#
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

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**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