Research article

Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model

  • Received: 05 July 2021 Accepted: 22 September 2021 Published: 29 September 2021
  • We consider the mathematical model of chemotaxis introduced by Patlak, Keller, and Segel. Aggregation and progression waves are present everywhere in the population dynamics of chemotactic cells. Aggregation originates from the chemotaxis of mobile cells, where cells are attracted to migrate to higher concentrations of the chemical signal region produced by themselves. The neural net can be used to find the approximate solution of the PDE. We proved that the error, the difference between the actual value and the predicted value, is bound to a constant multiple of the loss we are learning. Also, the Neural Net approximation can be easily applied to the inverse problem. It was confirmed that even when the coefficient of the PDE equation was unknown, prediction with high accuracy was achieved.

    Citation: Sunwoo Hwang, Seongwon Lee, Hyung Ju Hwang. Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 8524-8534. doi: 10.3934/mbe.2021421

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  • We consider the mathematical model of chemotaxis introduced by Patlak, Keller, and Segel. Aggregation and progression waves are present everywhere in the population dynamics of chemotactic cells. Aggregation originates from the chemotaxis of mobile cells, where cells are attracted to migrate to higher concentrations of the chemical signal region produced by themselves. The neural net can be used to find the approximate solution of the PDE. We proved that the error, the difference between the actual value and the predicted value, is bound to a constant multiple of the loss we are learning. Also, the Neural Net approximation can be easily applied to the inverse problem. It was confirmed that even when the coefficient of the PDE equation was unknown, prediction with high accuracy was achieved.



    Cancer is a group of diseases involving abnormal cell growth [1,2]. Currently, despite great progresses of many newly developed therapeutic methods [3,4,5], chemotherapy is still a common treatment method for many cancers. Patients are often administrated with high-dose of cytotoxic chemotherapeutics trying to eliminate tumor cells as much as possible [6,7,8,9,10,11]. Nevertheless, it is difficult to determine the proper dosage of chemotherapy, low dosage is ineffective in killing tumor cells, whereas excessive dosage may result in additional toxicity that is intolerable to patients [6]. Clinically, patients are often dosed at maximum or near maximum tolerated dose, which is carefully determined in phase Ⅰ studies [7,12]. However, high level doses often induce series side-effects, increasing the chemotherapy dose (also the treatment cost) would not yield the decreasing of the recurrence rate [13,14]. The recurrent tumors often show drug resistance, which is a major cause of treatment failure in chemotherapeutic drugs [15,16,17].

    Drug resistance has been a major challenge in cancer therapy. The mechanisms of drug resistance are complex, and many reasons are involved, including cellular plasticity [18], heterogenous tumor cells [19], or therapy induced gene mutations [20,21]. In this study, we focus on a mechanism of drug resistance due to chemotherapy-induced genome instability. Chemotherapy agents are cytotoxic by means of interfering with cell division in a way of damaging or stressing cells, and leading to cell death through apoptosis. During the early stage of apoptosis, apoptotic chromosome fragmentation (C-Frag) are produced through the cleavage by caspase-3 activated DNase (CAD) [22]. Nevertheless, C-Frag does not always result in cell death, sometimes the chromosome fragments can randomly rejoin to form genome chaos so that the cells survive from crisis [23,24]. These survived cells carry non-clonal chromosome aberrations (NCCAs), the major form of genome variation and the key index of genome instability in cancer cells [25,26,27,28]. It was proposed that such genome instability induced by chemotherapy is a source of drug resistance in cancer therapy [26,27]; quantitative control of drug dosages is important for the long-term clinical effects.

    Mathematical modelling approaches have been widely used in cancer research from different aspects [2,29]. A variety of models have been established to study the mechanisms of drug resistance after chemotherapy [30,31,32,33,34]. However, there are rare quantitative studies on how tumor cells population change in response to chemotherapy and drug resistance due to therapy-induced chromosome recombination. The roles of NCCAs in tumor recurrence is still controversial [35].

    In this study, we intend to investigate how NCCAs may affect tumor growth, and present a mathematical model for chromosome recombination-induced drug resistance in cancer therapy. The model extends the previously well studied G0 cell cycle model [36,37,38,39], and includes cell survival from C-Frag [23,24]. We mainly study cell population responses to various doses of chemotherapy, and show that there is an optimal dose (within the maximum tolerated dose) so that the steady state tumor cell number is relative low after chemotherapy. Moreover, the model implies that persistent extreme high dose therapy may induce oscillations in cell number, which is clinically inappropriate and should be avoided.

    In this study, we model the process of tumor growth through formulations of stem cell regeneration. Here, we mainly consider chemotherapy for leukemia, and mathematical models of hematopoiesis are referred in our modeling. We refer the classical G0 cell cycle model that has long been studied in literatures [40,41,42] (Figure. 1(a)). In the model, leukemia stem cells are classified as either resting or proliferative phase cells. Resting phase cells (Q, cells/kg) either enter the proliferative phase at a rate β (day1), or be removed from the pool of resting phase due to differentiation, senescence, or death, at a rate κ (day1). The cells at the proliferative phase undergo apoptosis in a rate μ (day1), and the duration of the proliferative phase is τ (days), each survived cell divides into two daughter cells through mitosis at the end of the proliferative phase. These processes can be described by a delay differential equation model [36,37,38,39]:

    dQdt=(β(Q)+κ)Q+2eμτβ(Qτ)Qτ. (2.1)
    Figure 1.  The G0 cell cycle model of leukemia stem cell regeneration. (a) Leukemia stem cell regeneration without chemotherapy. All stem cells are classified into the resting and the proliferative phase. During stem cell regeneration, resting phase cells either enter the proliferating phase with a rate β, or be removed from the resting pool with a rate κ due to differentiation, senescence, or death. The proliferating cells undergo apoptosis with a rate μ. At the end of the resting phase, each cell divides into two daughter cells through mitosis. (b) Leukemia stem cell regeneration under chemotherapy. Chemotherapy agents interfere the process of cell division and promote apoptosis [49]. During apoptosis, after the early stage of chromosome fragmentation, some cells survive from crisis (with a probability q(μ1)) through chromosome recombination, and the survived cells re-enter the G0 phase. Here the extra apoptosis rate μ1 is a parameter associated with the chemotherapy dose.

    Here, the subscript means the time delay, i.e., Qτ=Q(tτ). The equation (2.1) has been widely applied in the study of hematopoietic stem cell regeneration dynamics and blood disease [43,44,45,46], as well as the hematopoietic responses to chemotherapy [47,48].

    The proliferation rate β is a function of the resting phase cell number Q, indicating the regulation of cell proliferation through cytokines secreted from all stem cells. Normally, the proliferation rate is a decrease function of the cell number, and approaches 0 when the cell number Q is large enough [50]. Nevertheless, in the situation of cancer, the function can be non-monotonic because cancer cells can evade growth suppressors, and produce self-sustaining proliferative signaling [51].

    Now, we consider the effects of chemotherapy, which often promote cell death during the proliferative phrase due to the toxicity [49]. Hence, we write the apoptosis rate as μ=μ0+μ1, where μ0 represents the baseline apoptosis rate in the absence of chemotherapy, and μ1 the extra apoptosis rate due to treatment stress (the rate μ1 is often increase with the chemotherapy dose, hence we also refer μ1 as the dose for short). At the early apoptosis stage, the cells undergo chromosome fragmentation (C-Frag) (cell number given by (1eμτ)β(Qτ)Qτ). However, in a small population of cells with C-Frag, the fragments can rejoin to yield chromosome recombination and the cells survive from crisis and re-enter the G0 phase; other cells continue the apoptosis process (Figure. 1(b)). We assume that the probability of chromosome recombination, q(μ1) (0q(μ1)<1), is dependent on the chemotherapy dose, so that q(μ1) is an increase function. Therefore, we modify the above G0 cell cycle model (2.1) to include chromosome recombination, the cell number Q satisfies the following delay differential equation:

    dQdt=(β(Q)+κ)Q+(2eμτ+(1eμτ)q(μ1))β(Qτ)Qτ, (2.2)

    where

    μ=μ0+μ1. (2.3)

    Hereafter, we always assume that β(Q) is a decrease function, and q(μ1) is an increase function.

    For model simulation, we refer the classical models for hematopoietic stem cells (Table 1), and take the proliferation rate β(Q) and the probability of chromosome recombination q(μ1) as Hill type functions [44,50]:

    β(Q)=β0θnθn+Qn+β1,q(μ1)=q1μm1em+μm1. (2.4)
    Table 1.  Default parameter values. The parameter values for hematopoietic stem cells are referred to [44,50,52], and β1 is set to 0 for default, other parameters for the chemotherapy effect are taken arbitrary.
    Parameter Value Unit Source
    Q 1.53 ×106 cells/kg [44,52]
    β0 8.0 day1 [44,50]
    β1 0 day1 -
    θ 0.096 ×106 cells/kg [44,50]
    n 2 - [44]
    q1 1 day1 -
    e 0.34 day1 -
    m 4 - -
    κ 0.02 day1 [44,50]
    τ 2.8 days [44,52]
    μ0 0.001 day1 [44,50]

     | Show Table
    DownLoad: CSV

    From (2.2), the steady state Q(t)Q is given by the equation

    (β(Q)+κ)Q+(2eμτ+(1eμτ)q(μ1))β(Q)Q=0.

    Obviously, there is a zero solution Q=0. When the proliferation rate β(Q) is a decrease function, there is a positive steady state Q>0 if and only if the condition

    β0>κ2eμτ1+(1eμτ)q(μ1)β1>0 (3.1)

    is satisfied, and the steady state is given by the root of

    β(Q)=κ2eμτ1+(1eμτ)q(μ1). (3.2)

    From (3.1), there exists a positive steady state when the recombination rate q(μ1) satisfies

    κ(β0+β1)(1eμτ)<q(μ1)+2eμτ11eμτ<κβ1(1eμτ). (3.3)

    Specifically, if

    q(μ1)+2eμτ11eμτ<κ(β0+β1)(1eμτ), (3.4)

    the system has only zero solution steady state, and the zero solution is global stable which means the situation of no cells; and if

    q(μ1)+2eμτ11eμτ>κβ1(1eμτ), (3.5)

    the zero solution is unstable and all positive solutions approach to +, which means the situation of uncontrolled cell growth.

    Next, we consider the stability of the steady states. Let x(t)=Q(t)Q and linearize the equation (2.2) at x=0, we obtain

    dxdt=ax+bxτ, (3.6)

    where

    a=β(Q)+κ+β(Q)Q,b=[2eμτ+(1eμτ)q(μ1)][β(Q)+β(Q)Q].

    The zero solution of equation (3.6) is stable if and only if the coefficients a and b take values from the region S defined as

    S={(a,b)R2|asecωτ<b<a,where ω=atanωτ,a>1τ,ω(0,πτ)}. (3.7)

    For the zero solution Q(t)Q=0, we have

    a=β0+κ>0, b=(2eμτ+(1eμτ)q(μ1))β0>0.

    Hence, the zero solution is stable if and only if b<a, i.e., the inequality (3.4) is satisfied.

    For the positive steady state Q(t)=Q>0, we have

    a=ˉβ+κˆβ, b=(2eμτ+(1eμτ)q(μ1))(ˉβˆβ), (3.8)

    where ˉβ=β(Q)>0 and ˆβ=β(Q)Q>0. Hence, when the condition (3.1) is satisfied, the equation (3.7) gives the Hopf bifurcation curve for the positive steady state

    {ˆβcrit=[2eμτ+(1eμτ)q(μ1)](secωτ1)2eμτsecωτ+(1eμτ)q(μ1)ˉβ,ω=ˉβ[2eμτ+(1eμτ)q(μ1)][2eμτ1+(1eμτ)q(μ1)]2eμτsecωτ+(1eμτ)q(μ1)tanωτ, (3.9)

    where ω can be solved from the second equation of (3.9), and

    ω[1τarccos12eμτ+(1eμτ)q(μ1),πτ]. (3.10)

    When ˆβ<ˆβcrit, the positive steady state solution of equation (2.2) is stable, and when ˆβ>ˆβcrit, the steady state becomes unstable. Biologically, the Hopf bifurcation curve (3.9) gives the critical proliferation rate ˆβcrit, so that the steady state is stable when the proliferation rate is less than the critical rate, but when the proliferation rate is larger than the critical rate, the steady state becomes unstable and the system exhibits oscillatory dynamics, which can be a source of dynamical blood diseases [42].

    In summary, we have the following conclusion:

    Theorem 1. Consider the model equation (2.2), where β(Q) (β0>β(Q)>β1) is a decrease function, and 0q(μ1)<1. The equation always has zero steady state Q(t)0; if and only if the condition

    β0>κ2eμτ1+(1eμτ)q(μ1))β1>0 (3.11)

    is satisfied, equation (2.2) has a unique positive steady state solution Q(t)=Q, which is given by

    β(Q)=κ2eμτ1+(1eμτ)q(μ1)). (3.12)

    Moreover,

    (1) the zero steady state is stable if and only if

    q(μ1)+2eμτ11eμτ<κ(β0+β1)(1eμτ); (3.13)

    (2) when (3.11) is satisfied and let ˉβ=β(Q), ˆβ=β(Q)Q, for any μ1>0, there is a critical proliferation rate ˆβcrit>0, defined by (3.9), so that the positive steady state is stable if and only if ˆβ<ˆβcrit.

    When the functions β(Q) and q(μ1) are defined by (2.4), equation (3.2) gives

    ˉβ=κf,f=2eμτ1+(1eμτ)q(μ1), (3.14)

    and

    ˆβ=β(Q)Q=n(ˉββ1)(1ˉββ1β0). (3.15)

    Applying the default parameter values in Table 1, when μ1 varies from 0 to 1.2, the critical proliferation rate ˆβcrit (black line, given by (3.9)) and proliferation rate at the steady state ˆβ (red line, given by (3.15)) are shown in Figure 2(a). Here, ˆβ>ˆβcrit implies the parameter region with unstable steady state, and there are oscillatory solutions due to Hopf bifurcation (Figure 2(b)).

    Figure 2.  Bifurcation analysis of the model system. (a) Dependence of the bifurcation curve ˆβcrit (black line) in equation (3.9) and ˆβ (red line) in equation (3.15) on the parameter μ1. (b) Sample dynamics of cell population. Different color lines are obtained from different μ1 values (same color dots in (a)): μ1=0.1(blue line), μ1=0.3(green line), μ1=0.5(magenta line), and μ1=1.1(black line). Other values are the same as default values in Table 1. In simulations, we first set the parameters as their default values, Q(t)0.5 for t<0, and solve the equation to t=300 day.

    From Figure 2(a), when the drug dose is low, we have ˆβ<ˆβcrit, and (2.2) has a stable positive steady state (Figure 2(b), blue line). The steady state cell number depends on the dose μ1, and reach a local minimum when μ1 takes intermediate values (green dot in Figure 2(a), also referred to Figure 3). When the drug dose further increase to a high level (black dot), the steady state of (2.2) becomes unstable, and the cell populations show oscillatory dynamics (Figure 2(b), black line).

    Figure 3.  Dependence of the steady state Q with chemotherapy dose μ1. Black lines show the steady state, with solid line for the stable steady state, and dashed line for the unstable steady state. Red lines show the upper and lower bounds of the oscillation solutions when the steady state is unstable.

    From the above analyses, when

    f=2eμτ1+(1eμτ)q(μ1)>0, β1<κf<β0+β1, (3.16)

    the equation (2.2) has a unique positive steady state Q(t)=Q. Moreover, let ˉβ=β(Q), then ˉβ=κ/f, and

    0<ˉββ1<β0.

    Hence, the steady state can be expressed explicitly as

    Q=θnβ0ˉββ11. (3.17)

    Equation (3.17) shows that the steady state Q is dependent on the chemotherapy dose parameter μ1 through ˉβ=κ/f. The following theorem shows that under certain conditions, there is an optimal dose (within a tolerated dose) so that the steady state cell number reach a local minimum.

    Theorem 3.2. Consider the equation (2.2), if the following conditions are satisfied

    (1) the functions β(Q) and q(μ1) are continuous, and satisfy

    β0+β1>β(Q)>β10, 1q(μ1)0, β(Q)<0, q(μ1)>0,

    (2) the parameters (β0,β1,κ,μ0, and τ) satisfy the condition (3.16) (here we note μ=μ0+μ1),

    (3) both q(0) and q(0) are sufficiently small,

    (4) there exists μ1>0 so that conditions (1)-(2) are satisfied when μ1(0,μ1), and

    q(μ1)+q(μ1)τ1(e(μ0+μ1)τ1)>2,

    the steady state Q reaches a local minimum value at an optimal dose μ1=ˆμ1(0,μ1).

    Proof. When the conditions (1) and (2) are satisfied, equation (2.2) has a unique positive steady state Q(t)Q, which is given by (3.17). Hence, we have

    dQdμ1=θn(β0ˉββ11)1n1β0(ˉββ1)2dˉβdμ1=θn(β0ˉββ11)1n1β0(ˉββ1)2κf2dfdμ1=θn(β0ˉββ11)1n1β0κ(ˉββ1)2f2dfdμ1,

    and

    dfdμ1=τeμτ(q(μ1)+q(μ1)τ1(eμτ1)2). (3.18)

    Hence, the sign of dQdμ1 is determined by the sign of dfdμ1.

    The condition (3) implies dfdμ1|μ1=0<0, and condition (4) implies dfdμ1|μ1=μ1>0. Hence, there exists an optimal value ˆμ1(0,μ1), so that dQdμ1|μ1=ˆμ1=0, thus Q reaches a local minimum value at μ1=ˆμ1.

    Figure 3 shows the dependence of steady state with the dose μ1 when we take parameters from Table 1, which exhibits a local minimum at about ˆμ1=0.3. Here we note that the steady state cell number Q decrease with further increasing of μ1>0.5 and approaches zero when μ1 is large enough. However, clinically, higher level μ1 may exceed the maximum tolerated dose, and have a risk to induce oscillatory dynamics (Figure 2, and the discussion below). Hence, there is an optimal dose within a certain region (for example, μ1<0.5) so that the cell number reaches a relative low level.

    Above analyses show that chemotherapy may induce oscillatory dynamics through Hopf bifurcation when μ1 takes proper values. Here, we further analyze the condition for Hopf bifurcation, and identify how clinical conditions may affect the chemotherapy-induced oscillations [46]. Hematopoiesis can exhibit oscillations in one or several circulating cell types and show symptoms of periodic hematological diseases [43,53]. Here, we show the conditions to induce oscillations, and try to identify the criteria to avoid therapy-induced oscillations in cancer treatments.

    Using the previously introduced notations ˉβ and f in (3.14), the bifurcation curve (3.9) can be rewritten as

    {ˆβcrit=(f+1)(secωτ1)f+1secωτˉβ,ω=ˉβf(f+1)tanωτf+1secωτ, (3.19)

    and hence

    ˆβcrit=ω1cosωτfsinωτ. (3.20)

    Moreover, from equation (3.15), we have

    ˉβ=κf,ˆβ=β(Q)Q=n(κfβ1)(1κ/fβ1β0). (3.21)

    Hence, from (3.19)–(3.21), and the bifurcation condition ˆβcrit=ˆβ, we obtain the bifurcation curve in terms of f and κ (defined by a parameter ω>0) as

    {ω1cosωτfsinωτ=n(κfβ1)(1κ/fβ1β0),ω=κ(f+1)tanωτf+1secωτ. (3.22)

    Thus, given the parameter β0,β1,n,τ, equation (3.22) defines a bifurcation curve in the κ-f plane when ω varies over the region in (3.10).

    Figure 4 shows the bifurcation curves obtained from (3.22). For any κ>0, there exist an upper bound ¯f and a lower bound f_, so that the positive steady state (if exists) is stable when f_<f<¯f. Moreover, when the self-sustained proliferation rate β1 increases, the lower bound f_ is independent of β1, while the upper bound ¯f decreases with β1. Here, we note that f depends on the dose parameter μ1 through (3.14), these results provide a strategy to quantitatively control the chemotherapy dose to avoid therapy-induced oscillation, i.e., try to take values from the gray region for different differentiation rate κ.

    Figure 4.  Hopf bifurcation curve in the κ-f plane. The black solid lines are obtained from (3.22) with β0=0, the blue dashed line is obtained from (3.22) with β1=0.05. Gray shadow shows the parameter region of stable steady state.

    The era of chemotherapy has been ruled by the routine use of dose-intense protocols based on the "maximum-tolerate dose" concept. This protocol plays a prominent role in veterinary oncology, however there are many debates on using the high dose chemotherapy because of the side effects and recurrence rates [54]. Chromosome recombination induced by chemotherapy is a source to produce chaotic genome in cancer cells and may lead to drug resistance. Here, we present a mathematical model to consider cell population dynamics in response to chemotherapy with occasional occurrence of chromosome recombination. Model analyses show that maximum-tolerate dose may not always result in the best outcome when the probability of chromosome recombination is dependent on the dose; there is an optimal dose within a tolerated dose so that the steady state tumor cell number reaches a relative low level. Moreover, sustained administration of high dose chemotherapy may induced oscillatory cell population dynamics through Hopf bifurcation. Clinically, the long period oscillation can be considered as the frequent recurrence of tumor cells, which is a result of drug-resistant or chemotherapy-induced dynamical disease. For instance, the periodic chronic myelogenous leukemia show obvious oscillations in circulating blood cells [55]. We identify the parameter regions for the occurrence of oscillation dynamics, which is valuable when we try to avoid the frequent recurrence. To our knowledge, this is the first try to quantitatively study the tumor cell population dynamics in response to chemotherapy when chromosome recombination is induced. Genome chaotic has been widely reported in studies of genomic structure of cancer cells, however how genome chaotic play roles in cancer development and drug resistance is not well documented. The current study is the first try to consider this issue. Here, the proposed model only consider the main effect of cell survival from C-Frag through chromosome recombination. Nevertheless, many other effects should be included for a complete understanding of drug resistance due to the chromosome changes induced by cancer therapy. Moreover, the cell heterogeneity can also play important roles in drug resistance. These effects should be extended to the current model in order to obtain an optimal dose of chemotherapy.

    This work was supported by grant from National Natural Science Foundation of China (91730301, 11762011).

    No potential conflicts of interest were disclosed.



    [1] P. Devreotes, C. Janetopoulos, Eukaryotic chemotaxis: Distinctions between directional sensing and polarization, J. Biol. Chem., 278 (2003), 20445–20448. doi: 10.1074/jbc.R300010200
    [2] D. V. Zhelev, A. M. Alteraifi, D. Chodniewicz, Controlled pseudopod extension of human neutrophils stimulated with different chemoattractants, Biophys. J., 87 (2004), 688–695. doi: 10.1529/biophysj.103.036699
    [3] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415. doi: 10.1016/0022-5193(70)90092-5
    [4] L. A. Segel, A. Goldbeter, P. N. Devreotes, B. E. Knox, A mechanism for exact sensory adaptation based on receptor modification, J. Theor. Biol., 120 (1986), 151–179. doi: 10.1016/S0022-5193(86)80171-0
    [5] J. A. Sherratt, Chemotaxis and chemokinesis in eukaryotic cells: The Keller-Segel equations as an approximation to a detailed model, Bull. Math. Biol., 56 (1994), 129–146. doi: 10.1007/BF02458292
    [6] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311–338. doi: 10.1007/BF02476407
    [7] E. F. Keller, L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225–234. doi: 10.1016/0022-5193(71)90050-6
    [8] E. F. Keller, L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. doi: 10.1016/0022-5193(71)90051-8
    [9] H. Jo, H. Son, H. J. Hwang, E. H. Kim, Deep neural network approach to forward-inverse problems, Networks Heterog. Media, 15 (2020), 247. doi: 10.3934/nhm.2020011
    [10] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations, arXiv: 1711.10561.
    [11] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707. doi: 10.1016/j.jcp.2018.10.045
    [12] X. Li, Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer, Neurocomputing, 12 (1996), 327–343. doi: 10.1016/0925-2312(95)00070-4
    [13] J. Soler, J. A. Carrillo, L. L. Bonilla, Asymptotic behavior of an initial-boundary value problem for the Vlasov–Poisson–Fokker–Planck system, SIAM J. Appl. Math., 57 (1997), 1343–1372. doi: 10.1137/S0036139995291544
    [14] F. Bouchut, J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials, Differ. Integral Equat., 8 (1995), 487–514.
    [15] J. Han, A. Jentzen, E. Weinan, Solving high-dimensional partial differential equations using deep learning, Proceed. Nat. Aca. Sci., 115 (2018), 8505–8510. doi: 10.1073/pnas.1718942115
    [16] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric & Functional Analysis GAFA, 3 (1993), 209–262.
    [17] H. J. Hwang, J. W. Jang, H. Jo, J. Y. Lee, Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach, J. Comput. Phys., 419 (2020), 109665. doi: 10.1016/j.jcp.2020.109665
    [18] Clawpack Development Team, Clawpack software, http://www.clawpack.org, Version 5.5.0, (2018).
    [19] K. T. Mandli, A. J. Ahmadia, M. Berger, D. Calhoun, D. L. George, Y. Hadjimichael, et al., Clawpack: Building an open source ecosystem for solving hyperbolic PDEs, PeerJ Comput. Sci., 2 (2016), e68. doi: 10.7717/peerj-cs.68
    [20] R. Tyson, L. G. Stern, R. J. LeVeque, Fractional step methods applied to a chemotaxis model, J. Math. Biol., 41 (2000), 455–475. doi: 10.1007/s002850000038
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