1 Introduction
One of the fundamental characteristics of living organisms is their ability to move in response to external signals. Chemotaxis, which is the oriented motion induced by chemical gradients, has a critical role in many biological and medical fields. It is encountered in several selforganization phenomena such as: aggregation of bacteria, embryogenesis, ecology, tumor growth, etc.
In 1970, Keller and Segel Keller1 introduced a chemotaxis model to describe the aggregation of slim molds. Since then, several variants of the KellerSegel model have been developed to model phenomena in a variety of fields where chemotaxis is involved (see for more details Painter ). The model is still very popular, it reads in a generalized form
(1.1) 
where denotes the cell density at spatial position and time , is the chemoattractant concentration, and respectively define cells diffusion and chemical diffusion coefficients, is the chemotactic sensitivity, is a function describing cell growth and death, and is a kinetic function that describes production and degradation of the chemical signal.
Several numerical methods for KellerSegel systems have been proposed and analyzed. The interested reader may be referred for instance to the introduction of Akhmouch1 , and to Jungel ; Huang ; Xiao ; Zhang1 ; Sulman ; Zhang2 ; Liu ; Li ; Dehghan for more recent works.
In this paper, we focus on three models derived from (1.1). The first one has been proposed in the framework of pattern formation in embryology, the model reads:
(1.2) 
with zeroflux boundary and initial conditions
(1.3) 
Here and in the rest of this paper, and are positive constants, , is an open bounded subset, denotes the final time, is the outward unitnormal on the boundary , and is a nonnegative function.
The above model is a simpler version of the original parabolicparabolic model proposed by Oster and Murray Oster1 , in which the second equation of the system is elliptic, using the reasonable assumption that the chemoattractant diffuses much faster than cells. In Murray1 , it has been demonstrated that the model is able to generate stripe patterns similar to that of embryonic American alligators.
In Akhmouch2 , a semiimplicit finite volume scheme with an additional correction term has been proposed by Akhmouch and the author to solve the system (1.2)–(1.3). Moreover, a convergence analysis has been performed in the case of a nonnegative initial cell density , and it has been established that the numerical solution of the proposed scheme converges to a weak solution of (1.2)–(1.3) under some conditions. In this paper, a particular attention will be given to this analysis.
The second KellerSegel system considered in this paper is the following chemotaxisgrowth model:
(1.4) 
with zeroflux boundary conditions and suitable initial conditions. Herein, is a positive constant and is a polynomial growth term. This model has the ability to reproduce different bacterial pattern formations through the coupled effect of chemotaxis and growth (see, e.g., Aida ; Strehl1 ; Strehl2 ; Akhmouch1 ; Akhmouch3 ). It is well known that when , the solution of the system (called minimal model) may blowup in finite time for initial cell density with sufficiently large mass, which is an interesting mathematical phenomenon. Several works were devoted to study this system (see for more details Horstmann1 ; Hillen1 ; Perthame1 ). However, from biological point of view, the occurrence of finitetime blowup is considered as a pathological behavior of the model. An adequate choice of the reaction term can prevent this blowup Winkler .
The last KellerSegel system considered in the current paper reads:
(1.5) 
endowed with zeroflux boundary conditions and initial conditions. In the above model, is a function which satisfies the conditions: , there exists such that and for . The system (1.5), called volumefilling chemotaxis model, has been introduced by Hillen and Painter in Hillen2 , where the existence of solutions that are bounded is proved. The adopted choice of chemotactic sensitivity implies that cells stop to accumulate when the threshold is reached by , which prevents overcrowding. Numerical simulations of the model are performed in Hillen3 , and in Andreianov1 ; Chamoun1 for more general diffusion term.
The three chemotaxis models presented are highly nonlinear coupled systems due to the chemotaxis term ( for the volumefilling model). Moreover, the cell density reaction term of the second equation of (1.2), the growth term of the first equation of (1.4), and the convective term of the first equation of (1.5), are also nonlinear. These nonlinearities are an important factor for the choice of the time discretization approach when numerically solving such systems. The most used approaches in the literature are: explicit, implicit and semiimplicit schemes. The main drawback of explicit schemes is the too restrictive CFL condition necessary to ensure stability of the solution. For implicit schemes, the CFL constraint is avoided. But, on the other hand, a large nonlinear algebra system must be solved at each timestep, which is too timeconsuming. The third time discretization which is also widely used for solving nonlinear systems is the semiimplicit discretization. Semiimplicit schemes are linearized implicit schemes, so they only require to solve decoupled linear algebra systems at each time level. Regarding numerical methods for KellerSegel systems, Euler semiimplicit scheme is used in several works to obtain fully linear implicit schemes (see, e.g., Strehl2 ; Akhmouch3 ; Liu ; Saito1 ; Saito2 ). In other works, it has only been used to avoid the severe nonlinearity caused by the chemotaxis term and thus obtain decoupled nonlinear schemes Andreianov1 ; Chamoun1 ; Ibrahim1 . As proved in the convergence analysis carried out by Zhou and Saito Saito2 for the parabolicelliptic minimal model, by Akhmouch and the author Akhmouch3 for parabolicelliptic chemotaxis growth model and by Andreianov et al. for degenerate volumefilling chemotaxis model Andreianov1 , no CFL condition is required to obtain the convergence of Euler semiimplicit finite volume schemes for such systems. Nonetheless, despite the advantages of such strategy, these schemes generally suffer from a lack of accuracy.
In this work, a novel approach to develop linear implicit schemes is presented. More precisely, two methods relying on a basic singlelayer neural network to linearize schemes are proposed: the each step training linearized implicit (ESTLI) method, and the selected steps training linearized implicit (SSTLI) method which is based also on a selected steps training (STP) algorithm. The developed methods will be used to numerically solve the systems (1.2), (1.4) and (1.5). The objective is to build a method which has the following attractive benefits:

It must only require to solve decoupled linear systems at each timestep.

It should be significantly more accurate than semiimplicit method.

In terms of computational cost, the method should be nearly as efficient as semiimplicit method, specially for timeconsuming numerical experiments.

As semiimplicit method, it is required to be stable even for large timestep sizes.

It should be easily applied to any equation or system which can be solved by semiimplicit scheme.
Concerning the spatial approximation, a finite volume scheme similar to that of Akhmouch and Benzakour Akhmouch2 is adopted. We mention that finite volume method has been first introduced by Filbet Filbet1 for KellerSegel models, where a fully implicit finite volume scheme has been proposed for the parabolicelliptic minimal model.
As mentioned above, the approach presented in this article uses artificial neural networks (ANN) only as a linearization technique in a numerical scheme. We mention, however, that several interesting works were devoted to use ANN as a main tool to approximate solutions of partial differential equations, so we briefly review some of them. In Lagaris1
, multilayer perceptron was used by Lagaris et al. to solve boundary value problems. Lagaris et al.
Lagaris2developed also a method based on multilayer perceptron and radial basis function networks. In
Shirvany , a novel method was presented using the same networks, with application to the nonlinear Schrodinger equation. We note that several works proposed radial basis function networks to solve elliptic equations, see for instance MaiDuy ; Aminataei ; Jianyu . The reliability of other methods based on singlelayer Bernstein neural network Sun and singlelayer Chebyshev neural network Mall , was also demonstrated by application to elliptic equations. In Rudd , a method combining Galerkin methods and ANN was developed for the approximation of the solution of hyperbolic and parabolic partial differential equations. ANN were also used to solve systems of partial differential equations Beidokhti , and high dimensional partial differential equations Sirignano ; Weinan .This paper is organized as follows. In Sect. 2, a semiimplicit Euler scheme in time and finite volume in space for the the initialboundary value problem (1.2)–(1.3) is presented. The convergence of the scheme is also discussed in this section on the basis of the analysis performed in Akhmouch2 . Sect. 3 introduces ESTLI and SSTLI methods to solve (1.2)–(1.3). The STP algorithm is also presented in this section. Numerical tests applied to the models presented in this section are performed in Sect. 4, in order to compare the efficiency and accuracy of the elaborate schemes with that of semiimplicit schemes. Finally, Sect. 5 presents a conclusion.
2 The semiimplicit finite volume scheme
2.1 Definitions and notations
We assume that is an open bounded polygonal subset. An admissible finite volume mesh of in the sens of Definition 9.1 in Eymard1 is given by:

A family of control volumes (disjoint open and convex polygons) denoted by .

A family of edges, where is the set of edges of the control volume .

A family of points such that . The particularity of an admissible mesh is that the straight line going through and must be orthogonal to the common edge of and denoted .
We denote by m, the Lebesgue measure in or . Denoting by d the Euclidean distance, and for all , we define by:
Let be the number of timesteps to reach the final time . We set , where is the timestep size: .
Let be the set of functions from to which are constant over each control volume of the mesh. Let , for , the classical discrete norm reads
where for all and for all . We define also the discrete seminorm and the discrete norm:
where for all , if and otherwise, with .
2.2 Presentation of the scheme
A semiimplicit finite volume scheme associated to the problem (1.2)–(1.3) is given by:
for all and ,
(2.3)  
(2.4) 
endowed with the discrete initial condition
(2.5) 
The function is defined by
(2.6) 
where is a small nonnegative constant (), and if and otherwise.
In the above scheme, and denote respectively the approximations of the mean value of and on at time . As we can see, the difference between this scheme and a fully implicit one is the discretization of the reaction term , which is evaluated at the previous timestep. This strategy allows us to build a decoupled linear scheme: at each timestep, we solve (2.4) to obtain and then, we solve (2.3) to compute . The discretization used for the chemotaxis term is identical to the first order upwind scheme when , and to the second order central difference scheme otherwise. It is equivalent to that of Spalding Spalding1 when .
2.3 Convergence of the scheme
The scheme (2.3)–(2.5) is similar to that of Akhmouch2 , where a corrected decoupled scheme is proposed for the problem (1.2)–(1.3). The only difference between the two schemes is a bounded term which was added to the second equation of the scheme:
we refer to Section 2.2 of Akhmouch2 for the definition of . The numerical analysis of the two schemes is quite similar, so based on the results obtained for the corrected decoupled scheme, we briefly discuss the convergence of (2.3)–(2.5).
The existence, uniqueness and nonnegativity of the finite volume solution of (2.3)–(2.5) can be proved by an Mmatrix analysis, following exactly the proof of Proposition 3.1 in Akhmouch2 .
Now, to prove the convergence of the scheme to a weak solution of (1.2)–(1.3
) in the sense of Definition 2.1, we first need a priori estimates on the discrete solutions. For the finite volume approximation of
, and discrete estimates can be obtained directly from the equation (2.3). We refer to Proposition 4.1 of Akhmouch1 , where the same equation is analyzed.For the cell density finite volume solution, only estimate is obtained for the corrected decoupled scheme without establishing the boundedness of the solution, since it is supposed that . Two approaches have been proposed to obtain this estimate. The first one, detailed in the proof of Lemma 4.1 in Akhmouch2 , needs that , and holds also for the semiimplicit scheme (2.3)–(2.5). The main tool of this approach is the discrete GagliardoNirenbergSobolev inequality Chatard2 , which requires the following constraint on the mesh: there exists such that The second approach (Proposition 4.3 in Akhmouch2 ) is based on the discrete Gronwall inequality, and requires a timestep condition which can be relaxed in the case of the current semiimplicit scheme.
In the following proposition, estimate on the approximate solution of the semiimplicit scheme is established if is a bounded function, which is the most relevant case for numerical purposes.
Proposition 2.0.
Proof.
Let be the control volume for which , which implies that for all . Using the identity , we can easily see that for all , we have
From (2.3), and using the above equality, we find that
where
From (2.6), we can easily see that if . In the case when , we have
In view of the choice of , we deduce that
Next, multiplying (2.4) by , we have
Collecting the previous inequalities we obtain
Using the facts that is nonnegative, and is a bounded nonnegative function, the desired result follows by induction on . ∎
The estimate for the cell density can be now obtained by following the proof of Proposition 4.2 in Akhmouch2 . The rest of the convergence analysis of the scheme (2.3)–(2.5), including the compactness of the sequence of approximate solutions and the pass to the limit in the scheme, can be performed exactly as in Section 5 in Akhmouch2 (see also Hillairet1 ; Filbet1 ).
3 Linearization via a singlelayer neural network
In this section, for all family of elements , we denote by
the vector whose components are the elements of the family.
3.1 The ESTLI method
The idea of the ESTLI method is to replace (2.4) in the semiimplicit scheme by the equation:
(3.1) 
where is an approximation of , obtained through the use of an ANN. The training data are obtained from the previous values of the finite volume solution, which implies that the numerical solution must be computed by an other method for the first timesteps.
It is easy to see that the ESTLI scheme consumes more time than the semiimplicit scheme (2.3)–(2.4). Indeed, at each time step, in addition to the time required for forming and solving linear systems, a computational time is needed to train the neural network and to compute for all
. As training singlelayer neural networks already requires much computational time, and keeping in mind that the scheme must be efficient (see the objectives listed in Sect. 1), hidden layers are avoided. The network proposed is a singlelayer feedforward neural network which consists of two inputs, weights, a bias and an output. The activation function will be the identity activation function.
The structures of the ANN used in the training phase and prediction phase are presented in Figs. 1 and 2 respectively. As we can see, at each timestep with , the network is trained using the two inputs and
, and the target output
. The semiimplicit scheme (2.3)–(2.4) will be used for the two first timesteps ( and ) to obtain the numerical solution. The values of the discrete solution at each control volume are used in the training process. After completing the training, the final weights and the bias are used to compute , so for all :(3.2) 
The training process adopted is a standard one: the weights and the bias are updated until the error reaches the desired threshold . Denoting by the vector of weights and the bias at the time step and the iteration (related to the updating process), the update rule has the following form:
(3.3) 
where the expression of is determined according to the method selected as learning algorithm: gradient descent method, conjugate gradient method, LevenbergMarquardt method, etc. We refer to Yu ; Tan for an overview of update rules related to several optimization methods. The error function adopted to evaluate the training process is the mean squared error function :
(3.4) 
where is the computed output at the time step and the iteration .
One of the advantages of using ANN to build implicit linear schemes is the compatibility between this approach and the implicit stepbystep resolution. Indeed, if for we need to generate a randomly initial vector of weights and the bias, we will set starting from the fourth timestep. Here, is the vector of weights and the bias at the timesteps . This is because it is expected that the behavior of the solution does not change much between two successive timesteps, and therefore will be very close to . This will reduce the number of iterations needed to obtain the final bias and weights, specially for small timesteps.
3.2 The SSTLI method
The SSTLI method is an efficient version of the ESTLI method in which training of the network does not occur at each timestep, but on selected timesteps. The SST algorithm is developed to this purpose. Two criteria must be satisfied to train the network at the time step :

The performance required is not reached for the timestep : It is widely expected that the approximation of obtained by the use of ANN is better than simply use of a semiimplicit linearization, which means that ( is defined by (3.2) and is computed by the scheme (2.3)–(3.1)). To measure the performance of the scheme, the approach proposed is to introduce a parameter , and depending on the choice of this parameter, the scheme (2.3)–(3.1) will be said to satisfy the required performance at the timestep if:
(3.5) In this case, the network is not trained for the following timestep, and we set in order to compute .

is a better approximation for than : Defining by, for all , , the condition must be satisfied to train the network at the timestep . Otherwise, will be used to compute and .
The detailed steps of the SST algorithm are presented in the following. The algorithm stops when the the number of timesteps to reach the final time is exceeded.

If , set and go to the step 6. Otherwise, go to the next step.

If , set and go to the step 9. Otherwise, set , train the neural network (Fig. 1) until the error and obtain , go to the next step.

Set . If go to the step 9. Otherwise go to the next step.
4 Numerical experiments
This section mainly deals with the comparison between the Euler semiimplicit method, the ESTLI method, and the SSTLI method. In all numerical tests presented in this section, target outputs are normalized to the range of ( by considering the positivity of the numerical solution). The LevenbergMarquard method is chosen as learning algorithm, and the threshold mean squared error required is . Finally, for the parameter defined in (3.5), we take .
4.1 KellerSegel model for embryonic pattern formation
In this test, we consider the system (1.2)–(1.3). The spatial domain is with a uniform mesh of control volumes, and the final time is . We adopt the following initial condition:
(4.1) 
where
is a positive perturbation which is constant on each control volume. It is equal on each cell of the mesh to the average of ten uniformly distributed random values in
. For the model parameters, we take: and .The numerical cell density obtained by the SSTLI method with is presented in Fig. 3. We can observe from this figure the ability of the studied model to generate stripe patterns. We can see also from the threedimensional plot on the right that the obtained numerical solution is positive and is oscillation free. In Fig. 4, we present threedimensional plots of the cell density using SSTLI and ESTLI methods with a very large timestep size . As we can see, both numerical solutions are free from negative values or any instabilities.
In order to compare the accuracy of the three approaches presented in this paper, and since the exact solution of the KellerSegel system (1.2)–(1.3) is unavailable, we use a reference solution computed by the corrected decoupled scheme Akhmouch2 on a very fine timestepping . This reference solution is used to compute the relative errors at final time (see Table 1) for several timestep sizes. Computational cost (CPU) of the semiimplicit scheme (2.3)–(2.5), and the increase in computational cost (denoted as ) of SSTLI and ESTLI methods over the semiimplicit scheme are also presented in Table 1. From this table, we can observe that SSTLI method is slightly more accurate than ESTLI method except for . Moreover, both methods are in general four to five times more accurate than Euler semiimplicit scheme. Concerning the computational time, the presented results show that ESTLI method is considerably more timeconsuming than semiimplicit scheme. On the contrary, the increase in computational cost over semiimplicit scheme is generally weak in the case of SSTLI method, specially for the three finest timesteps (less than ). Finally, in Fig. 5, it can be seen that the three numerical approaches are firstorder accurate.
4.2 KellerSegel model with quadratic growth
In this subsection, we are concerned with the system (1.4) endowed with zeroflux boundary conditions. Herein, the computational domain is with a uniform mesh grid ., , , , and . The following initial conditions are considered:
and . The perturbation is defined as in (4.1). The scheme adopted is:
(4.2)  
(4.3) 
when using the semiimplicit method, is replaced by , for ESTLI method is computed by an ANN similar to that of Figs. 1 and 2. In the case of SSTLI method, SST algorithm is used.
The numerical cell density computed via SSTLI method with is plotted in Fig. 6. The continuous rings observed in the figure are similar to the patterns formed by Salmonella typhimurium bacteria Woodward1 . Numerical solutions computed by the SSTLI and ESTLI methods with are shown in Fig. 7. We can see that even for a large timestep, the stability and the positivity of the solution are assured.
The reference solution used is computed using the corrected decoupled scheme with . For a sequence of decreasing time stepping, relative errors and computational costs of the studied numerical methods are presented in Table 2, convergence speeds are presented in Fig. 8. Table 2 shows that SSTLI and ESTLI methods are much more accurate than semiimplicit method. The obtained results do not allow us to conclude which of the two proposed methods is more accurate, but it can be observed from the table that SSTLI give better results for the two smallest timesteps. In terms of computational cost, the increase in computational time over semiimplicit scheme is very marginal for the three most timeconsuming tests when using SSTLI method (less than ), which is not the case for the ESTLI method.
error  ()  error  ()  error  CPU (s)  

SSTLI  ESTLI  semiimplicit  
4.3 KellerSegel model with volumefilling
Here, we focus on the model (1.5) with zeroflux boundary conditions. The computational domain and the initial chemoattractant concentration are taken as in the previous subsection, and we prescribe the following initial cell density
is defined as in (4.1). Besides, we set and . The following scheme is used:
where the value of is determined as in (4.2)–(4.3), and when .
Numerical solution computed by SSTLI method with is plotted in Fig. 9. We observe from the figure that the maximal density of cells does not exceed even if the solution concentrates at the center, which is consistent with our choice of the function .
To compare relative errors for the three numerical methods studied in this paper, the reference solution of this test is computed using the SSTLI method with . Table 3 shows that relative errors for SSTLI and ESTLI methods are very close with a slight advantage for SSTLI method in the case of small timesteps. The two methods are about three to four times more accurate than semiimplicit scheme, and the three numerical approaches are firstorder accurate (see Fig. 10). Table 3 also shows that, except for , the increase in computational cost over semiimplicit scheme is weak to insignificant for SSTLI method. In terms of error, we observe in Table 4 that SSTLI and ESTLI methods are four to six times more accurate than semiimplicit scheme.
error  ()  error  ()  error  CPU (s)  

SSTLI  ESTLI  semiimplicit  
error  error  error  

SSTLI  ESTLI  semiimplicit  
4.4 KellerSegel model with cubic growth
The purpose of this test is to investigate the ability of SSTLI and ESTLI methods to capture some bacterial patterns which can be reproduced by the model (1.4). More precisely, we focus on bacterial honeycomb pattern, reported for example in Thar , and symmetrical spot pattern generated for instance by Escherichia coli budrene2 .
In this subsection, , and . Initial conditions and computational domain are the same as in Subsection 4.2. The scheme adopted is similar to (4.2)–(4.3), but since the logistic source term is now cubic, in (4.2) will be replaced by . The timestep size used for computations is .
For , the computed solutions at are shown in Fig. 11. The honeycomb pattern is observed for both ESTLI and SSTLI numerical solutions. When , we see from Fig. 12 that the computed solutions exhibit a very spiky behavior at , which means that spot pattern appears (see Fig. 13). Despite this fact, the two methods remain positivity preserving which confirms their robustness and reliability.
5 Conclusion
In this paper, two linearized implicit methods relying on a singlelayer neural network are developed to solve twodimensional KellerSegel systems: ESTLI and SSTLI methods. In terms of accuracy, the numerical tests performed demonstrate the superiority of the developed methods on semiimplicit method. However, concerning computational cost, only SSTLI method is nearly as efficient as semiimplicit scheme for timeconsuming numerical tests. Moreover, the two numerical methods are robust and easily applicable to any problem which can be solved by the semiimplicit approach.
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