Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (4): 597-619.doi: 10.1007/s42967-019-00031-y

• ORIGINAL PAPER • 上一篇    下一篇

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

Zhiping Mao1, Zhen Li1, George Em Karniadakis1,2   

  1. 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA;
    2 Pacifc Northwest National Laboratory, Richland, WA 99354, USA
  • 收稿日期:2018-10-26 修回日期:2018-12-26 出版日期:2019-12-30 发布日期:2019-10-16
  • 通讯作者: Zhen Li, Zhiping Mao E-mail:zhen_li@brown.edu;zhiping_mao@brown.edu
  • 基金资助:
    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).

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

Zhiping Mao1, Zhen Li1, George Em Karniadakis1,2   

  1. 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA;
    2 Pacifc Northwest National Laboratory, Richland, WA 99354, USA
  • Received:2018-10-26 Revised:2018-12-26 Online:2019-12-30 Published:2019-10-16
  • Contact: Zhen Li, Zhiping Mao E-mail:zhen_li@brown.edu;zhiping_mao@brown.edu
  • 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).

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摘要： 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.

关键词: Fractional PDEs, Gaussian process, Bayesian optimization, Fractional Laplacian, Conservation laws

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

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