The climate system can been described by a dynamical system and its associated attractor. The dynamics of this attractor depends on the external forcings that influence the climate. Such forcings can affect the mean values or variances, but regions of the attractor that are seldom visited can also be affected. It is an important challenge to measure how the climate attractor responds to different forcings. Currently, the Euclidean distance or similar measures like the Mahalanobis distance have been favored to measure discrepancies between two climatic situations. Those distances do not have a natural building mechanism to take into account the attractor dynamics. In this paper, we argue that a Wasserstein distance, stemming from optimal transport theory, offers an efficient and practical way to discriminate between dynamical systems. After treating a toy example, we explore how the Wasserstein distance can be applied and interpreted to detect non-autonomous dynamics from a Lorenz system driven by seasonal cycles and a warming trend.

If the climate system is viewed as a complex dynamical system
yielding a strange attractor, i.e., a highly complicated object around which
all trajectories wind up

In addition to climate internal variability, external forcings (either
natural or anthropogenic) perturb the climate system dynamics by introducing
a time dependence of the attractor. This is the cause of non-stationary
behavior of the climate system. At first order, this can translate into a
general shift of the underlying attractor

Classical distances, like the Euclidean distance are often used to measure attractor differences. The goal of our paper
is to present a framework, embedded in optimal transport theory

The paper is organized as follows. In Sect.

To characterize changes in the properties of the attractor of a dynamical system, the first step of our
methodology is to determine how two measures (or distributions of mass) differ. The idea is to derive a cost function for
transporting one mass distribution onto the other. As a simple example, we consider the three mass distributions shown in
Fig.

This mathematical problem traces back to

Computing the right-hand side of Eq. (

In our example (Fig.

A dynamical system can be defined by the action of an ordinary differential equation

For chaotic dynamical systems, trajectories

We now focus on the

We propose to discriminate two attractors based on
Eq. (

Then we compute the empirical measures associated with the snapshot attractors by discretizing the phase space
(approximated by the box

The difference between the summer and winter attractors is evaluated by

We detect a difference between the winter and summer of Lorenz 84 systems if

The estimation of the Wasserstein distance between attractors obviously depends on the number of available samples

The complete procedure to obtain an empirical probability distribution of Wasserstein distances, depending on the sample
size

box plots of distances computed using the Wasserstein distance

The probability distributions of the Wasserstein distance for

The distribution of the distances between winter (resp., summer) snapshot
attractors decreases to 0 (the expected asymptotic value) when

Kolmogorov–Smirnov test applied between distribution

The distance between winter and summer attractors (black box plots) decrease
with

This protocol was also applied for bin sizes of

For illustration purposes, we compute Euclidean distances between the same
snapshot attractors (Fig.

In this section, we test whether it is possible to differentiate between attractors if only partial information is available. This can happen if one or more variables of the system are omitted (projection onto the remaining variables) or if variables are censored (truncation of the values of a variable), or a combination of both. The motivation in atmospheric sciences is that the underlying dynamical system is defined in three spatial dimensions (on the sphere), and that observables of the attractor of this system are generally obtained over a limited area (censoring of the rest of the globe) and a fixed pressure level (projection).

It has been proven that a sequence of observables of a dynamical system
convey the same dynamics as the whole system

For the Lorenz 84 attractors, a first reduction of information is performed
by projecting the systems onto their

The Wasserstein distance distribution (Fig.

The same experiment is conducted with the Euclidean distance
(Fig.

We conclude that the Wasserstein distance has a high capacity of
discriminating different attractors coming from this dynamical system, even
with a partial information. It is particularly promising in atmospheric
sciences, where analyses are performed on truncated variables (e.g., a surface
field on a limited area: transformation

Snapshots of 10 000 points from the Lorenz 84 defined by Eq. (

We now focus on a time-varying dynamical system that mimics variability
around a seasonal cycle, and a monotonic forcing that plays after a
triggering time

The snapshot attractors of this system were investigated by

Such a coupled behavior is present in most regional temperature time series
at the decadal or centennial scales. The periodic part of the forcing

In this section, we want to quantify the change of the whole dynamics of the ensemble of snapshot attractors with the Wasserstein distance, and assess the detectability of changes from small numbers of observations.

Yearly averages Wasserstein distance between the reference attractor before forcing, and all other attractors.
The

We compute snapshot attractors for each time step, for

The empirical measure of the snapshot attractors is estimated at each time
step

The yearly averages of the distances to the four reference attractors are
shown in Fig.

We find that the variability of the distance variations decrease with the
number

In this example, the distances of the snapshot attractors to winter and
spring reference attractors increase with time after

Those results are consistent with those of

The same experiment is conducted with the Euclidean distance. For

The Wasserstein distance appears to be efficient to
measure changes in the dynamics in time evolving systems even with a
relatively low number of points (e.g.,

The other caveat of this approach is its computational cost. The minimization
of the cost function, constrained by the estimated measures, has to be
implemented by network simplex algorithms

The next research challenge is to adapt this method to climate model simulations (e.g. from the Coupled Model Intercomparison Project (Taylor et al., 2012)). The Wasserstein distance could be computed to discriminate between the atmospheric attractors from control, historical and scenario runs.

The Lorenz model simulations data files can be obtained upon request to the first author.

We just give here the general idea to compute Wasserstein distance with the
network simplex algorithm. We want to transport the measure

These constraints define a polyhedral convex set in the space of

Finding a first extremal point.

Iterate over the face of polyhedra (the simplex) until the minimal solution is reached.

Because the number of extremal points increases exponentially with the size of
data, this algorithm has an exponential complexity. But, in practice the
iterations over simplex are made in the direction of an optimal solution.
Thus, it has been found that the complexity of the algorithm is polynomial in
practice. Currently, we use a C

Besides the simulations studied in the previous sections, it is possible to
theoretically justify the use of the Wasserstein distance for non-autonomous
dynamical systems. Any dynamical system defined from an ordinary differential
equation, say

The minimization is done over all vector fields

YR conducted the computations and produced the figures. The idea of the experiments were formulated by the three authors. The three authors contributed to writing the manuscript.

The authors declare that they have no conflict of interest.

This work is supported by ERC grant no. 338965-A2C2. We thank Davide Faranda and Ara Arakelian for useful discussions. Edited by: Vicente Perez-Munuzuri Reviewed by: Valerio Lucarini and one anonymous referee