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

作     者：Zhiping Mao Zhen Li George Em Karniadakis 

作者机构：Division of Applied MathematicsBrown UniversityProvidenceRI02912USA Pacific Northwest National LaboratoryRichlandWA 99354USA 

出 版 物：《Communications on Applied Mathematics and Computation》 (应用数学与计算数学学报（英文）)

年 卷 期：2019年第1卷第4期

页      面：597-619页

核心收录：

学科分类：08[工学] 0818[工学-地质资源与地质工程] 0903[农学-农业资源与环境] 0714[理学-统计学（可授理学、经济学学位）] 0802[工学-机械工程] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 080201[工学-机械制造及其自动化] 

基　　金：Office of the Secretary of Defense, OSD Multidisciplinary University Research Initiative, MURI, (W911NF-15-1-0562) Army Research Office, ARO U.S. Department of Energy, USDOE, (DE-SC0019453) 

主　　题：Fractional PDEs Gaussian process Bayesian optimization Fractional Laplacian Conservation laws 

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 *** employ an agent-based 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 *** macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic *** 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 agent-based *** demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one-and two-dimensional nonlocal flocking *** 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 *** trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained *** can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian *** show in one-and two-dimensional 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 agent-based system solved by the particle *** proposed method offers 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 effective governing equations for nonlocal flocking dynamics directly from particle trajectories.