ODEs learn to walk ODE-Net based data-driven modeling for crowd dynamics Chen Cheng1and Jinglai Li2

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ODEs learn to walk: ODE-Net based data-driven modeling for

crowd dynamics

Chen Cheng1and Jinglai Li2∗

1School of Mathematical Sciences,

Shanghai Jiao Tong University, Shanghai 200240, China

2School of Mathematics,

University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Abstract

Predicting the behaviors of pedestrian crowds is of critical

importance for a variety of real-world problems. Data driven

modeling, which aims to learn the mathematical models from

observed data, is a promising tool to construct models that

can make accurate predictions of such systems. In this work,

we present a data-driven modeling approach based on the

ODE-Net framework, for constructing continuous-time mod-

els of crowd dynamics. We discuss some challenging issues in

applying the ODE-Net method to such problems, which are

primarily associated with the dimensionality of the under-

lying crowd system, and we propose to address these issues

by incorporating the social-force concept in the ODE-Net

framework. Finally application examples are provided to

demonstrate the performance of the proposed method.

Keywords: crowd dynamics, data-driven modeling,

ODE-Net, social force

1. Introduction

Collective motion of pedestrians is a highly common phe-

nomenon in urban life, and understanding the dynamics of

pedestrian crowds is essential for a large variety of applica-

tions, ranging from safety management [1, 2] to robot nav-

igation [3, 4]. Modeling the behaviors of pedestrian crowds

has attracted considerable attention in multiple disciplines

such as physics, social science and artiﬁcial intelligence, and

various models have been proposed in the past decades. Due

to the complexity of the crowd dynamics, driving the mathe-

matical models that can accurately predict the crowd behav-

iors is an extremely challenging task. To this end a partic-

ularly promising remedy is to develop mathematical models

with the assistance of related data, an approach often re-

ferred to as data-driven modeling [5].

Within the context of crowd dynamics modeling, we here

discuss two main strategies behind the data driven meth-

ods. The ﬁrst strategy assumes that the crowd dynamics

follows a speciﬁc mathematical model that is usually derived

based on physics, but all or some of the model parameters

are not available; one then estimates these parameters by ﬁt-

ting the observation data into the model. Examples of such

methods include [6–9], among some others. While this type

of methods are conceptually straightforward and relatively

easy to implement, their performance is ultimately limited

by the mathematical models adopted. The second strategy

oﬀers more ﬂexibility: namely it does not impose a speciﬁc

mathematical model; rather, it learns the model (often rep-

resented by an artiﬁcial neural network) directly from the

∗j.li.10@bham.ac.uk

data with machine learning techniques. While their imple-

mentation is usually more complicated, the machine-learning

based methods are much less restrictive than the ﬁrst kind

and can potentially obtain very accurate model, provided

that high-quality data are available.

In the past a few years, various eﬀorts have been made

to the machine learning based data driven modeling, e.g.,

[10–13]. To the best of our knowledge, most of these exist-

ing methods are designed to learn crowd dynamics models

that are discrete in time, largely because the discrete-time

models can be naturally formulated with a deep neural net-

work such as the recurrent neural network (RNN). On the

other hand, there is strong desire to develop continuous-time

models, as they can be used to predict the crowd behaviors

at any time of interest. The ODE-Net method, ﬁrst proposed

in [14], has gained attention as a tool to learn continuous-

time models of physical systems [15–18]. Simply speaking,

ODE-Net formulates the system of interest as an ordinary

diﬀerential equation system, which is represented by a deep

neural network, and learned from the data. The ODE-Net

method, however, can not be directly applied to the crowd

dynamics, and we summarise three main challenges of it, all

associated with the crowd size (or equivalently the dimen-

sionality of the system): ﬁrst and foremost, due to the high

training cost, ODE-Net generally has diﬃculty dealing with

systems of high dimensions, rendering it especially unsuited

for large-size crowds; secondly, in reality the size of a crowd

may vary in time, with pedestrians entering or leaving the

scene of interest, and such a system can not be easily mod-

eled by ODE-Net; ﬁnally, the model obtained by ODE-Net

cannot be used to predict crowds whose size is diﬀerent from

the training system, which makes its application very lim-

ited. In this work we propose an ODE-Net based method

to learn the crowd dynamics models from data, where the

aforementioned issues are addressed by incorporating under-

lying physical knowledge of the dynamics into the ODE-Net

model. In particular, we adopt the concept that the crowd

is a physical system driven by social and psychological forces

as is in the so-called social force model (SFM) [19], and

then learn those force functions from data. The resulting

social force based method allows one to learn the models

from data for large-scale and variable-size crowds, and also

use the learned models to predict the behaviors of crowds of

any sizes.

The rest of the paper is organized as follows. In Section 2

we present the social force based ODE-Net method, and in

Section 3 we demonstrate the performance of the proposed

method by applying it to data generated from two commonly

used crowd dynamics models. Finally Section 4 oﬀers some

conclusions and discussions.

arXiv:2210.09602v1 [cs.LG] 18 Oct 2022

2. Methodologies

2.1. ODE-Net for crowd dynamics

We start by introducing the ODE-Net from a deep neural

network perspective. Traditional deep neural networks, such

as residual networks, build complicated transformations by

composing a sequence of transformations to a hidden state:

zt+δt−zt

δt

=ht(zt), δt= 1,(2.1)

where ht(zt)is a function parameterized by a neural net-

work. These iterative updates can be interpreted as an Euler

discretization of a continuous transformation. In contrast to

traditional deep neural networks where δt= 1 is ﬁxed, ODE-

Net [14] introduced a continuous version in which δt→0.

As a result, Eq. (2.1) becomes

dz(t)

dt=h(z(t), t).(2.2)

In this continuous framework, training the networks becomes

to learn the function h(z, t)and next we will discuss how to

learn this function.

First we assume that the function h(z, t)is represented

by a neural network hθ(z, t)parameterized by θ, and we

have observed data at t0and t1, denoted as ˆ

z(t0)and ˆ

z(t1)

respectively. Starting from the input layer ˆ

z(t0), the output

layer z(t1)can be deﬁned by the solution to this ODE initial

value problem at some time t1:

z(t1) = ˆ

z(t0) + Zt1

hθ(z(t), t)dt, (2.3)

and the time from t0to t1is referred to as the integration

time of the data point. Eq. (2.3) can be computed using an

oﬀ-the-shelf diﬀerential equation solver and we write it as,

z(t1) = ODESolve (ˆ

z(t0), hθ, t0, t1).(2.4)

The network parameters θare computed by iteratively mini-

mizing a prescribed loss function L(ˆ

z(t1),z(t1)), which mea-

sures the diﬀerence between the observed data ˆ

z(t1)and

the model prediction z(t1). An interesting feature of this

method is that the gradient of the loss function with respect

to θcan be computed using the adjoint sensitivity method,

which is more memory eﬃcient than directly backpropagat-

ing through the integrator [14].

As has been discussed earlier, ODE-Net allows us to con-

struct a continuous-time model for the crowd dynamics.

Namely, let z(t)represents the state of the crowd at time

tand as a result Eq. (2.2) becomes the governing equation

of the crowd dynamics; suppose that we have observations of

the crowd ﬂow ˆ

z(t), and we can use the training process de-

scribed above to learn the function h(z, t)(or more precisely

its neural network representation hθ(z, t)).

Though the application of ODE-Net to crowd dynamics is

conceptually straightforward, the implementation is highly

challenging. When applied to crowd dynamics, zrepresents

the state of motion of the entire crowd that may consist of

a large number of particles (i.e., pedestrians, and through-

out the paper we use these two terms interchangeably), and

it follows that zcan be of very high dimensions since the

dimensionality of zis proportional to the size of the crowd.

In this case, learning a high-dimensional function h(z, t)can

be prohibitively diﬃcult: it may require a massive amount

of training data which may not be available in practice, and

the computational cost for training such a complex model

can be exceedingly high. In addition, as one can see, in the

formulation described above, the dimensionality of zneeds

to be ﬁxed, which often does not meet the reality, as in most

situations people may enter or leave the scene of interest and

the dimensionality of zvaries over time. More importantly,

as the dimensionality of zis ﬁxed, once the model is learned

from the data, it can only be used to predict systems of the

same number of particles, a serious limitation of the use-

fulness of the method. To address these issues, we propose

to address the dimensionality issue by incorporating the so-

cial force (SF) concept into the ODE-Net method, which is

detailed in Section 2.2.

2.2. Social-force based ODE-Net

Suppose that we consider a crowd of Nparticles and we can

write the state variable z= (z1, ..., zN)Twhere znrepresents

the state of motion of particle nfor each n= 1...N. In partic-

ular we have zn= (xn, vn)where xnand vnare respectively

the position and the velocity of particle n. We also introduce

the notations x= (x1, ..., xN)Tand v= (v1, ..., vN)T. Now

according to the Newton’s second law, model (2.2) can be

re-written as ˙

v=v

M−1f,(2.5)

where

f(x,v) = 



f1(x,v)

...

fN(x,v)





with fn(x,v)being the force applied to particle nand

M= diag[m1, ..., mN]with mnbeing the “mass” of particle

n. With formulation (2.5), the original ODE-Net problem

is turned into learning the force function f(x,v)and esti-

mating the mass matrix M, where one can see that learning

function f(x,v)is by far the more challenging task here.

It is important to note that in such problems fand M

should not be understood as the usual physical forces and

masses respectively. Rather, following the assumption of the

social force model [19], frepresents the socio-psychological

forces driven by personal motivations and environmental

constraints, and the mass matrix Mcharacterizes how easy

or diﬃcult to change the velocity of each pedestrian. At

this point the force ﬁeld f(x,v)is still a high dimensional

function for large crowd size N, and further simpliﬁcation is

needed to make the learning problem feasible.

We now introduce further assumptions to simplify the

force function. First we assume that the total force applied

to each particle/pedestrian consists of two parts:

fn=fmot

n+fint

n,(2.6)

where fmot

nis the force generated by personal motivation to

reach certain desired state of motion, and fint

nis the force

caused by the interactions with other particles and the en-

vironments (e.g., obstacles). The total interaction force is

further written as,

fint

j(6=n)=1

nj +

w=1

nw,(2.7)

where fp

nj is the interaction force between pedestrians nand

jand fo

nw between pedestrian nand the w-th obstacles (as-

suming there are Wobstacles in total). We now need to

deal with both the motivation and the interaction forces.

We ﬁrst assume that the personal motivation force depends

on the particle’s state of motion:

fmot

n=fmot

θ(xn, vn, d),(2.8)

where drepresents some environmental factors that also af-

fect the motivation force, and fmot

θ(·)is an artiﬁcial neural

network parametrized by θ. Next we consider the interac-

tion force fint. To this end, it is common to assume that

pedestrians psychologically tend to keep a distance between

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ODEslearntowalk:ODE-Netbaseddata-drivenmodelingforcrowddynamicsChenCheng1andJinglaiLi2*1SchoolofMathematicalSciences,ShanghaiJiaoTongUniversity,Shanghai200240,China2SchoolofMathematics,UniversityofBirmingham,Edgbaston,BirminghamB152TT,UKAbstractPredictingthebehaviorsofpedestriancrowdsisofcriticalimp...

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ODEs learn to walk ODE-Net based data-driven modeling for crowd dynamics Chen Cheng1and Jinglai Li2

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