Symbolic Derivation of Mean-Field PDEs from Lattice-Based Models

Transportation processes, which play a prominent role in the life and social sciences, are typically described by discrete models on lattices. For studying their dynamics a continuous formulation of the problem via partial differential equations (PDE) is employed. In this paper we propose a symbolic computation approach to derive mean-field PDEs from a lattice-based model. We start with the microscopic equations, which state the probability to find a particle at a given lattice site. Then the PDEs are formally derived by Taylor expansions of the probability densities and by passing to an appropriate limit as the time steps and the distances between lattice sites tend to zero. We present an implementation in a computer algebra system that performs this transition for a general class of models. In order to rewrite the mean-field PDEs in a conservative formulation, we adapt and implement symbolic integration methods that can handle unspecified functions in several variables. To illustrate our approach, we consider an application in crowd motion analysis where the dynamics of bidirectional flows are studied. However, the presented approach can be applied to various transportation processes of multiple species with variable size in any dimension, for example, to confirm several proposed mean-field models for cell motility.

[1]  Clemens G. Raab.  Integration of unspecified functions and families of iterated integrals , 2013, ISSAC '13.

[2]  Graham H. Campbell.  Symbolic integration of expressions involving unspecified functions , 1988, SIGS.

[3]  R. Risch.  The problem of integration in finite terms , 1969 .

[4]  Joel Moses,et al.  Symbolic integration: the stormy decade , 1966, CACM.

[5]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[6]  C. Cercignani.  Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations , 2000 .

[7]  Matthew J Simpson,et al.  Models of collective cell spreading with variable cell aspect ratio: a motivation for degenerate diffusion models. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  François Lemaire,et al.  On the integration of differential fractions , 2013, ISSAC '13.

[9]  Matthew J Simpson,et al.  Nonlinear diffusion and exclusion processes with contact interactions. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Angela Stevens,et al.  The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems , 2000, SIAM J. Appl. Math..

[11]  Victor J. Blue,et al.  Cellular automata microsimulation for modeling bi-directional pedestrian walkways , 2001 .

[12]  Bruno Buchberger,et al.  Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases , 2012, ArXiv.

[13]  C. Schmeiser,et al.  A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion , 2009, Journal of mathematical biology.

[14]  Karl Oelschläger,et al.  On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes , 1989 .

[15]  Matthew J Simpson,et al.  Simulating invasion with cellular automata: connecting cell-scale and population-scale properties. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  E. Zeydan,et al.  On the Integration of AI/ML-based scaling operations in the 5Growth platform , 2020, 2020 IEEE Conference on Network Function Virtualization and Software Defined Networks (NFV-SDN).

[17]  A. Schadschneider,et al.  Simulation of pedestrian dynamics using a two dimensional cellular automaton , 2001 .

[18]  W. Marsden.  I and J , 2012 .

[19]  M. Wolfram,et al.  Nonlinear Poisson–Nernst–Planck equations for ion flux through confined geometries , 2012 .

[20]  M. Burger,et al.  Continuous limit of a crowd motion and herding model: Analysis and numerical simulations , 2011 .

[21]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[22]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[23]  E. Kolchin.  Differential Algebra and Algebraic Groups , 2012 .

[24]  M. Burger,et al.  Mean field games with nonlinear mobilities in pedestrian dynamics , 2013, 1304.5201.

[25]  Markus Rosenkranz,et al.  Integro-differential polynomials and operators , 2008, ISSAC '08.

[26]  François Lemaire,et al.  An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data , 2014, CASC.

[27]  Ayse Humeyra Bilge.  A REDUCE program for the integration of differential polynomials , 1992 .

[28]  Karl Oelschl iger.  On the Derivation of Reaction-Diffusion Equations as Limit Dynamics of Systems of Moderately Interacting Stochastic Processes , 2004 .

[29]  Kerry A Landman,et al.  Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .

[31]  C. Landim,et al.  Scaling Limits of Interacting Particle Systems , 1998 .

[32]  Martin Burger,et al.  Nonlinear Cross-Diffusion with Size Exclusion , 2010, SIAM J. Math. Anal..