*3.3.2 Hybrid ANN models*

The following assumptions are made during the batch/fed-batch operation of the bioreactor. The reactor is completely mixed and the feed flow rate (*F*) is known. Measurements for biomass concentration (*X*), Substrate concentration (*S*), product

The first principles model consists of 4 state equations for biomass concentration X (g/l), Substrate concentration S (g/l), Product concentration P (g/l) and volume

*V*

*<sup>V</sup>* <sup>þ</sup> *<sup>K</sup>* 

> *qp Yp=<sup>s</sup>*

*X* (8)

(10)

),

). No

*<sup>V</sup>* ð Þ *Sin* � *<sup>S</sup>* (9)

*dt* <sup>¼</sup> *<sup>F</sup>* (11)

), *K* is the product decay constant (h�<sup>1</sup>

*μ* ¼ *ffuzzy*ð Þ *S*, *X* (13) *qp* ¼ *ffuzzy*ð Þ *S*, *X* (14)

), *F* represents substrate feed rate, *Sin* is

þ *msX* (12)

*dt* <sup>¼</sup> *<sup>μ</sup><sup>X</sup>* � *<sup>F</sup>*

*X* þ *F*

*dt* <sup>¼</sup> *qpX* � *<sup>P</sup> <sup>F</sup>*

*dV*

the substrate concentration in the feed, *qs* is the substrate consumption rate (h�<sup>1</sup>

direct measurements were made for these kinetic parameter rates. Therefore, a

*Yx=<sup>s</sup>*ð Þ *Ks* <sup>þ</sup> *<sup>S</sup>* <sup>þ</sup>

The kinetic parameters *μ* and *qp* depends on *X* and *S* and their values are unknown. Hence, fuzzy models are developed to estimate their values. An extended Kalman filter is designed to obtain the estimated values of parameters. The filter is tuned by fixing the process noise covariance matrix Q as [0.001,0.001, 0.001, 0.054,0.003] and measurement error covariance matrix R as [0.1, 0.05, 0.03, 0.1]. The filter performance is evaluated by stability border criterion *λ* and significance level, *L<sup>α</sup>* (**Table 2**). Smaller values of these two criterions represent good tuning of the Kalman filter. From the table, it is observed that the filter is tuned properly. In this work, the fuzzy sub-model identification for specific growth rate and product formation rate are done with fuzzy clustering. The basic idea is to form clusters (similar groups) with the available experimental data. Each cluster exhibit an independent rule in the rule

*λ* 0.32 *L<sup>α</sup>* for *X* 5 *L<sup>α</sup>* for *S* 3 *L<sup>α</sup>* for *P* 7

concentration (*P*) and volume (*V*) are available. The sampling interval is

*dx*

*ds dt* ¼ �*qs*

*dP*

30 minutes for these measurements.

*Biotechnological Applications of Biomass*

of the bioreactor V (l) as defined in Eq. (8–11).

where *μ* is the specific growth rate (h�<sup>1</sup>

fuzzy model structure is represented as in Eq. (12–14).

where *μm*, *Yx=<sup>s</sup>*, *Yp=<sup>s</sup>*, *Ks*, *ms* are constants.

**Table 2.**

**612**

*Results of Kalman filter.*

*qs* <sup>¼</sup> *<sup>μ</sup>mS*

*qp* is the product formation rate (h�<sup>1</sup>

Hybrid ANN models are combination of first principles and ANN wherein the ANNs are used to estimate the kinetic parameters (black box models) [32]. The hybrid model shown in **Figure 4** is a combination of neural network estimator with the Mass Balance equations (Mathematical model). The neural network estimator is capable of estimating the process parameters from the real time measurements and these kinetic parameters (μ and Ys/x) are updated in the mass balance equations to give the value of the state variables in the next time instant.

In general, the kinetic parameters are determined offline from experimental data. Due to the ability of neural networks to learn and model non-linear relationships, the parameter values can be estimated after proper training. Neural network with varying number of hidden neurons has been trained and MSE between the actual data and estimated data are calculated. Network with less MSE has been selected to find optimal hidden neurons. In this case study, a neural network structure comprising of two layer feed-forward network with sigmoid hidden neuron and linear output neuron is used. The state variables Xt *X t*ð Þ and *S t*ð Þ S tð Þ are the inputs and the parameters *μ*^ and *Y*^ *<sup>S</sup>=<sup>X</sup>* are the outputs of the neural network. The parameters are found using the Eq. (15) and Eq. (16) given below for every time instant.

$$
\mu = \mu\_{\text{max}} \frac{\mathbf{S(t)}}{k \mathbf{S} + \mathbf{S(t)}} \tag{15}
$$

where S(t) is the substrate concentration, *kS* – saturation constant which is found experimentally to be 0.01 g/L.

$$
\hat{Y}\_{\text{S/X}} = \frac{\mathbf{S}(t) - \mathbf{S}(t-1)}{\mathbf{X}(t) - \mathbf{X}(t-1)}
$$

$$
\mathbf{Y}\_{\text{S/X}} = \frac{\mathbf{S}\_{\text{t}} - \mathbf{S}\_{\text{t}-1}}{\mathbf{X}\_{\text{t}} - \mathbf{X}\_{\text{t}-1}} \tag{16}
$$

**Figure 2.** *Fuzzy model for kinetic parameters μ and* Yp/s.

**Figure 3.** *Performance of hybrid model for biomass(a), substrate(b) and product (c) concentrations.*

Where *S t*ð Þ and *S t*ð Þ � 1 are the present and past substrate concentrations, *X t*ð Þ Xt and Xt�<sup>1</sup> *X t*ð Þ � 1 are the present and past biomass concentrations respectively. For generalization, the experimental data is divided for training, testing and validation in the ratio 70:15:15. The training of neural network has been carried out in MATLAB. Levenberg–Marquardt algorithm is used with single hidden layer. Sigmoid and Linear activation functions are used for hidden and output layers respectively. Mean Square Error is the performance evaluation criterion and accordingly the number of hidden neurons is chosen to be 15. The number of iterations is fixed at 1000.

The response from the hybrid model obtained for training input is shown in **Figure 5(a)** and test input is shown in **Figure 5(b)**.

**3.4 NARX models**

**Figure 6.**

**615**

**Figure 4.**

**Figure 5.**

*Validation of hybrid model with training and testing data set.*

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production*

*DOI: http://dx.doi.org/10.5772/intechopen.96027*

*Variation of kinetic parameters μ and YX/S with respect to time.*

*Hybrid model structure.*

Non-linear regressive models with exogenous input (NARX) are found to be effective for non-linear system identification, as they have good predictive capability. To predict the system behavior without a deep mathematical knowledge [33, 34],

The process parameters μ and YX/S change with respect to time and the corresponding state variable measurements. The time varying natures of the parameters are shown in **Figure 6**.

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production DOI: http://dx.doi.org/10.5772/intechopen.96027*

**Figure 4.** *Hybrid model structure.*

**Figure 5.** *Validation of hybrid model with training and testing data set.*

**Figure 6.** *Variation of kinetic parameters μ and YX/S with respect to time.*

#### **3.4 NARX models**

Non-linear regressive models with exogenous input (NARX) are found to be effective for non-linear system identification, as they have good predictive capability. To predict the system behavior without a deep mathematical knowledge [33, 34],

Where *S t*ð Þ and *S t*ð Þ � 1 are the present and past substrate concentrations, *X t*ð Þ Xt and Xt�<sup>1</sup> *X t*ð Þ � 1 are the present and past biomass concentrations respectively. For generalization, the experimental data is divided for training, testing and validation in the ratio 70:15:15. The training of neural network has been carried out in MATLAB. Levenberg–Marquardt algorithm is used with single hidden layer. Sigmoid and Linear activation functions are used for hidden and output layers respectively. Mean Square Error is the performance evaluation criterion and accordingly the number of hidden neurons is chosen to be 15. The number of

*Performance of hybrid model for biomass(a), substrate(b) and product (c) concentrations.*

The response from the hybrid model obtained for training input is shown in

The process parameters μ and YX/S change with respect to time and the corresponding state variable measurements. The time varying natures of the

iterations is fixed at 1000.

**Figure 3.**

**614**

parameters are shown in **Figure 6**.

*Biotechnological Applications of Biomass*

**Figure 5(a)** and test input is shown in **Figure 5(b)**.

model identification with input–output measurements is generally used. For closer approximations of actual process, a NARX model is commonly employed [35]. The NARX is a recurrent dynamic network, with feedback connections enclosing several layers of the network. The defining equation for the NARX model is given in Eq. (17).

$$y(t) = \left\{ f(y(t-1), y(t-2), y(t-3), \dots, y(t-ny), u(t-1)\dots u(t-nu)) + e(t) \right\} \tag{17}$$

where *y t*ð Þ and *u t*ð Þ are output and input signal, '*f*' is a nonlinear function, *ny* and *nu* are the output and input delays of nonlinear model and *e t*ð Þ is the error term. The next value of the dependent output signal is regressed on previous values of the output signal and previous values of an independent (exogenous) input signal. The successful estimation of process state by soft sensor greatly depends on input output data set. The input variable should be chosen such that it has a direct or indirect relation with the estimation variable. The microbial cell metabolism is influenced by pH value, agitation speed and substrate concentration inside the bioreactor vessel, hence it is directly related to the biomass concentration [36].

As a case study [37], a NARX model is developed for estimation of biomass concentration using the dataset of pH, agitation speed and substrate concentration values starting from the time of inoculation till the end of fed batch process. The experimental data is divided into training, testing and validation in the ratio 70:15:15. The inputs for the NARX model are present values of pH, substrate concentration (S), agitation speed and previous sampling instant biomass concentration, X(k-1) and the output is the estimated biomass concentration, X(k) as shown in **Figure 7**.

capacitance probe is utilized in microbial mode and the biomass is measured at frequencies 0.6 MHz and 15 MHz. The 15 MHz reading is used as a form of auto zero and subtracted from the 0.6 MHz. Therefore, the capacitance of background matter is automatically subtracted from the signal. A resolution of 0.1 pF/cm on the instrument typically represents 106 Cells/ml, or 0.5 grams per liter. The dynamic NARX network are trained with different sets of input–output data. The response of

**Parameters Value**

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production*

Number of iterations 1000 Number of hidden layer 2 Number of hidden neurons in first layer 18 Number of hidden neurons in second layer 10 Bias 1 Activation function (hidden layer) Sigmoid Activation function (Output) Linear Performance Evaluation Mean Square Error

Architecture Dynamic neural network (NARX) Training Algorithm Levenberg–Marquardt algorithm

It is inferred from **Figure 8**, that the trained dynamic NARX network can be used in the place of biosensor. The error response for single NARX network is

The performance of NARX network is analyzed based on the performance criteria, Root Mean Square Error (RMSE) and coefficient of determination (R<sup>2</sup>

s

X*<sup>n</sup> i*¼1

where *N* is the length of data, *yi is* the predict value and *ti is* the target value (Rafsanjani *et al.* 2016). The RMSE and R2 values of NARX network are obtained as 0.01 and 0.8789 respectively and the correlation graph is shown in **Figure 10**.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *yi* � *ti* <sup>2</sup> *n*

*RMSE* ¼

*Comparison of NARX model output with experimental biomass concentration data.*

).

(18)

NARX network to test inputs is shown in **Figure 8**.

shown in **Figure 9**.

**Figure 8.**

**617**

**Table 3.**

RMSE is calculated by the Eq. (18).

*ANN parameters for NARX model development.*

*DOI: http://dx.doi.org/10.5772/intechopen.96027*

To obtain best performance from NARX model, two hidden layers are used and the numbers of hidden neurons in each layer are chosen based on Mean Square Error (MSE). ANN parameters used in the NARX model development are listed in **Table 3**.

Experimental validation of NARX Model is done with the help of capacitance probe. The annular dielectric probe when inserted into the bioreactor, gives a capacitance value that can be directly related to the concentration of biomass. The probe is useful particularly during fungal biomass cultivation due to the nonexistence of accurate offline measurements [38]. As the probe measures only the viable cells excluding the dead cells and insoluble substrates, it is an optimal choice for process validation. The probe generates a dielectric spectrum at 2 different frequencies given, based on cell size, morphology etc. [39]. In this case study, the

**Figure 7.** *NARX model for estimation of biomass concentration.*

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production DOI: http://dx.doi.org/10.5772/intechopen.96027*


#### **Table 3.**

model identification with input–output measurements is generally used. For closer approximations of actual process, a NARX model is commonly employed [35]. The NARX is a recurrent dynamic network, with feedback connections enclosing several layers of the network. The defining equation for the NARX model is given in Eq. (17).

*y t*ðÞ¼ *f yt* ð ð Þ � 1 , *y t*ð Þ � 2 , *y t*ð Þ � 3 , *::* … *y t*ð Þ � *ny* , *u t*ð Þ � 1 … *u t*ð Þ � *nu* Þ þ *e t*ð Þ

where *y t*ð Þ and *u t*ð Þ are output and input signal, '*f*' is a nonlinear function, *ny* and *nu* are the output and input delays of nonlinear model and *e t*ð Þ is the error term. The next value of the dependent output signal is regressed on previous values of the output signal and previous values of an independent (exogenous) input signal. The successful estimation of process state by soft sensor greatly depends on input output data set. The input variable should be chosen such that it has a direct or indirect relation with the estimation variable. The microbial cell metabolism is influenced by pH value, agitation speed and substrate concentration inside the bioreactor vessel,

As a case study [37], a NARX model is developed for estimation of biomass concentration using the dataset of pH, agitation speed and substrate concentration values starting from the time of inoculation till the end of fed batch process. The experimental data is divided into training, testing and validation in the ratio 70:15:15. The inputs for the NARX model are present values of pH, substrate concentration (S), agitation speed and previous sampling instant biomass concentration, X(k-1) and the output is the estimated biomass concentration, X(k) as shown

To obtain best performance from NARX model, two hidden layers are used and the numbers of hidden neurons in each layer are chosen based on Mean Square Error (MSE). ANN parameters used in the NARX model development are listed in

Experimental validation of NARX Model is done with the help of capacitance probe. The annular dielectric probe when inserted into the bioreactor, gives a capacitance value that can be directly related to the concentration of biomass. The probe is useful particularly during fungal biomass cultivation due to the nonexistence of accurate offline measurements [38]. As the probe measures only the viable cells excluding the dead cells and insoluble substrates, it is an optimal choice for process validation. The probe generates a dielectric spectrum at 2 different frequencies given, based on cell size, morphology etc. [39]. In this case study, the

hence it is directly related to the biomass concentration [36].

*Biotechnological Applications of Biomass*

in **Figure 7**.

**Table 3**.

**Figure 7.**

**616**

*NARX model for estimation of biomass concentration.*

(17)

#### *ANN parameters for NARX model development.*

capacitance probe is utilized in microbial mode and the biomass is measured at frequencies 0.6 MHz and 15 MHz. The 15 MHz reading is used as a form of auto zero and subtracted from the 0.6 MHz. Therefore, the capacitance of background matter is automatically subtracted from the signal. A resolution of 0.1 pF/cm on the instrument typically represents 106 Cells/ml, or 0.5 grams per liter. The dynamic NARX network are trained with different sets of input–output data. The response of NARX network to test inputs is shown in **Figure 8**.

It is inferred from **Figure 8**, that the trained dynamic NARX network can be used in the place of biosensor. The error response for single NARX network is shown in **Figure 9**.

The performance of NARX network is analyzed based on the performance criteria, Root Mean Square Error (RMSE) and coefficient of determination (R<sup>2</sup> ). RMSE is calculated by the Eq. (18).

$$RMSE = \sqrt{\sum\_{i=1}^{n} \frac{\left(yi - ti\right)^{2}}{n}} \tag{18}$$

where *N* is the length of data, *yi is* the predict value and *ti is* the target value (Rafsanjani *et al.* 2016). The RMSE and R2 values of NARX network are obtained as 0.01 and 0.8789 respectively and the correlation graph is shown in **Figure 10**.

**Figure 8.** *Comparison of NARX model output with experimental biomass concentration data.*

**Figure 9.** *Error plot for the NARX model.*

developed for the estimation of maximum specific growth rate is shown in **Figure 11**. The input gate, forget gate and output gate equations are given by

Eq. (23). The hidden state equation is represented in Eq. (24).

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production*

*DOI: http://dx.doi.org/10.5772/intechopen.96027*

*c* ð Þ*<sup>t</sup>* <sup>¼</sup> *<sup>f</sup>*

*c*ð Þ *<sup>t</sup>*�<sup>1</sup> , *c*ð Þ*<sup>t</sup>* , *c*ð Þ*<sup>t</sup>* Past, new and final memory *h*ð Þ *<sup>t</sup>*�<sup>1</sup> ,*h*ð Þ*<sup>t</sup>* Previous and current hidden state

ð Þ*t c*

*<sup>h</sup>*ð Þ*<sup>t</sup>* <sup>¼</sup> *<sup>o</sup>*ð Þ*<sup>t</sup>* tanh *<sup>c</sup>*

ð Þ *<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>i</sup>*

ð Þ*<sup>t</sup> c*

*i*

*LSTM model developed for maximum specific growth rate estimation.*

*f*

*c*

*x*ð Þ*<sup>t</sup>* Input word

ð Þ*<sup>t</sup>* Input state

ð Þ*<sup>t</sup>* forget state

The new memory cell and final memory cell equations are given as Eq. (22) and

ð Þ*<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup> <sup>W</sup>*ð Þ*<sup>i</sup> <sup>x</sup>*ð Þ*<sup>t</sup>* <sup>þ</sup> *<sup>U</sup>*ð Þ*<sup>i</sup> <sup>h</sup>*ð Þ *<sup>t</sup>*�<sup>1</sup> (19)

ð Þ*<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup> <sup>W</sup>*ð Þ*<sup>f</sup> <sup>x</sup>*ð Þ*<sup>t</sup>* <sup>þ</sup> *<sup>U</sup>*ð Þ*<sup>f</sup> <sup>h</sup>*ð Þ *<sup>t</sup>*�<sup>1</sup> (20)

*<sup>o</sup>*ð Þ*<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup> <sup>W</sup>*ð Þ*<sup>o</sup> <sup>x</sup>*ð Þ*<sup>t</sup>* <sup>þ</sup> *<sup>U</sup>*ð Þ*<sup>o</sup> <sup>h</sup>*ð Þ *<sup>t</sup>*�<sup>1</sup> (21)

ð Þ*<sup>t</sup>* <sup>¼</sup> tanh *<sup>W</sup>*ð Þ*<sup>c</sup> <sup>x</sup>*ð Þ*<sup>t</sup>* <sup>þ</sup> *<sup>U</sup>*ð Þ*<sup>c</sup> <sup>h</sup>*ð Þ *<sup>t</sup>*�<sup>1</sup> (22)

ð Þ*<sup>t</sup>* (23)

ð Þ*<sup>t</sup>* (24)

Eq. (19), (20) and (21) respectively.

where,

*i*

**Figure 11.**

*f*

**619**

**Figure 10.**

*Correlation graph between biomass concentrations monitored with real-time capacitance probe and estimated by the NARX model.*

It is observed that the NARX model has a low value of RMSE, a very high value of R<sup>2</sup> and good correlation to real time probe data, which confirms that this dynamic neural network soft sensor performs well in the estimation of biomass concentration.

#### **3.5 LSTM models**

Recurrent Neural Networks (RNNs) are a type of deep networks that are structured to capture the temporal dependencies of the process effectively [40]. Long Short-Term Memory (LSTM) networks are a type of recurrent neural network capable of learning order dependence in sequence prediction problems. The LSTM network was invented with the goal of addressing the vanishing gradients problem. The key insight in the LSTM design was to incorporate nonlinear, data-dependent controls into the RNN cell, which can be trained to ensure that the gradient of the objective function with respect to the state signal does not vanish [41] and hence LSTMs are well suited for classification and prediction problems. The LSTM model

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production DOI: http://dx.doi.org/10.5772/intechopen.96027*

**Figure 11.** *LSTM model developed for maximum specific growth rate estimation.*

developed for the estimation of maximum specific growth rate is shown in **Figure 11**. The input gate, forget gate and output gate equations are given by Eq. (19), (20) and (21) respectively.

The new memory cell and final memory cell equations are given as Eq. (22) and Eq. (23). The hidden state equation is represented in Eq. (24).

$$\dot{\mathbf{x}}^{(t)} = \sigma \left( \mathbf{W}^{(i)} \mathbf{x}^{(t)} + \mathbf{U}^{(i)} h^{(t-1)} \right) \tag{19}$$

$$f^{(t)} = \sigma\left(\mathcal{W}^{(f)}\mathcal{X}^{(t)} + U^{(f)}h^{(t-1)}\right) \tag{20}$$

$$\sigma^{(t)} = \sigma\left(\mathcal{W}^{(o)}\mathcal{X}^{(t)} + U^{(o)}h^{(t-1)}\right) \tag{21}$$

$$\overline{\mathfrak{L}}^{(t)} = \tanh\left(\mathcal{W}^{(c)}\mathfrak{x}^{(t)} + U^{(c)}h^{(t-1)}\right) \tag{22}$$

$$\mathcal{L}^{(t)} = f^{(t)} \overline{\mathcal{c}}^{(t-1)} + i^{(t)} \overline{\mathcal{c}}^{(t)} \tag{23}$$

$$h^{(t)} = \sigma^{(t)} \tanh\left(c^{(t)}\right) \tag{24}$$


It is observed that the NARX model has a low value of RMSE, a very high value of R<sup>2</sup> and good correlation to real time probe data, which confirms that this dynamic neural network soft sensor performs well in the estimation of biomass

*Correlation graph between biomass concentrations monitored with real-time capacitance probe and estimated*

Recurrent Neural Networks (RNNs) are a type of deep networks that are structured to capture the temporal dependencies of the process effectively [40]. Long Short-Term Memory (LSTM) networks are a type of recurrent neural network capable of learning order dependence in sequence prediction problems. The LSTM network was invented with the goal of addressing the vanishing gradients problem. The key insight in the LSTM design was to incorporate nonlinear, data-dependent controls into the RNN cell, which can be trained to ensure that the gradient of the objective function with respect to the state signal does not vanish [41] and hence LSTMs are well suited for classification and prediction problems. The LSTM model

concentration.

**618**

*by the NARX model.*

**Figure 10.**

**Figure 9.**

*Error plot for the NARX model.*

*Biotechnological Applications of Biomass*

**3.5 LSTM models**


**4. Conclusions**

concentration yield.

**Acknowledgements**

**Conflict of interest**

**Author details**

Chitra Murugan

**621**

Technology for their funding and support.

*DOI: http://dx.doi.org/10.5772/intechopen.96027*

The author declares no conflict of interest.

(MIT) Campus, Anna University, Chennai, India

provided the original work is properly cited.

\*Address all correspondence to: chitramurugan.g@gmail.com

Several modeling techniques that will aid in the monitoring and estimation of

fungal biomass in the presence of lignocellulosic substrates during fed-batch fermentation are discussed in this chapter. Moreover, the bioprocess models are validated with experimental data as discussed in case studies. The use of these soft sensors in industries with accompanying control system will improve the cellulase

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production*

The author acknowledge Anna University and Department of Science and

Department of Instrumentation Engineering, Madras Institute of Technology

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Similar to the development of NARX model, the modeling of LSTM network also includes data collection, parameter determination, training, testing, validation. The modeling can be done either in MATLAB or using python coding. As a case Study, consider a LSTM network in which two hidden layers are chosen and the number of neurons in each hidden layer is varied till the MSE reaches minimum value. The parameters chosen to frame the LSTM network are listed in **Table 4**. The training to testing ratio is chosen as 67:33. The predicted values of maximum specific growth rate calculated from the LSTM model are presented in **Figure 12**.

The performance of LSTM model is evaluated by the statistical measures RMSE, R2 and Accuracy factor (Af). The Af averages the distance between every point and the line of equivalence as a measure of finding the closeness between predicted and observed values. The RMSE, R2 , Af values of the LSTM model are found to be 0.011, 0.994 and 1.024 respectively. The RMSE and Af values are minimum which suggests that the LSTM predictive model fit well with the experimental data.


#### **Table 4.**

*Parameters for LSTM model development.*

**Figure 12.** *Comparison of LSTM predicted maximum specific growth rate data with experimental data.*

*Soft Sensors for Biomass Monitoring during Low Cost Cellulase Production DOI: http://dx.doi.org/10.5772/intechopen.96027*
