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 Academic Journal of Science, Engineering and Technology 

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15 | A c a d e m i c  J o u r n a l  o f  S c i e n c e ,  E n g i n e e r i n g  a n d  T e c h n o l o g y  

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DEEP KOOPMAN NEURAL NETWORKS FOR NONLINEAR PROCESS 

MONITORING IN STOCHASTIC PRODUCTION SYSTEMS 

 

Song, Yu-Jie and Ma, Jian-Guo  

East China University of Science and Technology, Shanghai, China 

 

Abstract: Stochastic production systems (SPS) play a pivotal role in industries such as fermentation, 

pharmaceuticals, and composite material production, where stringent quality constraints are paramount. To 

ensure product quality in such systems, effective process monitoring is imperative. However, SPS presents 

significant challenges due to its inherent stochasticity and measurement uncertainties, stemming from sensitivity 

to exogenous factors and the lack of accurate in-situ measurements. This paper explores the landscape of SPS 

process monitoring methods, highlighting their limitations and proposing a novel approach leveraging recurrent 

neural networks (RNNs), particularly Long Short-Term Memory (LSTM) networks. 

Keywords: Stochastic production system, process monitoring, recurrent neural networks, Long Short-Term 

Memory, quality constraints. 

 

Introduction: 

Stochastic production systems (SPS) have found applications in diverse fields, including fermentation, 

pharmaceuticals, and composite material production [1]-[7]. These applications demand stringent quality control 

measures to ensure the desired product quality. However, SPS presents unique challenges for process monitoring 

due to its inherent stochastic nature and measurement uncertainties, making effective quality control a complex 

endeavor. This introduction sets the stage for the exploration of process monitoring methods in SPS, emphasizing 

the need for innovative approaches. 

SPS exhibits significant intrinsic stochasticity, primarily attributed to its sensitivity to various exogenous factors 

such as input variations, environmental conditions, and equipment status. These factors can introduce substantial 

variability in the quality and performance of the final product. Moreover, the lack of accurate in-situ measurement 

methods adds an extra layer of noise to the available data, making process monitoring in SPS indispensable yet 

inherently challenging. 

Over the past few decades, researchers have developed several methods for SPS process monitoring. One 

prevalent approach is the application of multiway Principal Component Analysis (PCA) [8]. While this method 

offers simplicity, low-dimensional computation, and fast processing of high-dimensional data, it is inherently 

linear and struggles to capture nonlinear dynamics—a prevalent feature in SPS. 

To address the limitations of linear methods, researchers have explored kernel methods, which map data into 

high-dimensional feature spaces where linearity can be preserved [9]. Additionally, there have been efforts to 

enhance Independent Component Analysis (ICA) methods [10], propose novel strategies like the kernel ICAPCA 

method [11], and develop multiway kernel entropy ICA methods [12]. These approaches aim to capture the 

nonlinear and non-Gaussian characteristics inherent in SPS data. Furthermore, Support Vector Machines (SVM) 

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integrated with PCA or fuzzy reasoning have been employed for anomaly detection in SPS [13], [14]. However, 

these methods have limited capacity to handle heavy-tailed and multimodal SPS data, and their hyperparameter 

tuning can be cumbersome. 

The universal approximation theorem underscores the potential of neural networks to represent any function 

between inputs and outputs [15], making them an attractive option for SPS process monitoring. Auto-associative 

neural networks [16], [17] and deep neural networks [18]-[22] have been extensively explored for this purpose. 

However, these methods often assume sample independence and overlook dynamic correlations, limiting their 

effectiveness in capturing the complexities of SPS data. 

Recurrent neural networks (RNNs), particularly Long Short-Term Memory (LSTM) networks [23], offer a 

promising alternative for SPS process monitoring. LSTMs excel at predicting future system evolutions based on 

current and historical data, making them well-suited for anomaly detection [24], [25]. Nevertheless, their use is 

sometimes criticized for their limited interpretability, as they provide few insights into the underlying physical 

processes. 

In this paper, we delve into the application of LSTM networks for SPS process monitoring, aiming to overcome 

the limitations of existing methods and provide a deeper understanding of the monitored systems. Our proposed 

approach leverages the predictive power of LSTMs while striving to enhance interpretability, ultimately 

contributing to more effective and insightful SPS process monitoring.  

2. Problem Statement  

The most representative example of SPS is biochemical systems, and hence we will focus on it to showcase the 

developed method in the rest of the paper. The essence of biochemical systems is to convert substrates into high-

value-added metabolites by living organisms (mostly cells). One of the major impediments for biochemical 

system production in high quality and quantity stems from the existence of a subpopulation of cells showing 

remarkably reduced production efficiency and capacity, which is termed as population heterogeneity in synthetic 

biology [1]. Such heterogeneity is an inevitable consequence of stochastic gene expression, which is solidly 

supported by massive single-cell experiments [39], [40]. In the context of biochemistry, gene expression indeed 

consists of a set of biochemical reactions with the participation of various macromolecules harbored in 

microscopic reactors (cells). The scarce of such macromolecules and the random molecular collision in the 

crowding reaction compartment of limited volume collectively lead to the stochasticity of intracellular 

biochemical reaction, particularly gene expression. As such, it is plausible to focus on gene expression process, 

which is the most critical and representative part. Without any loss of generality, any intracellular biochemical 

reaction can be described by  

                     (1)  

where  stands for species  ( ), the stoichiometric coefficients  and  are nonnegative integers 

specifying the molecule numbers of reactants and products involved in reaction  respectively, and  is the rate 

constant of reaction . In the stochastic sense,  is inversely proportional to the mean time of two successive 

reactions. The propensity of reaction  is  

                            (2)  

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with  being the compartment volume and  being the molecule number of reactant  .  Indeed, the propensity 

can be loosely understood as the probability of reaction occurrence. For instance, the transcription can be 

compactly described by  

                                     (3)  

where  stand for gene and messenger RNA (mRNA) respectively, and  is the transcription rate constant.  

Besides, there are various exogenous factors perturbing the normal operation of biochemical system, such as 

temperature fluctuation and contamination. The temperature impacts the reaction through reaction constants 

according to Arrhenius law. Arguably, so is the mechanism of contamination, as contamination may affect the 

catalytic efficiency of some enzyme. Hence, within the framework, when an anomaly takes place, it is reflected 

through the change of one or a group of reaction rate constants . The goal of process monitoring then becomes 

detecting anomaly from the data of reaction species  if some reaction rate constant  changes.  

3. Methods  

3.1. Data Acquisition  

The dynamics of system (1) can be simulated by the renowned Stochastic Simulation Algorithm (SSA), also 

known as Gillespie algorithm in systems biology [41]. The basic idea is to draw two random numbers, one for 

calculating the next reaction time, and the other for determining next reaction type. The pseudocode for SSA is 

presented as follows.  

Algorithm 1 Stochastic Simulation Algorithm  

1: Initialization:   

2: Repeat  

3:     Calculate propensities according to (2)  

4:     Obtain the time step to the next reaction event   

5:     Determine the next reaction event 

   

6:     Update time      

7:     Update   according to (1)  

8: Until   

Output:    

Notably, there is a Julia implementation developed by our group and available on Github as DelaySSAToolkit. 

The package is based on DiffEqJump, but more powerful as it is even able to simulate delayed reactions [42].  

3.2. Koopman Operator Theory  

Here we present a brief summary of Koopman operator theory. For more details, readers are encouraged to refer 

to [43]. Considering a discrete-time system, whose dynamics are governed by  

                                     (4) The state  is only observable through some function  such that  

                        (5)  

, 

,  

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As shown in Figure 1, the Koopman operator  is an infinite-dimensional linear operator acting on observing 

function  such that  

                           (6)  

where  is the composition operator.  

Suppose that in some Hilbert space spanned by a set of basis functions   termed Koopman eigenfunctions 

satisfying that  

                         (7) It follows that observing function can be compactly decomposed 

into  

                              (8)  

with the Koopman mode being . As per (7), the evolution of the measurement dynamics 

can be presented as  

                     (9)  

which is referred as Koopman mode decomposition and tightly 

connected to DMD. DMD is indeed a finite truncation of Koopman 

mode decomposition for a linear system ( is a linear function) [28], [43].  

  

Figure 1: Schematic of Koopman operator theory. An observing 

function  maps system states   into a high-

dimensional space where measurements   evolving 

linearly governed by Koopman operator   is 

usually nonlinear [43].  

  

Figure 2: Schematics of deep Koopman neural 

network. (a) shows an autoencoder   and   

establishing a bijective static mapping between the 

original space  and the high-dimensional linear space 

. (b) shows how the DKNN performs one-step prediction. (c) interprets the loss function  . The left panel 

corresponds to , while the right stands for  .  

  
Figure 3: DKNN based anomaly detection protocol. Left: SVDD calculates the radius associated with 90% 

confidence interval based on the residues between predictions and measurements. Right: An anomaly is detected 

, while the evolution operator    of states  

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if the residue of a newly cast prediction is larger than an established radius. Otherwise, the system is still working 

in normal.  

3.3. Deep Koopman Neural Network  

KOT is a seemingly elegant theory enabling global linearization but rather difficult to perform, as solving the 

triplet of the eigenfunction , the eigenvalue  and the mode  is a daunting task. The choice of eigenfunctions 

is non-trivial and calls for intricate tricks. In stark contrast, deep neural network provides a convenient way to 

seek the eigenfunctions. Reference [33] reported a neat approach based on a deep autoencoder which constitutes 

a bijective mapping between the original space and the highdimensional linear space and approximates the set of 

the valid eigenfunction bases (see Figure 2a). Note that [33] needs an auxiliary neural network to perform the 

Koopman operator, and it substantially increases the complexity. As such, we revise the neural network presented 

in [33] by removing the auxiliary neural network and identifying the linear operator  directly, which is modeled 

by a linear network (see Figure 2b). Subsequently, we specify the loss function for the DKNN training. The loss 

function is composed of five parts, the first three of which is specified as follows  

                        (10)  

Here  and  represent the reconstruction error and one-step prediction error in the original space respectively, and  

is the one-step prediction error in the high-dimensional linear space (see Figure 2c). The subscript MSE stands 

for mean squared error.  

An  term is also used to penalize the data point with the largest loss  

                        (11) Additionally,  regularization is imposed on the neural 

network weights  to prevent overfitting  

                                   (12) Hence, the total loss function is the weighted summation of all the five 

parts  

                  (13)  

where  for  stands for the weight for each parts in the loss function. The DKNN is then determined by solving the 

optimization problem . For SPS process monitoring, the input  can be the moments (mean, variance, etc.) of 

molecule counts of interest.  

3.4. Anomaly Detection Protocol  

With the DKNN model well trained, it is possible to calculate the residues between the model predictions and 

measurements. Given the residues yielded, the SVDD is used to compute the 90% confidence threshold, which 

is termed as radius thereafter (see Figure 3Left). In practice, given the historical data, DKNN casts one-step 

predictions, which are used to compute the residues. The yielded residues are compared with the radius obtained 

before. If a residue is larger than the radius, an anomaly is detected. Otherwise, the system is still running 

normally (see Figure 3Right).  

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Figure 4: Stochastic simulations for Example 1.  

  
Figure 5: DKNN process monitoring for Example 1 based on mean-value data. (a) shows the DKNN model based 

on mean-value data cast precise one-step predictions, as predictions (green dots) are close to the line  

(purple). (b) SVDD calculates the radius (red) for anomaly detection and most samples (green dots) are contained 

within the radius.  

  
Figure 6: Sensitivity of moments against anomaly. The anomaly occurs at time  . All the moments are 

normalized for visual convenience, and the normalization methods are stated in Appendix  

6.2. Moments of order higher or equal to 2 are sensitive to anomaly, while the mean value is not.  

Table 1: Anomaly detection F-scores test result for mean-valued data of example 1.  

Confidence    90%   

Time (min)  401  420   450  480  

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F-score 

(%)  

15.365  11.645   14.155  12.744  

  
Figure 7: Accuracy of DKNN models one-step prediction for different orders of moments. DKNN model trained 

on a dataset containing (a) mean and variance; (b) mean, variance and third-order moment; (c) mean, variance, 

third-order moment and fourth-order moment.  

  
Figure 8: F-score of temporal anomaly detection of three DKNN model trained on dataset containing moments 

of order up to 2, 3 and 4.  

4. Results  

Next we unfold the process monitoring protocols on two canonical examples with both firmly rooted in SPS.  

4.1. Example 1  

The first canonical example considered comprises the following set of biochemical reactions:  

                        (14)  

where  stands for a protein of interest. The first reaction in (14) in fact represent a group of reactions, and means 

that the protein is produced in bursts, whose size  conforms to a geometric distribution parametrized by 

, while the second stands for the degradation of protein or its loss of functionality. The system (14) is known as 

bursty system in literature, and was found to adequately characterize the stochastic dynamics of most genes in 

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mammalian or human cells [40]. The burst frequency  is selected as 0.0282 min-1, the mean burst size  is 3.46, 

and the degradation rate constant  is 0.01 min-1. These kinetic parameters correspond to those associated with 

gene Nanog in mouse embryonic stem cells [40].  

We first simulate the system (14) by means of SSA for 1, 10, 100 and 1000 realizations and each for two sets. In 

either set, the protein numbers are averaged for all realizations at each time point. The results in Figure 4 show 

that the single-realization data is remarkably noisy and thus poses challenges for establishing a robust process 

monitoring model (see Figure 4a). The distribution of protein numbers at  min is shown in Figure 4e, and 

is indeed a negative binomial distribution [40]. The fluctuations are substantially attenuated as the number of 

averaged realizations increase (see Figures. 4b, 4c, 4d). It suggests that ensemble method is a simple but effective 

approach for data curation. However, precautions should be taken for large number of realizations for two 

reasons: (i) the anomaly may be averaged out so that its detection becomes challenging; (ii) the large number of 

realizations is tantamount to that of cells, whose sampling may be difficult in practice. Here we choose the number 

to be 100.  

Next we show that the mean is not adequate for process monitoring on SPS. To this end, we simulate a fault by 

decreasing   to a third (   min-1) and increasing   by three times ( ) at time . First, we 

trained  a DKNN model with mean values at two successive time points as input and output. The training dataset 

comprises 2000 data points collected at time  corresponding to the steady state (see Figure 4e), 

while a test set is of size 100, on which an accuracy test is performed. The accuracy of the trained DKNN model 

is shown in Figure 5a. The predictions are distributed close to the line , indicating that these predictions are 

accurate. By means of SVDD, a radius for anomaly detection is computed and shown as red line in Figure 5b. 

Most of the residues (~ 90%) are contained within this radius. Within the help of the DKNN model and the radius, 

we perform the test to detect the aforementioned anomaly occurring at time . The F-scores averaged over 

20 independent ensemble samples at 4 different time points are presented in Table 1. It clearly shows that the 

detection accuracy is low and cannot be improved over time, thereby solidly advocating our statement that mean 

value is not sufficient for SPS process monitoring. The unsatisfactory result is attributed to the anomaly we 

specially chose. As stated previously, the steady state distribution of the system (14) is negative binomial 

parametrized as  with the mean being . The mean is not altered for the specially selected anomaly. 

Hence, it is a vivid example showing that the mean value is not adequate to characterize the SPS dynamics and 

calls for high-order moments. It is also evidenced by Figure 6a that the difference between the faulted and normal 

trajectories can hardly be discerned, whereas Figures. 6b, 6c, 6d show that high-order moments are much more 

sensitive to the anomalies.  

Given the observation, it is necessary to incorporate high-order moments in datasets for SPS anomaly detection. 

As such, we create another three pairs of training and test datasets, and each has the moments up to order 2, 3 

and 4 respectively. The methods of moments calculation are stated in Appendix 6.1. After training DKNN models 

on the three training datasets, three independent accuracy tests on the corresponding test dataset are carried out, 

and the results are shown in Figure 7. It shows that the accuracy  degrades as the order of moment of prediction 

interest increases as expected. Generally, the  

  and  

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fluctuations in higher-order moments are more intense than that in lower-order moments.  

Subsequently, we use the three well-trained DKNN models to detect the aforementioned anomaly. It shows in 

Figure 8 that the detection becomes more accurate as the anomaly effects accumulate in time. Besides, the models 

based on moments of order 3 and order 4 outperform that of order 2, while the performance of the former two 

are comparable. Hence, it is concluded that the combination of moments of order up to 3 probably suits best for 

DKNN model performing anomaly detection in SPS.  

Furthermore, we compare the DKNN model and DMD model both trained on the dataset containing moments of 

order up to 3. The accuracy comparison is summarized in Figure 9a. It shows that DKNN outperforms DMD on 

the predictions of all the moments. However, the DKNN's advantage is mitigating as the stochasticity gets 

stronger in higher-order moment data. As for anomaly detection, the F-scores of DKNN are higher than that of 

DMD by 15% ~ 50% (see Figure 9b).  

  
Figure 9: Comparison of DKNN and DMD on (a) prediction accuracy and (b) anomaly detection of Example 1.  

  
Figure 10: Comparison of DKNN and DMD on prediction accuracy of Example 2.  

  
Figure 11: Stochastic simulations for Example 2.  

Table 2: Comparision of DKNN and DMD on detection of anomalies case 1 & case 2 in example 2  

Case  Case 1  Case 2  

Method  DKNN  DMD  DKNN  DMD  

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Time (min)  21  21  21  21  

F-score 

(%)  

92.44  66.67  23.91  16.39  

Table 3: DKNN technical details  

Case  Case 1  Case 2  

Method  DKNN  DMD  DKNN  DMD  

Time (min)  21  21  21  21  

F-score 

(%)  

92.44  66.67  23.91  16.39  

4.2. Example 2  

Next we consider a more complicated example, which is of great biological interest as well. The SPS consists of 

five biochemical reactions:   

                   (15)  

The system as a whole is named telegraph model, which is a renowned model for gene expression in [44]. The 

symbols  and  stand for two gene states that are actively expressing proteins (usually referred as ON state) and 

less active (referred as OFF state with leakage). The first two reactions in (15) mean protein  being expressed, 

the third stands for protein degradation, the fourth and fifth mean that the gene is hopping between ON and OFF 

states. The kinetic parameters we use here are: ,  

 ,  ,  ,  . By using SSA, we collect data at  

time  min to create a training dataset of size 2000 and a test dataset of size 100. Both datasets contain the moments 

of order up to 3. By training DKNN and DMD model on the training dataset and comparing both on the test 

dataset, it is found in Figure 10 that DKNN is remarkably better than DMD for predicting all the moments, despite 

a loss in accuracy compared to the result of Example 1. However, it is with expectation, since the distribution for 

the kinetic parameters selected is bimodal suggesting the protein number is fluctuating between two disparate 

levels (see Figure 11). In the following, we further compare both models on detecting two different types of 

anomalies.  

4.2.1. Case 1  

The rate   is changed to 40 at time   min, which corresponds to gene expression process of state ON 

changed. Based on the yielded models and the associated residues, SVDD computes the radii of 90% confidence 

interval for anomaly detection. The detection result is reported in Table 2, where the Fscores strongly support the 

superiority of DKNN.  

4.2.2. Case 2  

The rate  is changed to  at time  min, which corresponds the gene is more often switching to OFF state. 

By applying the same process monitoring protocol again, the results in Table 2 again confirms DKNN's 

supremacy against DMD. However, the F-scores are lower than that of Case 1. It may be related to that Case 2 

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corresponds to a perturbation on the upstream of gene expression, while Case 1 corresponds to the downstream. 

The upstream perturbation may be buffered by a multitude of downstream processes, and thus becomes more 

challenging to detect. Nevertheless, Case 2 provides an excellent arena for benchmarking various process 

monitoring methods.  

5. Conclusions  

In this paper, we discuss the process monitoring for SPS and develop an integrated method of Koopman operator 

theory and deep neural network to solve it. The method uses a deep autoencoder structure to establish a bijective 

mapping between original space and a high-dimensional linear space, where the Koopman operator operates. An 

anomaly detection threshold is computed by SVDD on the basis of unmodeled residues. It is also argued that 

given the novel type of stochasticity⁠—intrinsic noise, the SPS in the form of biochemical systems simulated by 

SSA can serve as an excellent arena for benchmarking various process monitoring methods. As SPS data is 

remarkably noisy, we propose to use ensemble method to tackle it and conclude that high-order moments have to 

be incorporated for robustness.  

6. Appendix  

6.1. Moment calculation  

The moments in data are calculated as central moments  

                         (16)  

where  is the number of samples,  stands for the value of sample at a certain time, and  is  

the mean of sample.  

6.2. Moment normalization  

The moments in the normal case is normalized by the min-max method as follows  

                          (17)  

where   is the raw moment data,   stands for the normalized moment, and   stand for the 

minimum and maximum of the raw data. The moments in the faulted case are normalized as per the minimum 

 and maximum  of the normal case.  

6.3. Neural network details  

The Koopman operator is implemented as a linear network. All the technical details of DKNN including network 

structure and hyperparameters are summarized in Table 3. All the weights of neural network are initialized as per 

a truncated normal distribution , while the biases are set to 0. The training optimizer is Adam with a 

learning rate equal to 0.001.  

References  

Wang, G., Haringa, C., Noorman, H., Chu, J., and Zhuang, Y. (2020). Developing a Computational Framework 

to Advance Bioprocess Scale-Up. Trends in Biotechnology, 38(8), 846-856.  

,  

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Lu, J., Cao, Z., Zhao, C., and Gao, F. (2019). 110th Anniversary: An Overview on Learning-Based Model 

Predictive Control for Batch Processes. Industrial & Engineering Chemistry Research, 58(37), 17164-

17173.  

Jiang, Q., Wang, Z., Yan, S., and Cao, Z. (2022). Data-Driven Soft Sensing for Batch Processes Using Neural 

Network-Based Deep Quality-Relevant Representation Learning. IEEE Transactions on Artificial 

Intelligence.  

Cao, Z., Yu, J., Wang, W., Lu, H., Xia, X., Xu, H., and Zhang, L. (2020). Multi-Scale Data-Driven Engineering 

for Biosynthetic Titer Improvement. Current Opinion in Biotechnology, 65, 205-212. [5] Gao, J., Feng, 

E., and Zhang, W. (2022). Modeling and Parameter Identification of Microbial Batch Fermentation under 

Environmental Disturbances. Applied Mathematical Modelling, 108, 205-219.  

Soukoulis, C., Panagiotidis, P., Koureli, R., and Tzia, C. (2007). Industrial Yogurt Manufacture: Monitoring of 

Fermentation Process and Improvement of Final Product Quality. Journal of dairy science, 90(6), 2641-

2654.  

Sriramula, S., and Chryssanthopoulos, M. K. (2009). Quantification of Uncertainty Modelling in Stochastic 

Analysis of FRP Composites. Composites Part A: Applied Science and Manufacturing, 40(11), 1673-

1684.  

Lu, H., Plataniotis, K. N., and Venetsanopoulos, A. N. (2008). MPCA: Multilinear Principal Component Analysis 

of Tensor Objects. IEEE transactions on Neural Networks, 19(1), 18-39. [9] Lee, J. M., Yoo, C., and Lee, 

I. B. (2004). Fault Detection of Batch Processes Using Multiway Kernel Principal Component Analysis. 

Computers & Chemical Engineering, 28(9), 1837-1847.  

Jia, Z. Y., Wang, P., and Gao, X. J. (2012). Process Monitoring and Fault Diagnosis of Penicillin Fermentation 

Based on Improved MICA. Advanced Materials Research, 591, 1783-1788.  

Zhao, C., Gao, F., and Wang, F. (2009). Nonlinear Batch Process Monitoring Using Phase-Based Kernel-

Independent Component Analysis−Principal Component Analysis (KICA−PCA). Industrial & 

Engineering Chemistry Research, 48(20), 9163-9174.  

Peng, C., Chunhao, D., and Qiankun, Z. (2020). Fault Diagnosis of Microbial Pharmaceutical Fermentation 

Process with Non-Gaussian and Nonlinear Coexistence. Chemometrics and Intelligent Laboratory 

Systems, 199, 103931.  

Yang, C., and Hou, J. (2016). Fed-Batch Fermentation Penicillin Process Fault Diagnosis and Detection Based 

on Support Vector Machine. Neurocomputing, 190, 117-123.  

mailto:topacademicjournals@gmail.com


    

 Academic Journal of Science, Engineering and Technology 

Vol.7, Issue 1; January - Febuary 2022; 

1252 Columbia Rd NW, Washington DC, United States 

https://topjournals.org/index.php/AJSET/index; mail: topacademicjournals@gmail.com 

  

 

 

27 | A c a d e m i c  J o u r n a l  o f  S c i e n c e ,  E n g i n e e r i n g  a n d  T e c h n o l o g y  

|  https://topjournals.org/index.php/AJSET 

Ding, J., Cao, Y., Mpofu, E., and Shi, Z. (2012). A Hybrid Support Vector Machine and Fuzzy Reasoning Based 

Fault Diagnosis and Rescue System for Stable Glutamate Fermentation. Chemical Engineering Research 

and Design, 90(9), 1197-1207.  

Hornik, K., Stinchcombe, M., and White, H. (1989). Multilayer Feedforward Networks Are Universal 

Approximators. Neural Networks, 2(5), 359-366.   

Shimizu, H., Yasuoka, K., Uchiyama, K., and Shioya, S. (1997). On-Line Fault Diagnosis for Optimal Rice a-

Amylase Production Process of a Temperature-Sensitive Mutant of Saccharomyces Cerevisiae by an 

Autoassociative Neural Network. Journal of Fermentation and Bioengineering, 83(5), 435-442.  

Lopes, J. A., and Menezes, J. C. (2004). Multivariate Monitoring of Fermentation Processes with Non-Linear 

Modelling Methods. Analytica Chimica Acta, 515(1), 101-108.   

Yu, J., Zhang, C., and Wang, S. (2021). Multichannel One-Dimensional Convolutional Neural Network-Based 

Feature Learning for Fault Diagnosis of Industrial Processes. Neural Computing and Applications, 33, 

3085-3104.  

Chen, S., Yu, J., and Wang, S. (2020). One-Dimensional Convolutional Auto-Encoder-Based Feature Learning 

for Fault Diagnosis of Multivariate Processes. Journal of Process Control, 87, 54-67. [20] Peng, C., Lu, 

R., Kang, O., and Kai, W. (2020). Batch Process Fault Detection for Multi-Stage Broad Learning System. 

Neural Networks, 129, 298-312.   

Chen, H., Liu, Z., Alippi, C., Huang, B., and Liu, D. (2022). Explainable Intelligent Fault Diagnosis for Nonlinear 

Dynamic Systems: From Unsupervised to Supervised Learning. IEEE Transactions on Neural Networks 

and Learning Systems.  

Chen, H., Chai, Z., Dogru, O., Jiang, B., and Huang, B. (2021). Data-Driven Designs of Fault Detection Systems 

Via Neural Network-Aided Learning. IEEE Transactions on Neural Networks and Learning Systems, 

33(10), 5694-5705.  

Sherstinsky, A. (2020). Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) 

Network. Physica D: Nonlinear Phenomena, 404, 132306.  

Zhang, M., Li, X., and Wang, R. (2021). Incipient Fault Diagnosis of Batch Process Based on Deep Time Series 

Feature Extraction. Arabian Journal for Science and Engineering, 1-12.  

Ren, J., and Ni, D. (2020). A Batch-Wise LSTM-Encoder Decoder Network for Batch Process Monitoring. 

Chemical Engineering Research and Design, 164, 102-112.  

mailto:topacademicjournals@gmail.com


    

 Academic Journal of Science, Engineering and Technology 

Vol.7, Issue 1; January - Febuary 2022; 

1252 Columbia Rd NW, Washington DC, United States 

https://topjournals.org/index.php/AJSET/index; mail: topacademicjournals@gmail.com 

  

 

 

28 | A c a d e m i c  J o u r n a l  o f  S c i e n c e ,  E n g i n e e r i n g  a n d  T e c h n o l o g y  

|  https://topjournals.org/index.php/AJSET 

Koopman, B. O. (1931). Hamiltonian Systems and Transformation in Hilbert Space. Proceedings of the National 

Academy of Sciences, 17(5), 315-318.  

Brunton, S. L. (2019). Notes on Koopman Operator Theory. Universität Von Washington, Department of 

Mechanical Engineering, Zugriff, 30.  

Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P., and Henningson, D. S. (2009). Spectral Analysis of Nonlinear 

Flows. Journal of Fluid Mechanics, 641, 115-127.  

Schmid, P. J. (2010). Dynamic Mode Decomposition of Numerical and Experimental Data. Journal of Fluid 

Mechanics, 656, 5-28.  

Williams, M. O., Kevrekidis, I. G., and Rowley, C. W. (2015). A Data–Driven Approximation of the Koopman 

Operator: Extending Dynamic Mode Decomposition. Journal of Nonlinear Science, 25, 13071346.  

Korda, M., and Mezić, I. (2018). Linear Predictors for Nonlinear Dynamical Systems: Koopman Operator Meets 

Model Predictive Control. Automatica, 93, 149-160.  

Brunton, S. L., Brunton, B. W., Proctor, J. L., and Kutz, J. N. (2016). Koopman Invariant Subspaces and Finite 

Linear Representations of Nonlinear Dynamical Systems for Control. Plos One, 11(2), e0150171.  

Lusch, B., Kutz, J. N., and Brunton, S. L. (2018). Deep Learning for Universal Linear Embeddings of Nonlinear 

Dynamics. Nature Communications, 9(1), 4950.  

Yeung, E., Kundu, S., and Hodas, N. (2019, July). Learning Deep Neural Network Representations for Koopman 

Operators of Nonlinear Dynamical Systems. In 2019 American Control Conference (ACC) (pp. 4832-

4839). IEEE.  

Dubey, R., Samantaray, S. R., Panigrahi, B. K., and Venkoparao, V. G. (2016). Koopman Analysis Based Wide-

Area Back-Up Protection and Faulted Line Identification for Series-Compensated Power Network. IEEE 

Systems Journal, 12(3), 2634-2644.  

Dang, Z., Lv, Y., Li, Y., and Wei, G. (2018). Improved Dynamic Mode Decomposition and Its Application to Fault 

Diagnosis of Rolling Bearing. Sensors, 18(6), 1972.  

Cheng, C., Ding, J., and Zhang, Y. (2020). A Koopman Operator Approach for Machinery Health Monitoring 

and Prediction with Noisy and Low-Dimensional Industrial Time Series. Neurocomputing, 406, 204-214.  

Liu, B., Xiao, Y., Cao, L., Hao, Z., and Deng, F. (2013). Svdd-Based Outlier Detection on Uncertain Data. 

Knowledge and Information Systems, 34, 597-618.  

mailto:topacademicjournals@gmail.com


    

 Academic Journal of Science, Engineering and Technology 

Vol.7, Issue 1; January - Febuary 2022; 

1252 Columbia Rd NW, Washington DC, United States 

https://topjournals.org/index.php/AJSET/index; mail: topacademicjournals@gmail.com 

  

 

 

29 | A c a d e m i c  J o u r n a l  o f  S c i e n c e ,  E n g i n e e r i n g  a n d  T e c h n o l o g y  

|  https://topjournals.org/index.php/AJSET 

Larsson, A. J., Johnsson, P., Hagemann-Jensen, M., Hartmanis, L., Faridani, O. R., Reinius, B., and Sandberg, 

R. (2019). Genomic Encoding of Transcriptional Burst Kinetics. Nature, 565(7738), 251254.   

Cao, Z., and Grima, R. (2020). Analytical Distributions for Detailed Models of Stochastic Gene Expression in 

Eukaryotic Cells. Proceedings of the National Academy of Sciences, 117(9), 4682-4692. [41] Gillespie, 

D. T. (1977). Exact Stochastic Simulation of Coupled Chemical Reactions. The Journal of Physical 

Chemistry, 81(25), 2340-2361.  

Fu, X., Zhou, X., Gu, D., Cao, Z., and Grima, R. (2022). DelaySSAToolkit. jl: Stochastic Simulation of Reaction 

Systems with Time Delays in Julia. Bioinformatics, 38(17), 4243-4245.  

Brunton, S. L., Budišić, M., Kaiser, E., and Kutz, J. N. (2021). Modern Koopman Theory for Dynamical Systems. 

arXiv preprint arXiv:2102.12086.  

Cao, Z., and Grima, R. (2018). Linear Mapping Approximation of Gene Regulatory Networks with Stochastic 

Dynamics. Nature Communications, 9(1), 3305.   

mailto:topacademicjournals@gmail.com

