# Study of Burgers–Huxley Equation Using Neural Network Method

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## Abstract

**:**

## 1. Introduction

## 2. Basic Idea of a Lie-Series-Based Neural Network Algorithm

#### 2.1. Differential Forms and Lie Series Solution

#### 2.2. Algorithm of a Lie-Series-Based Neural Network

**Theorem**

**1.**

**Proof.**

Algorithm 1: A Lie-series-based neural network algorithm for problem (3) |

Require Determine the operator D according to (3), and solve it with the decomposed part ${D}_{1}$ to obtain $\overline{u}$.Begin1. Consider a uniformly spaced distribution of discrete points within the initial condition ${\xi}_{\ell}(\ell =1,2,\dots ,\lambda )$.2. Determining the structure of a neural network. (The number of hidden layers and the number of neurons, the selection of the activation function $\sigma $.)3. Initialization of the neural networks parameters $\mathbf{W}$, $\mathbf{b}$.5. Minimize the loss function $\mathbb{L}\left(\theta \right)$.End |

#### 2.3. The General Structure of the Neural Network

## 3. Lie-Series-Based Neural Network Algorithm for Solving Burgers Huxley Equation

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Comparison of the ${\overline{u}}_{1}\left(\xi \right)$ solution of the Burgers–Huxley equation with the exact solution $u\left(\xi \right)$.

**Figure 4.**Schematic diagram of a Lie series-based neural network algorithm for solving Burgers–Huxley equation.

**Figure 5.**(

**Left**) Comparison of solution ${\widehat{u}}_{1}\left(\xi \right)$ with the exact solution $u\left(\xi \right)=\frac{1}{2}\left(1-tanh\left(\frac{\xi}{4}\right)\right)$ of (13) in the training set. (

**Right**) Comparison of solution ${\widehat{u}}_{1}\left(\xi \right)$ with the exact solution $u\left(\xi \right)=\frac{1}{2}\left(1-tanh\left(\frac{\xi}{4}\right)\right)$ of (13) in the test set.

**Figure 7.**(

**Top**) The true solution $u(t,x)=\frac{1}{2}\left(1-tanh\left(\frac{x}{4}+\frac{3t}{8}\right)\right)$ of the Burgers–Huxley equation is on the left, the predicted solution $\widehat{u}(t,x)$ is on the right. (

**Bottom**) Comparison of predicted and exact solutions at time $t=0.3$, $0.5$, and $0.8$. (The dashed blue line indicates the exact solution $u(t,x)$, and the solid red line indicates the predicted solution $\widehat{u}(t,x)$).

**Figure 8.**Contour plot of the Burgers–Huxley equation with respect to the solution ${\widehat{u}}_{1}(t,x)$ and the exact solution $u(t,x)$.

**Figure 9.**(

**Left**) Comparison of solution ${\widehat{u}}_{1}\left(\xi \right)$ with the exact solution $u\left(\xi \right)=\frac{1}{2}\left(1+tanh\left(\frac{\xi}{4}\right)\right)$ of (15) in the training set. (

**Right**) Comparison of ${\widehat{u}}_{1}\left(\xi \right)$ with the exact solution $u\left(\xi \right)=\frac{1}{2}\left(1+tanh\left(\frac{\xi}{4}\right)\right)$ of (15) in the test set.

**Figure 11.**Comparison of the ${\overline{u}}_{1}\left(\xi \right)$ solution of the Burgers–Fisher equation with the exact solution $u\left(\xi \right)$.

**Figure 12.**(

**Top**) The true solution $u(t,x)=\frac{1}{2}\left(1+tanh\left(\frac{x}{4}+\frac{5t}{8}\right)\right)$ of the Burgers–Fisher equation is on the left, the predicted solution $\widehat{u}(t,x)$ is on the right. (

**Bottom**) Comparison of predicted and exact solutions at time $t=0.3$, $0.5$, and $0.8$. (The dashed blue line indicates the exact solution $u(t,x)$, and the solid red line indicates the predicted solution $\widehat{u}(t,x)$).

**Figure 13.**(

**Left**) Comparison of solution ${\widehat{u}}_{1}\left(\xi \right)$ with the exact solution $u\left(\xi \right)=\frac{1}{2}\left(1+tanh\left(\frac{\sqrt{2}\xi}{4}\right)\right)$ of (17) in the training set. (

**Right**) Comparison of solution ${\widehat{u}}_{1}\left(\xi \right)$ with the exact solution $u\left(\xi \right)=\frac{1}{2}\left(1+tanh\left(\frac{\sqrt{2}\xi}{4}\right)\right)$ of (17) in the test set.

**Figure 15.**(

**Top**) The true solution $u(t,x)=\frac{1}{2}\left(1+tanh\left(\frac{\sqrt{2}x}{4}-\frac{t}{4}\right)\right)$ of the Huxley equation is on the left, the predicted solution $\widehat{u}(t,x)$ is on the right. (

**Bottom**) Comparison of predicted and exact solutions at time $t=0.3$, $0.5$, and $0.8$. (The dashed blue line indicates the exact solution $u(t,x)$, and the solid red line indicates the predicted solution $\widehat{u}(t,x)$).

**Figure 16.**Contour plot of the Huxley equation with respect to the solution ${\widehat{u}}_{1}(t,x)$ and the exact solution $u(t,x)$.

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**MDPI and ACS Style**

Wen, Y.; Chaolu, T.
Study of Burgers–Huxley Equation Using Neural Network Method. *Axioms* **2023**, *12*, 429.
https://doi.org/10.3390/axioms12050429

**AMA Style**

Wen Y, Chaolu T.
Study of Burgers–Huxley Equation Using Neural Network Method. *Axioms*. 2023; 12(5):429.
https://doi.org/10.3390/axioms12050429

**Chicago/Turabian Style**

Wen, Ying, and Temuer Chaolu.
2023. "Study of Burgers–Huxley Equation Using Neural Network Method" *Axioms* 12, no. 5: 429.
https://doi.org/10.3390/axioms12050429