# Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

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## Abstract

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## 1. Introduction

## 2. Statement of the Inverse Problem and a Gradient Method of Its Solution

**Remark**

**1.**

- Set $s:=0$ and ${q}^{\left(0\right)}\left(x\right)$ as an initial guess.
- Find the solution ${u}^{\left(s\right)}(x,t)$ of the direct problem:$$\left\{\begin{array}{c}\epsilon \frac{{\partial}^{2}{u}^{\left(s\right)}}{\partial {x}^{2}}-\frac{\partial {u}^{\left(s\right)}}{\partial t}=-{u}^{\left(s\right)}\frac{\partial {u}^{\left(s\right)}}{\partial x}+{u}^{\left(s\right)},\phantom{\rule{1.em}{0ex}}x\in (0,1),\phantom{\rule{1.em}{0ex}}t\in (0,T],\hfill \\ {u}^{\left(s\right)}(0,t)={u}_{left}\left(t\right),\phantom{\rule{1.em}{0ex}}{u}^{\left(s\right)}(1,t)={u}_{right}\left(t\right),\phantom{\rule{1.em}{0ex}}t\in [0,T],\hfill \\ {u}^{\left(s\right)}(x,0)={q}^{\left(s\right)}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in (0,1).\hfill \end{array}\right.$$
- Find the solution ${\psi}^{\left(s\right)}(x,t)$ of the adjoint problem:$$\left\{\begin{array}{c}\epsilon \frac{{\partial}^{2}{\psi}^{\left(s\right)}}{\partial {x}^{2}}+\frac{\partial {\psi}^{\left(s\right)}}{\partial t}={u}^{\left(s\right)}\frac{\partial {\psi}^{\left(s\right)}}{\partial x}+{\psi}^{\left(s\right)}+\hfill \\ \phantom{\rule{34.14322pt}{0ex}}+2\theta (t-{t}_{0})\delta (x-{{f}_{1}}_{{\delta}_{1}}\left(t\right))({u}^{\left(s\right)}(x,t)-{{f}_{2}}_{{\delta}_{2}}\left(t\right)),\phantom{\rule{1.em}{0ex}}x\in (0,1),\phantom{\rule{1.em}{0ex}}t\in [0,T),\hfill \\ {\psi}^{\left(s\right)}(0,t)=0,\phantom{\rule{1.em}{0ex}}{\psi}^{\left(s\right)}(1,t)=0,\phantom{\rule{1.em}{0ex}}t\in [0,T],\hfill \\ {\psi}^{\left(s\right)}(x,T)=0,\phantom{\rule{2.em}{0ex}}x\in (0,1).\hfill \end{array}\right.$$Here $\delta \left(x\right)$ is the Dirac delta function., $\theta \left(t\right)$ is the Heaviside step function.
- Find the gradient of the functional (4):$${J}^{\prime}\left[{q}^{\left(s\right)}\right]\left(x\right)=-{\psi}^{\left(s\right)}(x,0).$$
- Find an approximate solution at the next step of the iteration:$${q}^{(s+1)}\left(x\right)={q}^{\left(s\right)}\left(x\right)-{\beta}_{s}{J}^{\prime}\left[{q}^{\left(s\right)}\right]\left(x\right),$$
- Check a condition for stopping the iterative process. If it is satisfied, we put ${q}^{inv}\left(x\right):={q}^{(s+1)}\left(x\right)$ as a solution of the inverse problem. Otherwise, set $s:=s+1$ and go to step 2.
- (a)
- In the case of experimental data measured with errors ${\delta}_{1}$ and ${\delta}_{2}$, the stopping criterion is$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \underset{{t}_{0}}{\overset{T}{\int}}\left[{\left({x}_{t.p.}\left(t;{q}^{(s+1)}\right)-{{f}_{1}}_{{\delta}_{1}}\left(t\right)\right)}^{2}dt+{\left(u\left({{f}_{1}}_{{\delta}_{1}}\left(t\right),t;{q}^{(s+1)}\right)-{{f}_{2}}_{{\delta}_{2}}\left(t\right)\right)}^{2}\right]dt\le \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{273.14662pt}{0ex}}\le {\delta}_{1}^{2}+{\delta}_{2}^{2}.\hfill \end{array}$$Here ${x}_{t.p.}\left(t;q\right)$ is the position of the reaction front determined by the direct problem (1) for a given function $q\left(x\right)$.
- (b)
- In the case of exact input data, the iterative process stops when $J\left[{q}^{\left(s\right)}\right]$ is less than the error of the finite-difference approximation.

**Remark**

**2.**

**Remark**

**3.**

#### Examples of Numerical Calculations

## 3. Deep Machine Learning Method

#### Examples of Numerical Calculations

## 4. Discussion

- The question of the theoretical justification of the uniqueness and stability of the solution of the considered inverse problem remains open. This may be the subject for a separate work. In this article, we limited ourselves to testing the effectiveness of the proposed approach using numerical experiments.
- When constructing the objective functional, it is possible to use additional smoothing terms (for example, in the form of Tikhonov’s functional [53]). We have limited ourselves to considering the cost functional that determines the least squares method, since its use has already given rather good results.
- Applying deep machine learning, we aimed to demonstrate the fundamental possibility of solving problems of the considered type with limited experimental data using this method. In this regard, we used a fairly good dataset to train the neural network. The question of choosing the optimal neural network configuration remains open. This issue is of significant interest and may be the topic of a separate work.
- The methods of asymptotic analysis were used only to determine the function ${x}_{t.p.}\left(t\right)$ in the formulation of the direct problem. However, other equivalent ways of defining this function are possible that will not affect the quality of the recovered solution.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Formulations of the inverse problem of recovering the initial condition: (

**a**) formulation with data at the final time moment (an inverse backward problem), (

**b**) formulation with data on the position of a reaction front measured with a time delay.

**Figure 3.**The form of a solution to the direct problem $u(x,{t}_{m})$ at some fixed set of time instants ${t}_{m}\in [0,T]$.

**Figure 4.**(

**a**) Exact model function ${f}_{1}\left(t\right)$ and noisy function ${{f}_{1}}_{{\delta}_{1}}\left(t\right)$; (

**b**) exact model function ${f}_{2}\left(t\right)$ and noisy function ${{f}_{2}}_{{\delta}_{2}}\left(t\right)$; (

**c**) result of restoring the function ${q}^{inv}\left(x\right)$ for $\{{\delta}_{1},{\delta}_{2}\}=\{0.03,0.1\}$.

**Figure 5.**The dependence of the accuracy of the reconstruction of the approximate solution ${q}^{inv}\left(x\right)$ on the value ${t}_{0}$, which determines the time delay at the beginning of experimental measurements of the input data. The dashed curves mark the error of the reconstruction for the case when error “$\delta $” is equal to 0.

**Figure 6.**Dependence of the accuracy of the reconstruction of the approximate solution ${q}^{inv}\left(x\right)$ on the mean square error of the input data “$\delta $”.

**Figure 7.**An arbitrary example of a set of functions $\widehat{q}\left(x\right)$ used to train a neural network.

**Figure 8.**The result of restoring the function ${q}^{inv}\left(x\right)$ using deep machine learning in the case of “good” dataset used for training the neural network.

**Figure 9.**The result of restoring the function ${q}^{inv}\left(x\right)$ using deep machine learning in the case of “bad” dataset used for training the neural network.

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**MDPI and ACS Style**

Lukyanenko, D.; Yeleskina, T.; Prigorniy, I.; Isaev, T.; Borzunov, A.; Shishlenin, M.
Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay. *Mathematics* **2021**, *9*, 342.
https://doi.org/10.3390/math9040342

**AMA Style**

Lukyanenko D, Yeleskina T, Prigorniy I, Isaev T, Borzunov A, Shishlenin M.
Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay. *Mathematics*. 2021; 9(4):342.
https://doi.org/10.3390/math9040342

**Chicago/Turabian Style**

Lukyanenko, Dmitry, Tatyana Yeleskina, Igor Prigorniy, Temur Isaev, Andrey Borzunov, and Maxim Shishlenin.
2021. "Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay" *Mathematics* 9, no. 4: 342.
https://doi.org/10.3390/math9040342