#
Evolutionary Approaches to the Identification of Dynamic Processes in the Form of Differential Equations and Their Systems^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Evolutionary Approach EvolODE for the Identification of Dynamical Systems in the Form of an ODE

_{ti}and output y

_{i}characteristics of the process (where t is the number of inputs, i = 1, …, n, where n is sample size):

_{i}represents values from the original sample.

#### 2.2. The Evolutionary Approach EvolODES for the Identification of Dynamical Systems in the Form of an ODE System

_{ti}and output y

_{si}characteristics of the process (where t is the number of input and output variables, i = 1, …, n, where n is sample size):

## 3. Results

_{2}+ I

_{2}→ 2HI. A sample of temperature (T) and rate constant (k) values was applied to construct a model. The sample size was 12 points. As a result, the authors obtained a symbolic model. It corresponds to the well-known Arrhenius Equation (7):

_{a}is the activation energy, and R is the universal gas constant [26].

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Nesmachnow, S.; Massobrio, R.; Arrechea, E.; Mumfordb, C.; Olivera, A.C.; Pablo, A.; Vidalc, O.J.; Tchernykhd, A. Traffic lights synchronization for Bus Rapid Transit using a parallel evolutionary algorithm. Int. J. Transp. Sci. Technol.
**2019**, 8, 53–67. [Google Scholar] [CrossRef] - Tretyakova, A.; Seredynski, F. Application of evolutionary algorithms to maximum lifetime coverage problem in wireless sensor networks. In Proceedings of the IEEE 27th International Parallel and Distributed Processing Symposium Workshops and PhD Forum, IPDPSW, Boston, MA, USA, 20–24 May 2013. [Google Scholar] [CrossRef]
- Bagheri, S.; Wu, N.; Filizadeh, S. Application of artificial intelligence and evolutionary algorithms in simulation-based optimal design of a piezoelectric energy harvester. Smart Mater. Struct.
**2020**, 29, 105004. [Google Scholar] [CrossRef] - Lee, S.-J.; Yoon, Y. Electricity Cost Optimization in Energy Storage Systems by Combining a Genetic Algorithm with Dynamic Programming. Mathematics
**2020**, 8, 1526. [Google Scholar] [CrossRef] - Parmar, G.; Prasad, R.; Mukherjee, S. Order reduction of linear dynamic systems using stability equation method and GA. Int. J. Comput. Inf. Eng.
**2007**, 1, 26–32. [Google Scholar] - Chen, Y.; Luo, Y.; Liu, Q.; Xu, H.; Zhang, D. Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE). Phys. Rev. Res.
**2022**, 4, 023174. [Google Scholar] [CrossRef] - Cao, H.; Kang, L.; Chen, Y.; Yu, J. Evolutionary modeling of systems of ordinary differential equations with genetic programming. Genet. Program. Evol. Mach.
**2000**, 1, 309–337. [Google Scholar] [CrossRef] - Cottineau, C.; Reuillon, R.; Chapron, P.; Rey-Coyrehourcq, S.; Pumain, D. A Modular Modelling Framework for Hypotheses Testing in the Simulation of Urbanisation. Systems
**2015**, 3, 348–377. [Google Scholar] [CrossRef] - Ryzhikov, I.S. Automatic linear differential equation identification in analytical form. Vestnik SibSAU Aerosp. Tehnol. Control Syst.
**2014**, 1, 66–72. [Google Scholar] - Ryzhikov, I.; Brester, C. Restart operator for optimization heuristics in solving linear dynamical system parameter identification problem. In Proceedings of the 11th International Joint Conference on Computational Intelligence, Vienna, Austria, 17–19 September 2019. [Google Scholar]
- Gaucel, S.; Keijzer, M.; Lutton, E.; Tonda, A. Learning Dynamical Systems Using Standard Symbolic Regression. Lect. Notes Comput. Sci.
**2014**, 8599, 25–36. [Google Scholar] [CrossRef] - Iba, H. Inference of differential equation models by genetic programming. Inf. Sci.
**2008**, 178, 4453–4468. [Google Scholar] [CrossRef] - Medvedev, A.V.; Tereshina, A.V.; Yareshenko, D.I. Nonparametric modelling of multidimensional memoryless processes. In Proceedings of the Twelfth International Conference Computer Data Analysis and Modeling: Stochastics and Data Science, Minsk, Belarus, 18–22 September 2019. [Google Scholar]
- Medvedev, A.V.; Chzhan, E.A. On Nonparametric Modelling of Multidimensional Noninertial Systems with Delay. Bull. South Ural State Univ. Ser. Math. Model. Program. Comput. Softw.
**2017**, 10, 124–136. [Google Scholar] [CrossRef] - Sholokhova, A.A.; Ivanov, A.N. Modeling of dynamic systems based on polynomial neural networks. Model. Optim. Inf. Technol.
**2017**, 4, 23. [Google Scholar] - Ballesteros, M.; Polyakov, A.; Efimov, D.; Chairez, I.; Poznyak, A.S. Non-parametric identification of homogeneous dynamical systems. Automatica
**2021**, 129, 109600. [Google Scholar] [CrossRef] - Lee, C.-H.; Teng, C.-C. Identification and Control of Dynamic Systems Using Recurrent Fuzzy Neural Networks. IEEE Trans. Fuzzy Syst.
**2000**, 8, 349–366. [Google Scholar] - Merta, J.; Brandejsky, T. Lifetime adaptation in genetic programming for the symbolic regression. Adv. Intell. Syst. Comput.
**2019**, 1047, 339–346. [Google Scholar] - Karaseva, T.S. Automatic differential equations identification by self-configuring genetic programming algorithm. IOP Conf. Ser. Mater. Sci. Eng.
**2020**, 734, 12093. [Google Scholar] [CrossRef] - Das, S.S.; Mullick, S.S.; Suganthan, P.N. Recent Advances in Differential Evolution—An Updated Survey. Swarm Evol. Comput.
**2016**, 27, 1–30. [Google Scholar] [CrossRef] - Karaseva, T.S.; Semenkina, O.E. Hybrid approach to the dynamic systems identification based on the self-configuring genetic programming algorithm and the differential evolution method. IOP Conf. Ser. Mater. Sci. Eng.
**2021**, 1047, 12076. [Google Scholar] [CrossRef] - Kazakovtsev, L.; Rozhnov, I.; Shkaberina, G. Self-configuring (1 + 1)-evolutionary algorithm for the continuous p-median problem with agglomerative mutation. Algorithms
**2021**, 14, 130. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential Evolution—A Simple and Efficient Heuristic for global Optimization over Continuous Spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Meyer-Nieberg, S.; Beyer, H.-G. Self-Adaptation in Evolutionary Algorithms. Parameter Setting Evol. Algorithm
**2007**, 54, 47–75. [Google Scholar] - Karaseva, T.; Semenkin, E. On the automatic identification of differential equations using a hybrid evolutionary approach. In Proceedings of the 35th International Conference on Information Technologies (InfoTech 2021), Varna, Bulgaria, 16–17 September 2021. [Google Scholar] [CrossRef]
- The Effect of Temperature on the Rate of Chemical Reactions. Available online: http://chemnet.ru/rus/teaching/eremin/4.html (accessed on 13 June 2022).
- Gasull, A.; Kooij, R.; Torregrosa, J.E. Limit cycles in the Holling-Tanner model. Publ. Mat.
**1997**, 41, 149–167. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Scheme of the evolutionary approach EvolODE for the identification of dynamical systems in the form of ODE.

**Figure 4.**Scheme of the evolutionary approach EvolDES for the identification of dynamical systems in the form of an ODE system.

**Figure 5.**Initial sample fit with the output of the obtained model of the change in the rate of a chemical reaction on temperature.

**Figure 6.**Initial sample fit to the output of the obtained model of changes in the number of predators and prey over time.

**Figure 7.**Symbolic accuracy of the resulting ODEs: (

**a**) Data without noise; (

**b**) Data with 5% of noise; (

**c**) Data with 10% of noise.

**Figure 8.**Symbolic accuracy of the resulting ODE systems: (

**a**) Data without noise; (

**b**) Data with 5% of noise; (

**c**) Data with 10% of noise.

**Table 1.**Test tasks for the evolutionary approach to the identification of dynamical systems in the form of an ODE.

No | Differential Equation | Initial Sample Point |
---|---|---|

1 | ${y}^{\prime}=\frac{y+{x}^{2}\mathrm{cos}x}{x}$ | y(x_{0}) = 0.06 |

2 | ${y}^{\prime}=-\frac{y\mathrm{ln}y}{x}$ | y(x_{0}) = e |

3 | ${y}^{\u2033}=\frac{2{\left({y}^{\prime}\right)}^{2}}{y-1}$ | y(x_{0}) = 2 |

4 | ${y}^{\u2033}=6{y}^{\prime}-9y+6x{e}^{3x}$ | y(x_{0}) = 3 |

5 | ${y}^{\u2034}=4{y}^{\prime}+24{e}^{2x}-4\mathrm{cos}\left(2x\right)+8\mathrm{sin}\left(2x\right)$ | y(x_{0}) = 1 |

6 | ${y}^{IV}=y+2\mathrm{cos}\left(x\right)$ | y(x_{0}) = 0 |

7 | ${y}^{V}={y}^{IV}+2x+3$ | y(x_{0}) = 2 |

8 | ${y}^{\prime}=\frac{y+{x}_{1}^{2}\mathrm{cos}{x}_{2}}{{x}_{3}}$ | y($\overline{x}$_{0}) = 0.06 |

9 | ${y}^{\u2033}=\frac{{y}^{\prime}+{x}_{1}^{2}y{y}^{\prime}}{{x}_{2}^{2}}$ | y($\overline{x}$_{0}) = 0 |

10 | ${y}^{\u2033}=-2{y}^{\prime}-2y+2{x}_{1}^{2}+8{x}_{2}+6$ | y($\overline{x}$_{0}) = 3 |

**Table 2.**Testing results of the evolutionary approach to the identification of dynamical systems in the form of an ODE.

No | Sample Size | ||||||||
---|---|---|---|---|---|---|---|---|---|

150 | 100 | 50 | |||||||

Noise, % | |||||||||

0 | 5 | 10 | 0 | 5 | 10 | 0 | 5 | 10 | |

1 | 0 | 0 | 0.001 | 0 | 0 | 0 | 0 | 0 | 0.0001 |

2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0001 | 0.0001 |

3 | 0.0001 | 0.0008 | 0.0008 | 0.0001 | 0.0008 | 0.0008 | 0.0002 | 0.0009 | 0.0012 |

4 | 0.0002 | 0.0003 | 0.0003 | 0.0002 | 0.0003 | 0.0003 | 0.0003 | 0.0004 | 0.0005 |

5 | 0 | 0.0005 | 0.0007 | 0 | 0.0006 | 0.0007 | 0.0001 | 0.0008 | 0.0008 |

6 | 0 | 0.0001 | 0.0002 | 0 | 0.0002 | 0.0003 | 0.0001 | 0.0004 | 0.0005 |

7 | 0 | 0.0008 | 0.0009 | 0.0001 | 0.0009 | 0.0009 | 0.0001 | 0.0013 | 0.0014 |

8 | 0.0046 | 0.0055 | 0.0059 | 0.0074 | 0.0092 | 0.0104 | 0.0113 | 0.0201 | 0.0287 |

9 | 0.0017 | 0.0047 | 0.0053 | 0.002 | 0.0049 | 0.0051 | 0.0033 | 0.0058 | 0.0075 |

10 | 0.0011 | 0.0033 | 0.0041 | 0.0023 | 0.0037 | 0.0042 | 0.0024 | 0.0037 | 0.0039 |

**Table 3.**Test tasks for the evolutionary approach to the identification of dynamical systems in the form of an ODE system.

No | System of Differential Equations | Initial Sample Point |
---|---|---|

1 | $\{\begin{array}{c}\frac{d{y}_{1}}{dt}=2{y}_{1}-5{y}_{2}+3\\ \frac{d{y}_{2}}{dt}=5{y}_{1}-6{y}_{2}-1\end{array}$ | y_{1}(0) = 6y _{2}(0) = 5 |

2 | $\{\begin{array}{c}\frac{d{y}_{1}}{dt}=2{y}_{1}+{y}_{2}\\ \frac{d{y}_{2}}{dt}=3{y}_{1}+t{e}^{t}\end{array}$ | y_{1}(1) = 1y _{2}(1) = 2 |

3 | $\{\begin{array}{c}\frac{d{y}_{1}}{dt}={y}_{1}+2{y}_{2}+{e}^{-2t}\\ \frac{d{y}_{2}}{dt}=4{y}_{1}-{y}_{2}\end{array}$ | y_{1}(0.1) = 5y _{2}(0.1) = 8 |

4 | $\{\begin{array}{c}\frac{d{y}_{1}}{dt}={y}_{2}-{y}_{1}^{2}-{y}_{1}\\ \frac{d{y}_{2}}{dt}=3{y}_{1}-{y}_{2}^{2}-{y}_{1}\end{array}$ | y_{1}(0) = 0y _{2}(0) = 1 |

5 | $\{\begin{array}{c}\begin{array}{c}\frac{d{y}_{1}}{dt}=-{y}_{2}\\ \frac{d{y}_{2}}{dt}={y}_{1}\end{array}\\ \begin{array}{c}\frac{d{y}_{3}}{dt}={y}_{1}-{y}_{4}\\ \frac{d{y}_{4}}{dt}={y}_{2}+{y}_{3}\end{array}\end{array}$ | y_{1}(0) = 1y _{2}(0) = 0y _{3}(0) = 0y _{4}(0) = 0 |

6 | $\{\begin{array}{c}\begin{array}{c}\frac{d{y}_{1}}{dt}=2{y}_{1}-{y}_{2}-{y}_{3}\\ \frac{d{y}_{2}}{dt}=3{y}_{1}-2{y}_{2}-3{y}_{3}+2t\end{array}\\ \frac{d{y}_{3}}{dt}=2{y}_{3}-{y}_{1}-{y}_{2}-{t}^{2}\end{array}$ | y_{1}(0) = 2y _{2}(0) = 3y _{3}(0) = 2 |

7 | $\{\begin{array}{c}\begin{array}{c}\frac{d{y}_{1}}{dt}=-10{y}_{1}+10{y}_{2}\\ \frac{d{y}_{2}}{dt}=25{y}_{1}-{y}_{2}-{y}_{1}{y}_{2}\end{array}\\ \frac{d{y}_{3}}{dt}={y}_{1}{y}_{2}-10{y}_{3}\end{array}$ | y_{1}(0) = 1y _{2}(0) = 0y _{3}(0) = −1 |

8 | $\{\begin{array}{c}\frac{d{y}_{1}}{dt}=3{y}_{1}+2{y}_{2}\\ \frac{{d}^{2}{y}_{2}}{d{t}^{2}}=-10{y}_{1}-{y}_{2}\end{array}$ | y_{1}(0) = −1y _{2}(0) = 7 |

9 | $\{\begin{array}{c}\frac{{d}^{2}{y}_{1}}{d{t}^{2}}=-t{y}_{2}\\ \frac{{d}^{2}{y}_{2}}{d{t}^{2}}={y}_{2}-2\frac{d{y}_{1}}{dt}\end{array}$ | y_{1}(1) = 4y _{2}(1) = −4 |

10 | $\{\begin{array}{c}\frac{{d}^{2}{y}_{1}}{d{t}^{2}}=\frac{d{y}_{1}}{dt}-\frac{d{y}_{2}}{dt}+{e}^{-t}+\mathrm{cos}t\\ \frac{{d}^{2}{y}_{2}}{d{t}^{2}}=\frac{d{y}_{1}}{dt}-\frac{d{y}_{2}}{dt}-2{e}^{t}-\mathrm{sin}t\end{array}$ | y_{1}(0) = 2y _{2}(0) = 0 |

**Table 4.**Testing results of the evolutionary approach to the identification of dynamical systems in the form of an ODE system.

No | Sample Size | ||||||||
---|---|---|---|---|---|---|---|---|---|

150 | 100 | 50 | |||||||

Noise, % | |||||||||

0 | 5 | 10 | 0 | 5 | 10 | 0 | 5 | 10 | |

1 | 0.034 | 0.036 | 0.039 | 0.032 | 0.037 | 0.039 | 0.03 | 0.038 | 0.04 |

2 | 0.018 | 0.023 | 0.024 | 0.018 | 0.024 | 0.024 | 0.019 | 0.025 | 0.024 |

3 | 0.018 | 0.051 | 0.057 | 0.018 | 0.052 | 0.057 | 0.019 | 0.055 | 0.058 |

4 | 0 | 0 | 0.001 | 0 | 0 | 0.001 | 0.002 | 0.002 | 0.003 |

5 | 0 | 0 | 0.002 | 0 | 0 | 0.002 | 0 | 0.001 | 0.003 |

6 | 0.022 | 0.024 | 0.025 | 0.031 | 0.035 | 0.39 | 0.038 | 0.039 | 0.038 |

7 | 0.027 | 0.031 | 0.032 | 0.028 | 0.029 | 0.033 | 0.031 | 0.032 | 0.039 |

8 | 0 | 0 | 0.002 | 0 | 0.001 | 0.002 | 0.002 | 0.003 | 0.003 |

9 | 0 | 0.002 | 0.002 | 0 | 0.002 | 0.002 | 0.001 | 0.002 | 0.003 |

10 | 0.041 | 0.042 | 0.053 | 0.042 | 0.045 | 0.054 | 0.039 | 0.045 | 0.053 |

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**MDPI and ACS Style**

Karaseva, T.; Semenkin, E.
Evolutionary Approaches to the Identification of Dynamic Processes in the Form of Differential Equations and Their Systems. *Algorithms* **2022**, *15*, 351.
https://doi.org/10.3390/a15100351

**AMA Style**

Karaseva T, Semenkin E.
Evolutionary Approaches to the Identification of Dynamic Processes in the Form of Differential Equations and Their Systems. *Algorithms*. 2022; 15(10):351.
https://doi.org/10.3390/a15100351

**Chicago/Turabian Style**

Karaseva, Tatiana, and Eugene Semenkin.
2022. "Evolutionary Approaches to the Identification of Dynamic Processes in the Form of Differential Equations and Their Systems" *Algorithms* 15, no. 10: 351.
https://doi.org/10.3390/a15100351