Communications on Applied Mathematics and Computation >

2019 , Vol. 1 >Issue 4: 597 - 619

DOI: https://doi.org/10.1007/s42967-019-00031-y

Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations

Expand
  • 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA;
    2 Pacifc Northwest National Laboratory, Richland, WA 99354, USA

Received date: 2018-10-26

  Revised date: 2018-12-26

  Online published: 2019-10-16

Supported by

This work was supported by the OSD/ARO/MURI on “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications (W911NF-15-1-0562)” and the DOE PhILMs Project (DE-SC0019453).

Fold

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 agent-based model to describe the microscopic dynamics of each individual in a fock, and use a fractional partial diferential 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 focking dynamics, we learn the efective nonlocal infuence function in fPDEs directly from particle trajectories generated by the agent-based simulations. We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one- and two-dimensional nonlocal focking 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 feld data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the infuence function using a Gaussian process regression model implemented with the Bayesian optimization. We show in one- and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the fnite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method. The proposed method ofers new insights into how to scale the discrete agent-based models to the continuum-based PDE models, and could serve as a paradigm on extracting efective governing equations for nonlocal focking dynamics directly from particle trajectories.

Key words： Fractional PDEs; Gaussian process; Bayesian optimization; Fractional Laplacian; Conservation laws

Cite this article

Zhiping Mao, Zhen Li, George Em Karniadakis . Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations[J]. Communications on Applied Mathematics and Computation, 2019 , 1(4) : 597 -619 . DOI: 10.1007/s42967-019-00031-y

References

1. Antoniou, P., Pitsillides, A., Blackwell, T., Engelbrecht, A., Michael, L.:Congestion control in wireless sensor networks based on bird focking behavior. Comput. Netw. 57(5), 1167-1191 (2013)
2. Babaee, H., Perdikaris, P., Chryssostomidis, C., Karniadakis, G.E.:Multi-fdelity modelling of mixed convection based on experimental correlations and numerical simulations. J. Fluid Mech. 809, 895- 917 (2016)
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 feld study. Proc. Natl. Acad. Sci. USA 105(4), 1232-1237 (2008)
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)
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)
6. Bouchut, F., Jin, S., Li, X.:Numerical approximations of pressureless and isothermal gas dynamics. SIAM J. Numer. Anal. 41(1), 135-158 (2003)
7. Chen, D.X., Vicsek, T., Liu, X.L., Zhou, T., Zhang, H.T.:Switching hierarchical leadership mechanism in homing fight of pigeon focks. Europhys. Lett. 114, 60008 (2016)
8. Cristiani, E., Piccoli, B., Tosin, A.:Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints. In:Naldi, G., Pareschi, L., Toscani, G. (eds.) Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences, pp. 337-364. Springer, New York (2010)
9. Cucker, F., Smale, S.:Emergent behavior in focks. IEEE Trans. Autom. Control 52(5), 852-862 (2007)
10. Eftimie, R., de Vries, G., Lewis, M.A.:Complex spatial group patterns result from diferent animal communication mechanisms. Proc. Natl. Acad. Sci. USA 104(17), 6974-6979 (2007)
11. Figalli, Alessio, Kang, Moon-Jin:A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment. Anal. PDE 12(3), 843-866 (2019)
12. Giardina, I.:Collective behavior in animal groups:theoretical models and empirical studies. HFSP J. 2(4), 205-219 (2008)
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)
14. Jafry, S.W., Treur, J.:Agent-based and population-based modeling of trust dynamics. In:Nguyen, N.T. (ed.) Transactions on Computational Collective Intelligence, vol. IX, pp. 124-151. Springer, Berlin, Heidelberg (2013)
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)
16. Levine, H., Rappel, W.J., Cohen, I.:Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 017101 (2001)
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)
18. Lukeman, R., Li, Y.X., Edelstein-Keshet, L.:Inferring individual rules from collective behavior. Proc. Natl. Acad. Sci. USA 107(28), 12576-12580 (2010)
19. Mahmoodi, K., West, B.J., Grigolini, P.:Self-organizing complex networks:individual versus global rules. Front. Physiol. 8, 478 (2017)
20. Nagai, K.H.:Collective motion of rod-shaped self-propelled particles through collision. Biophys. Physicobiol. 15, 51-57 (2018)
21. Nagy, M., Ákos, Z., Biro, D., Vicsek, T.:Hierarchical group dynamics in pigeon focks. Nature 464, 890-893 (2010)
22. Niwa, H.-S.:Self-organizing dynamic model of fsh schooling. J. Theor. Biol. 171(2), 123-136 (1994)
23. Okubo, A.:Dynamical aspects of animal grouping:swarms, schools, focks, and herds. Adv. Biophys. 22, 1-94 (1986)
24. Pang, G., Perdikaris, P., Cai, W., Karniadakis, G.E.:Discovering variable fractional orders of advection-dispersion equations from feld data using multi-fdelity bayesian optimization. J. Comput. Phys. 348, 694-714 (2017)
25. Pu, H.T., Lian, J., Fan, M.Q.:Automatic recognition of fock behavior of chickens with convolutional neural network and kinect sensor. Int. J. Pattern Recognit. Artif. Intell. 32(7), 1850023 (2018)
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)
27. Rasmussen, C.E., Nickisch, H.:Gaussian processes for machine learning (GPML) toolbox. J. Mach. Learn. Res. 11, 3011-3015 (2010)
28. Shvydkoy, R., Tadmor, E.:Eulerian dynamics with a commutator forcing. Trans. Math. Appl. 1, 1-26 (2017)
29. Shvydkoy, R., Tadmor, E.:Eulerian dynamics with a commutator forcing ii:focking. Discr. Contin. Dyn. Syst. A 37(11), 5503-5520 (2017)
30. Tadmor, E., Tan, C.:Critical thresholds in focking hydrodynamics with non-local alignment. Philos. Trans. A Math. Phys. Eng. Sci. 372(20), 20130401 (2014)
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)
32. Tunstrom, K., Katz, Y., Ioannou, C.C., Huepe, C., Lutz, M.J., Couzin, I.D.:Collective states, multistability and transitional behavior in schooling fsh. PLoS Comput. Biol. 9(2), e1002915 (2013)
33. Wang, H., Du, N.:Fast solution methods for space-fractional difusion equations. J. Comput. Appl. Math. 255, 376-383 (2014)
34. Wang, H., Tian, H.:A fast and faithful collocation method with efcient matrix assembly for a twodimensional nonlocal difusion model. Comput. Methods Appl. Mech. Eng. 273, 19-36 (2014)
35. Wang, D., Tan, D., Liu, L.:Particle swarm optimization algorithm:an overview. Soft Comput. 22(2), 387-408 (2018)
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)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157