)

Oxidative ethanol metabolism:

Qe,ox = min

(Qe,up

(Qo,lim −Qs,oxYo/s)Ye/o

)(A.13)

Ethanol production rate:

Qe,pr = Ye/sQs,red (A.14)

Total specific growth rate:

µ = µox + µred + µe or

µ = Yoxx/sQs,ox + Y red

x/sQs,red + Yx/eQe,ox(A.15)

Q

Q

TI

S

C

VC

C

C

X

d(VCc)

dt= QcVX + kLac(C

∗c − Cc)V (A.3)

d(VCe)

dt= (Qe,pr −Qe,ox)VX (A.4)

d(VX)

dt= µVX (A.5)

dV

dt= F (A.6)

hereCs,Co,Cc,Ce,XandVare state variables which denoncentrations of glucose, dissolved oxygen, carbon diothanol and biomass, respectively;F is the feed rate which

he input of the system;m is the glucose consumption ratehe maintenance energy;Ye/s andYox

x/s are yield coefficientsLao andkLac are volumetric mass transfer coefficientsS0s the concentration of feed.

Glucose uptake rate:

s = Qs,maxCs

Ks + Cs(A.7)

Oxidation capacity:

o,lim = Qo,maxCo

Ko + Co(A.8)

Specific growth rate limit:

s,lim = µcr

Yoxx/s

(A.9)

Carbon dioxide production rate:

c = Yoxc/sQs,ox + Y red

c/sQs,red + Yc/eQe,ox (A.16)

Oxygen consumption rate:

o = Yo/sQs,ox + Yo/eQe,ox (A.17)

able A.2nitial conditions for dynamic simulation

tate variables Values

s 5 × 10−4

50000

e 0

o 2.4 × 10−4

c 00.54


Respiratory quotient:

RQ = Qc

Qo(A.18)

The model parameters and initial conditions that are usedfor dynamic simulations are listed inTables A.1 and A.2.

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modelling and optimization of fed-batch fermentation processes using dynamic neural networks and...

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