Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations
 49 Downloads
Abstract
Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agentbased model to describe the microscopic dynamics of each individual in a flock, and use a fractional partial differential equation (fPDE) to model the evolution of macroscopic quantities of interest. The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model. Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics, we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agentbased simulations. We demonstrate how the learning framework is used to connect the discrete agentbased model to the continuum fPDEs in one and twodimensional nonlocal flocking dynamics. In particular, a Cucker–Smale particle model is employed to describe the microscale dynamics of each individual, while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities. The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization. We show in one and twodimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agentbased system solved by the particle method. The proposed method offers new insights into how to scale the discrete agentbased models to the continuumbased PDE models, and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.
Keywords
Fractional PDEs Gaussian process Bayesian optimization Fractional Laplacian Conservation lawsMathematics Subject Classifcation
35R11 34K29 92D50 70E551 Introduction
Collective behavior is a widespread phenomenon in physical and biological systems, and also in social dynamics, such as collective motion of selfpropelled particles [20], bird flocking [1], fish schooling [22], swarms of insects [31], trails of foraging ants [4], herds of mammals [23], complex networks [19]. Although this collective dynamics is at very different scales and levels of complexity, the mechanism of selforganization, where local interactions for the individuals lead to a coherent group motion, is very general and transcends the detailed objects [12]. To this end, the simulation and modeling of both physical [16] and biological [5] systems have driven a rich field of research to explore how the individual behavior engenders the largescale collective motion. There are generally two different approaches to investigate the underlying mechanics: (i) at the microscopic level, agentbased models are developed to simulate dynamics of each individual in flocks, such as swarms, tori, and polarized groups; (ii) at the macroscopic level, the mathematical modeling approach is based on continuum models described by partial differential equations (PDEs). Agentbased models assume behavioral rules at the level of the individual, whose microscopic dynamics is governed by an evolution equation affected by the social forces, including alignment of velocities, attraction, and shortrange repulsion, acting on it [12]. Because the number of agents in a coupled dynamics is often large, the agentbased models cannot be solved exactly but can be easily implemented using numerical simulations. Agentbased simulation captures the detailed dynamics of each individual and can handle the increasing complexity of realworld flocking systems. However, for a flocking dynamics involving a large number of agents, the agentbased model becomes computationally expensive [14]. Alternatively, by assuming that a flocking group is already formed and considering a large enough number of agents to make the mean field approximation meaningful, Eulerian models can be derived by applying the continuum hypothesis to the microscopic dynamics, leading to PDEs for the macroscopic quantities, i.e., the mean velocity and the population density [12].
Advances in digital imaging [3] and highresolution lightweight GPS devices [21] allow gathering longtime and longdistance trajectories of individuals in flocks. For example, Ballerini et al. [3] used stereometric and computer vision techniques to measure 3D individual birds positions in huge starling flocks (up to 2 600 European starlings) in the field. Nagy et al. [21] employed highresolution lightweight GPS devices to collect track logs of homing pigeons flying in flocks of up to 13 individuals and analyzed the hierarchical group dynamics in the pigeon flocks. Lukeman et al. [18] recorded time series of 2D swimming flocks (up to 200 ducks) by oblique overhead photography and analyzed each individual’s position, velocity and trajectory. Tunstrom et al. [32] used automated tracking software for fish to obtain detailed data regarding the individual positions and velocities of schooling fish (up to 300 fish) over long periods of time. Despite significant development of these experimental ideas, a gap between flocking theory/modeling and experiment still exists. Due to the rich diversity of theoretical models with distinct forms of interaction [18], it is necessary to assess which of them correspond to actual behavior in specified collective motions in nature. Numerical simulations alone cannot tackle this problem because flocking patterns similar to experimental observations can be generated by significantly different model mechanisms. Although some mechanisms through which the collective motion is achieved have been qualitatively understood by simulation and modeling, a quantitative test of the model assumptions in realistic data is still challenging. Therefore, it is of great interest to establish a direct connection between the individual trajectories obtained in experiments and the effective governing equations in the mathematical approach.
For the agentbased system, we adopt the Cucker–Smale model with a nonlocal alignment term developed in [9]. Using a singular kernel makes the nonlocal alignment term to be a nonlinear function involving fractional Laplacian for the Euler equations which has mass and momentum conservations. The choice of the fractional order of the fractional Laplacian offers a flexibility in modeling the complicated interactions between agents. It is worth noting that the computation of the corresponding Euler equations with nonlocal forces is very challenging. The challenges are twofold due to the fact that the alignment term is not only nonlocal but also nonlinear. A typical challenge for nonlocal problems is the high computation cost of matrix vector multiplication, which is usually \(O(K^2)\) with K being the total number of degrees of freedom at each time step when using an explicit time scheme. We resolve this issue using the fast Fourier transform, resulting in a computation cost reduced to \(O(K\log K)\). For the issue of nonlinearity, we use a piecewise constant approximation, i.e., a finite volume scheme, which can significantly simplify the computation of the nonlinear nonlocal term. Moreover, we show that the proposed finite volume scheme preserves the mass and momentum, both in one and two space dimensions.
The remainder of this paper is organized as follows. In Sect. 2, we introduce a typical agentbased model and the corresponding Euler equations for flocking dynamics. In Sect. 3, we describe the details of numerical algorithms to solve the agentbased model and to discretize the fPDEs, including the statistical algorithm for sampling macroscopic quantities from discrete particle data, the velocity Verlet algorithm for time integration, and the finite volume approximation for the Euler system of equations. Subsequently, in Sect. 4, we present the numerical examples of one and twodimensional nonlocal flocking dynamics, and compare the results obtained by the agentbased simulations and by solving the fPDEs. In Sect. 5, we introduce the learning framework to infer the effective influence function from particle trajectories using a Gaussian process regression model implemented with the Bayesian optimization. Finally, we conclude with a brief summary and discussion in Sect. 6.
2 Mathematical Models
 i.Agentbased model The agent’s behavior is characterized by its position and velocity. We consider the following agentbased model, i.e., the C–S model with an alignment term [9], at the microscale:where \(\phi (\cdot )\) is a kernel denoting the influence function.$$\begin{aligned}\left \{ \begin{aligned}&\dot{\varvec{x}}_i = \varvec{v}_i, \\&\dot{\varvec{v}}_i = \frac{1}{N}\sum _{j=1}^N \phi (\varvec{x}_i \varvec{x}_j) (\varvec{v}_j  \varvec{v}_i), \quad (\varvec{x}_i,\varvec{v}_i) \in {\mathbb {R}}^n \times {\mathbb {R}}^n, \end{aligned} \right.\end{aligned}$$(1)
 ii.Euler equations For large crowds, i.e., \(N\gg 1\), using the mean field limit argument, the particle system governed by Eq. (1) can lead to the following macroscale Euler system of equations [11, 30]:where \(\rho \) is the density and \(\varvec{u}\) is the velocity of the macromodel, \([{\mathcal {L}},\varvec{u}](\rho ): = {\mathcal {L}}(\rho \varvec{u})  {\mathcal {L}}(\rho ) \varvec{u}\) is the commutator forcing and$$\begin{aligned}\left \{ \begin{aligned}&\rho _t + \nabla \cdot (\rho \varvec{u}) = 0,\\&\varvec{u}_t + \varvec{u}\cdot \nabla \varvec{u} = [{\mathcal {L}},\varvec{u}](\rho ), \; (\varvec{x},t ) \in \Omega \times {\mathbb {R}}^+, \end{aligned} \right.\end{aligned}$$(2)In this paper, we consider the singular kernel, i.e., \(\phi (r) := c_{n,\alpha }r^{(n+\alpha )}\), where \(c_{n,\alpha } = \frac{\alpha \Gamma (\frac{n+\alpha }{2})}{2\pi ^{\alpha +n/2}\Gamma (1\alpha /2)}\), which is associated with the action of the fractional Laplacian\({\mathcal {L}}(f) = (\Delta )^{\alpha /2}f,\, 0<\alpha <2\), namely$$\begin{aligned} {\mathcal {L}}(f)(\varvec{x}) := \int _{{\mathbb {R}}^n} \phi (\varvec{x} \varvec{y}) \left( f(\varvec{y})  f(\varvec{x}) \right) \mathrm{d}\varvec{y}. \end{aligned}$$where \(\text {p.v.}\) means the principle value. The corresponding forcing is given by the following singular integral:$$\begin{aligned} {\mathcal {L}}(f)(\varvec{x}) = \text {p.v.}\; c_{n,\alpha }\int _{{\mathbb {R}}^n} \frac{f(\varvec{y})  f(\varvec{x})}{\varvec{x}\varvec{y}^{n+\alpha }}\mathrm{d}\varvec{y}, \end{aligned}$$In the onedimensional case, the global regularity for \(\alpha \in [1,2]\) is proved by Shvydkoy and Tadmor [28]. Also, they proved fast velocity alignment as the velocity \(u(\cdot ,t)\) approaches a constant state, \(u\rightarrow {\tilde{u}}\), with exponentially decaying slope [29]. The Euler system of equations given by Eq. (2) preserves mass and momentum. In particular, the system (2) can be rewritten as$$\begin{aligned}{}[{\mathcal {L}},\varvec{u}](\rho ) = \text {p.v.}\; c_{n,\alpha }\int _{{\mathbb {R}}^n} \frac{u(\varvec{y})  u(\varvec{x})}{\varvec{x}\varvec{y}^{n+\alpha }}\rho (\varvec{y})\mathrm{d}\varvec{y}. \end{aligned}$$(3)Obviously, the conservation of mass can be obtained by integrating the first equation of the above system over \({\mathbb {R}}^n\). In addition, because the nonlocal operator \({\mathcal {L}}\) is assumed selfadjoint, i.e.,$$\begin{aligned} \left \{ \begin{aligned} &\rho _t + \nabla \cdot (\rho \varvec{u})= 0,\\ &(\rho \varvec{u})_t + \nabla \cdot (\rho \varvec{u}\otimes \varvec{u})= \rho {\mathcal {L}}(\rho \varvec{u})  \rho \varvec{u} {\mathcal {L}}(\rho ). \end{aligned}\right. \end{aligned}$$(4)we have the conservation of momentum by integrating the second equation of the above system over \({\mathbb {R}}^n\).$$\begin{aligned} \int _{{\mathbb {R}}^n}\left( \rho {\mathcal {L}}(\rho \varvec{u})  \rho \varvec{u} {\mathcal {L}}(\rho ) \right) \text{d}{\varvec{x}}= 0, \end{aligned}$$
3 Numerical Methods
3.1 Microscale: AgentBased Model
 Step 1. We first sample particles in the domain according to the initial density, i.e., give the initial position of each particle. To this end, we divide the domain \(\Lambda \) into P nonoverlapping subdomains \(\Lambda _p : = [x_{p1},x_{p}]\) with \(a = x_0< x_1<\cdots <x_{P} = b\) and compute the density in each subdomain \(\Lambda _p\):$$\begin{aligned} \rho _p = \int _{x_{p1}}^{x_{p}} \rho _0(x) \mathrm{d}x, \quad p = 1,2,\cdots , P. \end{aligned}$$

Step 2. Then we have that the number of particles sampled in each subdomain is \(N_p = N\cdot \rho _p, p =1,2,\cdots ,P\), where \(N\gg 1\) is the total number of sampled particles. For each subdomain \(\Lambda _p\), we divide it into \(N_p\) uniform subintervals \([x_{p,q}, x_{p,q+1}]\) with \(x_{p1} = x_{p,0}< x_{p,1}<\cdots <x_{p,N_p} = x_{p}\) and take the middle point of each subinterval, i.e., \((x_{p,q}+ x_{p,q+1})/2\), as the position of the particle.

Step 3. We now compute the initial velocity of each particle. Since we have already assigned an initial position for each particle, its initial velocity can be computed directly, i.e., for each particle \(x_i,\,i= 1,\cdots , N\), its initial velocity is given by \(v_i = v_0(x_i)\).
For the twodimensional case, the procedure is the same as the one for the onedimensional case. We thus omit the details here.
3.2 Macroscale: Euler System of Equations
In the last subsection, we solve the agentbased model using the particle method, while in this subsection, we use the finite volume method (FVM) to solve the macroscale model, i.e., the Euler equations (2).
3.2.1 OneDimensional Case
 i.
G is symmetric;
 ii.
G is a Toeplitz matrix.
Note that the matrix G is a Toeplitz matrix; therefore, this allows us to use the fast matrix vector procedure to compute \(G \varvec {m}\) or \(G \varvec {\rho }\) in \(O(K\log K)\) operation [33]. Furthermore, since the matrix G is symmetric, we can show that the above scheme possesses mass and momentum conservation laws. We state it in the following theorem:
Theorem 1
Proof
3.2.2 TwoDimensional Case
Since \(\varvec{G}\) is a blockToeplitz–Toeplitzblock matrix, the computation of \(\varvec{g}(\rho , m_1 , m_2)\) can be implemented in \(O(KL\log (KL))\) operations using the fast matrix vector multiplication [34].
Theorem 2
Proof
4 Numerical Examples
Example 1
For the agentbased model, the total number of sampled particles for the simulation is \(N = 1\,024\), while for the Euler equations the space step is \(\Delta x = 1/256\). The numerical solutions of the density and velocity at different times are shown in Figs. 2 and 3 for the value of fractional order \(\alpha = 0.5\) and \(\alpha = 1.2\), respectively. We see that the velocity tends to a constant value, which is in agreement with the analytical result given in [29]. Moreover, we observe from the comparison that the solutions of the microscale agentbased model coincide with solutions of the macroscale Euler equations. This means that the solution of the Euler system of equations can give a good prediction to the solution of the agentbased model.
Example 2
In this case, the total number of sampled particles for the simulation is \(N = 9\,976\) for the agentbased model, while for the Euler equations the space step is \(\Delta x = \Delta y = 1/64\). The numerical solutions of the density at different times are shown in Figs. 4 and 5 for the value of fractional order \(\alpha = 0.5\) and \(\alpha = 1.2\), respectively. The relative differences of the density between the solution of the micromodel and the macromodel are also shown in the third row of each figure. Again we observe that solutions of the microscale agentbased model are in good agreement with solutions of the macroscale Euler system of equations. This means that the numerical solution of the corresponding macromodel can yield correctly density distribution consistent with the collective behavior of the particle system.
5 Infer the Influence Function Using Gauss Process Machine Learning
5.1 Gaussian Process Regression
5.2 Bayesian Optimization
5.3 Numerical Examples
5.3.1 OneDimensional Case
Example 3
We first consider the onedimensional problem with the same influence function and initial and boundary conditions as that for Example 1.
Learned value of \(\alpha \) and relative error of the mean field \(F(\alpha )\) for the onedimensional case
Given \({\hat{\alpha }}\)  Learned value of \(\alpha \)  Output \(F(\alpha )\) 

0.5  0.480 3  \(1.217\,4\text{E}02\) 
1.2  1.165 1  \(7.966\,3\text{E}03\) 
5.3.2 TwoDimensional Case
Example 4
We now consider the twodimensional problem with the same influence function and initial and boundary conditions as that for Example 2.
Similarly as the onedimensional case, we obtain the data by solving the agentbased model (1) with the particle method for a given \({\hat{\alpha }}\) in the square domain \(\Omega \) generating the positions \((x_i^{{\hat{\alpha }}}(t),y_i^{{\hat{\alpha }}}(t)), \, i = 1,\cdots ,N\) and the corresponding velocities \(u_i^{{\hat{\alpha }},P}(t)\) and \(v_i^{{\hat{\alpha }},P}(t), \,i = 1,\cdots , N\) of the particles at time \(t = k \Delta T\), where \(\Delta T = 0.1, k = 5,6,\cdots , 20\), here we set \(N = 9\,976\).
Learned value of \(\alpha \) and relative error of the mean field \(F(\alpha )\) for the twodimensional case
Given \({\hat{\alpha }}\)  Learned value of \(\alpha \)  Output \(F(\alpha )\) 

0.5  0.513 4  \(2.142\,8\text{E}02\) 
1.2  1.200 9  \(2.023\,3\text{E}02\) 
6 Summary and Discussion
We presented a comparative study of nonlocal flocking dynamics using both the agentbased model and the continuum Eulerian model. Because animals in flocks generally do not interact mechanically and can be influenced by other individuals a certain distance away, we introduced nonlocal influence functions to consider the effects of animal communication in flocking dynamics. In particular, the microscopic dynamics of each individual in flocks is described by a Cucker–Smale particle model with nonlocal interaction terms, while the evolution of macroscopic quantities, i.e., the mean velocity and the population density, is modeled by the fractional partial differential equations (fPDEs). We performed agentbased simulations to generate the particle trajectories of each individual in flocking dynamics, and also solved the Euler equations with nonlocal influence functions using a finite volume scheme. In one and twodimensional benchmarks of nonlocal flocking dynamics, we demonstrated that, given specified influence functions, the Euler system of equations is able to capture the correct evolution of macroscopic quantities consistent with the collective behavior of the agentbased model.
Because experiments on flocking dynamics can get time series of trajectories of individuals in flocks using digital imaging or highresolution GPS devices, we used the trajectories generated by the agentbased simulations to mimic the field data of tracking logs that can be obtained experimentally. Subsequently, we proposed a learning framework to connect the discrete agentbased model to the continuum fPDEs for nonlocal flocking dynamics. Instead of specifying a phenomenological fPDE with an empirical fractional order, we learned the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agentbased simulations. More specifically, we employed a Gaussian process regression (GPR) model implemented with the Bayesian optimization to learn the fractional order of the influence function from the particle trajectories. We showed in both one and twodimensional examples that the numerical solution of the learned Euler system of equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behaviors of the discrete agentbased system. The relative error between the agentbased system and the learned Euler system of equations is about 1% for onedimensional cases and 2% for twodimensional cases.
Although we only demonstrated the effectiveness of the proposed learning framework in relatively simple cases, i.e., one and twodimensional nonlocal flocking dynamics, this method established a direct connection between the discrete agentbased models to the continuumbased PDE models, and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories. It is worth noting that the agentbased model we used in the present work does not consider stochastic terms, and thus the training data of particle trajectories do not contain noise. However, the experimental data of tracking logs obtained by digital imaging may include noise from measurement uncertainty, where a multifidelity framework proposed by Babaee et al. [2] can be used to handle different sources of uncertainties in the learning process. Moreover, in addition to the GPRbased learning method for connecting individual behavior to collective dynamics, it is also of interest to introduce deep learning strategies such as the CNN (convolutional neural network)based method [25] and the particle swarm optimization algorithm [35] to bridge the gap between flocking theory/modeling and experiments.
Notes
Acknowledgements
This work was supported by the OSD/ARO/MURI on “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications (W911NF1510562)” and the DOE PhILMs Project (DESC0019453).
References
 1.Antoniou, P., Pitsillides, A., Blackwell, T., Engelbrecht, A., Michael, L.: Congestion control in wireless sensor networks based on bird flocking behavior. Comput. Netw. 57(5), 1167–1191 (2013)CrossRefGoogle Scholar
 2.Babaee, H., Perdikaris, P., Chryssostomidis, C., Karniadakis, G.E.: Multifidelity modelling of mixed convection based on experimental correlations and numerical simulations. J. Fluid Mech. 809, 895–917 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Ballerini, M., Calbibbo, N., Candeleir, R., Cavagna, A., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini, A., Viale, M., Zdravkovic, V.: Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl. Acad. Sci. USA 105(4), 1232–1237 (2008)CrossRefGoogle Scholar
 4.Beekman, M., Sumpter, D.J., Ratnieks, F.L.: Phase transition between disordered and ordered foraging in pharaoh’s ants. Proc. Natl. Acad. Sci. USA 98(17), 9703–9706 (2001)CrossRefGoogle Scholar
 5.Bernardi, S., Colombi, A., Scianna, M.: A discrete particle model reproducing collective dynamics of a bee swarm. Comput. Biol. Med. 93, 158–174 (2018)CrossRefGoogle Scholar
 6.Bouchut, F., Jin, S., Li, X.: Numerical approximations of pressureless and isothermal gas dynamics. SIAM J. Numer. Anal. 41(1), 135–158 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 7.Chen, D.X., Vicsek, T., Liu, X.L., Zhou, T., Zhang, H.T.: Switching hierarchical leadership mechanism in homing flight of pigeon flocks. Europhys. Lett. 114, 60008 (2016)CrossRefGoogle Scholar
 8.Cristiani, E., Piccoli, B., Tosin, A.: Modeling selforganization in pedestrians and animal groups from macroscopic and microscopic viewpoints. In: Naldi, G., Pareschi, L., Toscani, G. (eds.) Mathematical Modeling of Collective Behavior in Socioeconomic and Life Sciences, pp. 337–364. Springer, New York (2010)CrossRefGoogle Scholar
 9.Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Eftimie, R., de Vries, G., Lewis, M.A.: Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci. USA 104(17), 6974–6979 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Figalli, Alessio, Kang, MoonJin: A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment. Anal. PDE 12(3), 843–866 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Giardina, I.: Collective behavior in animal groups: theoretical models and empirical studies. HFSP J. 2(4), 205–219 (2008)CrossRefGoogle Scholar
 13.Giuggioli, L., Potts, J.R., Rubenstein, D.I., Levin, S.A.: Stigmergy, collective actions, and animal social spacing. Proc. Natl. Acad. Sci. USA 110(42), 16904–16909 (2013)CrossRefGoogle Scholar
 14.Jaffry, S.W., Treur, J.: Agentbased and populationbased modeling of trust dynamics. In: Nguyen, N.T. (ed.) Transactions on Computational Collective Intelligence, vol. IX, pp. 124–151. Springer, Berlin, Heidelberg (2013)Google Scholar
 15.Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 16.Levine, H., Rappel, W.J., Cohen, I.: Selforganization in systems of selfpropelled particles. Phys. Rev. E 63, 017101 (2001)CrossRefGoogle Scholar
 17.Li, Z., Bian, X., Li, X., Deng, M., Tang, Y.H., Caswell, B., Karniadakis, G.E.: Dissipative particle dynamics: foundation, evolution, implementation, and applications. In: Bodnár, T., Galdi, G.P., Nečasová, Š. (eds.) Particles in Flows, pp. 255–326. Birkhäuser, Cham (2017)CrossRefGoogle Scholar
 18.Lukeman, R., Li, Y.X., EdelsteinKeshet, L.: Inferring individual rules from collective behavior. Proc. Natl. Acad. Sci. USA 107(28), 12576–12580 (2010)CrossRefGoogle Scholar
 19.Mahmoodi, K., West, B.J., Grigolini, P.: Selforganizing complex networks: individual versus global rules. Front. Physiol. 8, 478 (2017)CrossRefGoogle Scholar
 20.Nagai, K.H.: Collective motion of rodshaped selfpropelled particles through collision. Biophys. Physicobiol. 15, 51–57 (2018)CrossRefGoogle Scholar
 21.Nagy, M., Ákos, Z., Biro, D., Vicsek, T.: Hierarchical group dynamics in pigeon flocks. Nature 464, 890–893 (2010)CrossRefGoogle Scholar
 22.Niwa, H.S.: Selforganizing dynamic model of fish schooling. J. Theor. Biol. 171(2), 123–136 (1994)CrossRefGoogle Scholar
 23.Okubo, A.: Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. Adv. Biophys. 22, 1–94 (1986)CrossRefGoogle Scholar
 24.Pang, G., Perdikaris, P., Cai, W., Karniadakis, G.E.: Discovering variable fractional orders of advectiondispersion equations from field data using multifidelity bayesian optimization. J. Comput. Phys. 348, 694–714 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 25.Pu, H.T., Lian, J., Fan, M.Q.: Automatic recognition of flock behavior of chickens with convolutional neural network and kinect sensor. Int. J. Pattern Recognit. Artif. Intell. 32(7), 1850023 (2018)CrossRefGoogle Scholar
 26.Rasmussen, C.E.: Gaussian processes in machine learning. In: Bousquet, O., von Luxburg, U., Rätsch, G. (eds.) Advanced Lectures on Machine Learning, pp. 63–71. Springer, Berlin, Heidelberg (2004)CrossRefGoogle Scholar
 27.Rasmussen, C.E., Nickisch, H.: Gaussian processes for machine learning (GPML) toolbox. J. Mach. Learn. Res. 11, 3011–3015 (2010)MathSciNetzbMATHGoogle Scholar
 28.Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing. Trans. Math. Appl. 1, 1–26 (2017)MathSciNetzbMATHGoogle Scholar
 29.Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing ii: flocking. Discr. Contin. Dyn. Syst. A 37(11), 5503–5520 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 30.Tadmor, E., Tan, C.: Critical thresholds in flocking hydrodynamics with nonlocal alignment. Philos. Trans. A Math. Phys. Eng. Sci. 372(20), 20130401 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 31.Theraulaz, G., Bonabeau, E.: Modelling the collective building of complex architectures in social insects with lattice swarms. J. Theor. Biol. 177(4), 381–400 (1995)CrossRefGoogle Scholar
 32.Tunstrom, K., Katz, Y., Ioannou, C.C., Huepe, C., Lutz, M.J., Couzin, I.D.: Collective states, multistability and transitional behavior in schooling fish. PLoS Comput. Biol. 9(2), e1002915 (2013)MathSciNetCrossRefGoogle Scholar
 33.Wang, H., Du, N.: Fast solution methods for spacefractional diffusion equations. J. Comput. Appl. Math. 255, 376–383 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 34.Wang, H., Tian, H.: A fast and faithful collocation method with efficient matrix assembly for a twodimensional nonlocal diffusion model. Comput. Methods Appl. Mech. Eng. 273, 19–36 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 35.Wang, D., Tan, D., Liu, L.: Particle swarm optimization algorithm: an overview. Soft Comput. 22(2), 387–408 (2018)CrossRefGoogle Scholar
 36.Yang, Y., Wei, D., Shu, C.W.: Discontinuous Galerkin method for Krause’s consensus models and pressureless Euler equations. J. Comput. Phys. 252, 109–127 (2013)MathSciNetCrossRefzbMATHGoogle Scholar