# Koopman Learning with Episodic Memory

William T. Redman  Dean Huang  AIMdyn Inc. Maria Fonoberova  AIMdyn Inc. Igor
Mezić

###### Abstract

Koopman operator theory, a data-driven dynamical systems framework, has found
significant success in learning models from complex, real-world data sets,
enabling state-of-the-art prediction and control. The greater interpretability
and lower computational costs of these models, compared to traditional machine
learning methodologies, make Koopman learning an especially appealing
approach. Despite this, little work has been performed on endowing Koopman
learning with the ability to learn from its own mistakes. To address this, we
equip Koopman methods – developed for predicting non-stationary time-series –
with an episodic memory mechanism, enabling global recall of (or attention to)
periods in time where similar dynamics previously occurred. We find that a
basic implementation of Koopman learning with episodic memory leads to
significant improvements in prediction on synthetic and real-world data. Our
framework has considerable potential for expansion, allowing for future
advances, and opens exciting new directions for Koopman learning.

##  1 Introduction

Over the past decade, Koopman learning [1] has emerged as a leading framework
for modeling real-world dynamical systems. The development of numerical
methods to efficiently approximate spectral objects associated with the
Koopman operator from data [2, 3, 4, 5, 6] has led to high quality forecasting
of complex time-series, such as Earth’s magnetic field [7], highway traffic
patterns [8], climate indicators [9], and COVID-19 spreading [10]. The
generalization of Koopman operator theory to include controlled dynamical
systems [11, 12, 13, 14] has led to state-of-the-art control of soft robotics
[15, 16, 17, 18]. And the advancement of theoretical results [19, 20, 21, 4,
22], linking Koopman spectral objects to state-space geometry – as well as the
inherent linearity of the Koopman operator – has enabled the interpretation of
learned models to an unprecedented degree.

Unlike many traditional machine learning (ML) methods, such as convolutional
neural networks, that are trained to extract informative features of the data
(i.e., observable functions), Koopman learning identifies dynamical
relationships between observables. Thus, while requiring appropriate
observables to be chosen by a knowledgeable human or ML [23, 5, 24, 25],
Koopman learning can easily discover spatio-temporal interactions that are
crucial to understanding the behavior of the system under study.

Koopman learning’s data-driven, predictive, efficient, controllable,
interpretable, and dynamics-based properties make it well aligned with next
generation artificial intelligence. And yet, to-date, there has been almost no
work on enabling Koopman learning to learn from its previous mistakes, an
essential aspect of any “intelligent” framework. The need for such an approach
becomes clear when considering applying Koopman learning to non-autonomous
dynamical systems [such as the yearly flu cases in the United States (U.S.),
Fig. 1A], where temporally local Koopman models are generated over streaming
or sliding time windows [26, 8, 10]. While this mitigates the mixing of data
from distinct dynamical regimes, its implicit forgetting of the past requires
any repeated structure of the data (Fig. 1B) to be re-learned from scratch.

Figure 1: Example of repeated structure in non-stationary dynamical system.
(A) Number of flu cases per week in the U.S. from 1997 to 2022, as reported to
CDC by Public Health Laboratories. (B) Zoomed-in four year time windows of
(A). Years are colored by their qualitative similarities in shape and peak
amplitude.

Traditional ML methods have long wrestled with this problem, and – in their
application to time-series – the use of memory has proven to be useful. This
was done first by recurrent neural networks (RNNs) [27], whose within-layer
connectivity enables information to remain in the network through time,
affecting future internal states. The introduction of long short-term memory
(LSTM) networks [28] and liquid state machines [29] led to computationally
tractable models that became widely adopted. However, in these models, long
histories (or “memory traces”) are continually present, making training
challenging and susceptible to vanishing gradients. Over the past several
years, Transformer networks [30], which make use of the self-attention
mechanism – originally designed to improve upon RNNs used for machine
translation [31] – have performed dominantly in a number of domains, attesting
to the power selectively attending to global aspects of data can confer. In
the context of time-series prediction, where Transformers have become
increasingly utilized [32, 33, 34, 35], self-attention can be viewed as
recalling specific points in the past that are relevant to the present.
Despite these advances, the high computational costs required to train such
models, as well as the mixed performance gains reported when comparing
Transformers to linear models [36], suggests that there may be room for
improvement from alternative approaches.

Inspired by the integration of memory into traditional ML methods, we develop
an approach that saves spectral objects associated with temporally local
approximations of the Koopman operator, and utilizes this information to make
new predictions, that are informed from the past. Because the Koopman spectral
objects are computed over a time window, these compact representations of the
dynamics capture episodes of previously observed data. Therefore, we refer to
the general framework we introduce as Koopman learning with episodic memory
(Fig. 2). Through deployment on synthetic and real-world data sets, we show
that the specific realization of Koopman learning with episodic memory we
develop, called extended dynamic mode decomposition (EDMD) with memory, sees
large improvements in prediction accuracy. This comes in large part from the
ability of these models to not repeatedly make the same mistakes. We believe
that our work not only demonstrates a way in which to better applications of
Koopman operator theory to the prediction of non-autonomous dynamical systems,
but additionally advances the state of Koopman learning as an artificial
intelligence framework, opening up a wealth of future directions.

##  2 Koopman operator theory

The central object of interest in Koopman operator theory is the Koopman
operator, $U$, an infinite-dimensional linear operator that describes the time
evolution of observables (i.e., functions of the underlying state-space
variables) that live in the functional space $\mathcal{F}$ [37, 38, 19]. In
particular, after $t>0$ amount of time, which can be continuous or discrete,
the value of the observable $f\in\mathcal{F}$, which can be a scalar or a
vector valued function, is given by

$U^{t}f(x_{0})=f\left[T^{t}(x_{0})\right],$ (1)

where $T$ is the dynamical map evolving the system and $x_{0}$ is the initial
condition in state-space. For the remainder of the paper, it will be assumed
that the state-space being considered is of finite dimension and that
$\mathcal{F}$ is a suitably chosen space of functions in which the spectral
expansion exists [22].

The action of the Koopman operator on the observable $f$ can be decomposed as

$Uf(p)=\sum_{i=0}^{\infty}\lambda_{i}\phi_{i}(x_{0})v_{i},$ (2)

where the $\phi_{i}$ are eigenfunctions of $U$, with
$\lambda_{i}\in\mathbb{C}$ as their eigenvalues and $v_{i}$ as their Koopman
modes [19, 22].

Spectrally decomposing the action of the Koopman operator is powerful because,
for a discrete-time dynamical system, the value of $f$ at time step
$k\in\mathbb{Z}^{+}$ is given simply by

$f\left[T^{k}(x_{0})\right]=U^{k}f(x_{0})=\sum_{i=0}^{\infty}\lambda_{i}^{k}\phi_{i}(x_{0})v_{i}.$
(3)

From Eq. 3, we see that the dynamics of the system in the directions $v_{i}$,
scaled by $\phi_{i}(x_{0})$, are given by the magnitude of the corresponding
$\lambda_{i}$. For the remainder of the paper, we will consider the output of
the Koopman mode decomposition (KMD) being the Koopman eigenvalues,
$\lambda_{i}$, and the scaled Koopman modes, $\phi_{i}(x_{0})v_{i}$, which we
will refer to as “Koopman modes” for simplicity.

###  2.1 Extended dynamic mode decomposition

There exist many ways of numerically approximating the KMD from a data matrix
$X=[x_{0},x_{1},...,x_{m-1}]$, where $x_{i}\in\mathbb{C}^{n}$ are snapshots of
the underlying dynamical system. In the case that a single observable (i.e.,
$n=1$) is recorded from the system under study, time delays can be introduced,
so as to enlarge the set of observables [39]. A popular approach to compute
the KMD is dynamic mode decomposition (DMD) [40, 2]. DMD aims to find a matrix
$A\in\mathbb{C}^{n\times n}$, such that

$||AX-Y||_{F},$ (4)

is minimized with respect to the Frobenius norm, where $Y=[x_{1},...,x_{m}]$.
A solution to Eq. 4 is given by

$A=YX^{+},$ (5)

where $+$ is the Moore-Penrose pseudo-inverse [2, 40, 41]. Practically, $A$ is
often approximated by performing reduced singular value decomposition (SVD)
[40, 41]. While simple, this approach yields the KMD when certain conditions
are met [2]. These conditions are restrictive, and in general, are not
guaranteed to be satisfied. In addition, it is often the case that the action
of the Koopman operator, restricted to the identity observable (that is, the
original state-space variables), is not invariant. For these, and other
reasons, many variants to DMD have been proposed.

One such approach is extended dynamic mode decomposition (EDMD) [3], which
considers not the data matrices $X$ and $Y$, but a dictionary of functions
$\mathcal{D}=[\psi_{1},...,\psi_{k}]$ applied to $X$ and $Y$. That is, it
considers the functions $\Psi(x)=[\psi_{1}(x),...,\psi_{k}(x)]$, where
$\Psi:\mathbb{C}^{n}\rightarrow\mathbb{C}^{k}$. The Koopman operator is then
approximated as

$K=G^{+}A,$ (6)

where

$\centering G=\frac{1}{m}\sum_{i=1}^{m}[\Psi(x_{i})]^{*}\Psi(x_{i})\hskip
28.45274ptA=\frac{1}{m}\sum_{i=1}^{m}[\Psi(x_{i})]^{*}\Psi(y_{i}).\@add@centering$

EDMD introduces a choice of which set of functions to include in the
dictionary $\mathcal{D}$, which can be addressed from an understanding of the
nature of the data [42], as well as can be optimized via dictionary learning
[23]. In this paper, we consider a family of Gaussian radial basis functions
as the dictionary, but our approach generalizes beyond this decision.

##  3 Koopman learning with episodic memory

Figure 2: Overview of episodic memory for Koopman learning. For the current
state of the system under study, the data within the sliding window (inset
figure on left) is used to compute the KMD. The resulting Koopman spectra and
modes are then compared to a library (or memory bank) of saved Koopman spectra
and modes, associated with past data. If a match of both Koopman spectra and
modes is found (illustrated by the red box), then the data that occurred in
the past (dashed red line) is used to predict the future behavior of the
system (dashed green line). If no match is found, sliding EDMD is used to
generate a prediction.

A motivation for the integration of episodic memory in Koopman learning is the
existence of repeated structure in some non-stationary dynamical systems (Fig.
1B). These patterns can emerge when underlying interactions between components
of the system are constant (e.g., human contact patterns), as well as when
constraints of the system are constant (e.g., healthcare infrastructure and
supply chains).

In order to identify when data currently being observed has similar dynamical
features to data seen in the past, three ingredients are necessary. First, a
representation of the dynamical features of data is needed. Given that Koopman
operator theory decomposes data into Koopman eigenvalues and modes (Eq. 2),
the compact representation of the KMD satisfies this requirement. In this
paper, we use EDMD [3] (Eq. 6) to approximate the Koopman spectral objects,
but our approach applies broadly to other numerical methods [6]. Second, a way
to compare dynamical features is needed, so that the notion of “similarity”
can be quantified. Previous work comparing the Koopman spectra across
dynamical systems [43, 44] has used the Wasserstein distance, a metric from
optimal transport that measures how much one distribution has to be modified
to match another. We use the same approach here, although again, other choices
could be made. Finally, the dynamical features need to be saved for future
comparison. Here, we simply store the Koopman spectra and modes, associated
with all local time windows, in very basic data structures. Future
applications of this approach can make use of more sophisticated ways in which
to efficiently save the Koopman representations, as well as ways in which to
incorporate the current KMD with similar saved representations (see
Discussion).

###  3.1 Algorithm

Our implementation of Koopman learning with episodic memory (EDMD with
memory), on a data set $X=[x_{0},...,x_{T}]$, has four primary steps:

1) At time step $t$, EDMD is performed on the window of data
$\left[x_{t},...,x_{t+\omega}\right]$, where $\omega\in\mathbb{Z}^{+}$ is the
size of the sliding window. We refer to this approach as “sliding EDMD”.

2) The computed Koopman eigenvalues and modes are compared to the Koopman
eigenvalues and modes saved from all previous time windows, via the
Wasserstein distance.

3) The time window $[t_{\text{min}}-\omega,t_{\text{min}}]$, whose Koopman
eigenvalues and modes have the smallest Wasserstein distance with the current
Koopman eigenvalues and modes, is identified. If the Wasserstein distance is
smaller than a pre-determined threshold, $\varepsilon\in\mathbb{R}^{+}$, we
predict $\Delta$ steps into the future by directly using what previously
occurred after $t_{\text{min}}$. That is, we set
$\hat{x}_{t+\Delta}=x_{t_{\text{min}}+\Delta}$, where $\hat{x}_{t+\Delta}$ is
the prediction and $x_{t_{\text{min}}+\Delta}$ is the value of the data that
occurred after the previous time window with similar dynamics111Note that we
ensure $t_{\text{min}}+\Delta<t$, in order to avoid using data we do not have
at time $t$.. In this way, EDMD with memory “remembers” what happened last
time similar dynamics occurred. If the closest saved spectral objects have
Wasserstein distance greater than $\varepsilon$, then the standard sliding
EDMD prediction is used. In practice, different values of $\varepsilon$ can be
defined for the Koopman eigenvalues and modes ($\varepsilon_{\lambda}$ and
$\varepsilon_{\text{mode}}$, respectively).

4) After prediction, the computed Koopman eigenvalues and modes from the
current time window are stored with all the other previously saved spectral
objects.

Note that in our current implementation, we only allow information from one
memory to be used, and for the prediction to be exactly what occurred before
(up to scaling, see below). Combining information across multiple similar
memories is an exciting direction that we will pursue in future work.

##  4 Results

###  4.1 Synthetic piecewise exponential data

To probe the predictive capabilities of EDMD with memory, we first designed a
synthetic data set that standard sliding EDMD performed poorly on. Namely, we
generated data coming from a piecewise exponential dynamical system described
by

$x_{t}=\lambda_{t}x_{t-1}(1+\eta\xi),$ (7)

where $\lambda_{t}\in\\{-1.0101,-0.99,0.99,1.0101\\}$ and is pseudo-randomly
re-sampled every $10$ time steps222To mitigate numerical instability, we
enforced the condition that the sign of $\lambda$ should change at each
transition, so that the magnitude of $x_{t}$ does not explode., and $\xi$ is a
random variable drawn from a stationary Gaussian distribution of mean $0$ and
variance $1$, with $\eta$ acting as a magnitude of the noise.

Figure 3: Episodic memory improves the ability of Koopman learning to predict
synthetic non-stationary exponential data. (A) The predictions of sliding EDMD
$\Delta=1,...,5$ time steps (red dots), often poorly match the true data. (B)
Utilizing EDMD memory (blue line) leads to $650\%$ improvement on predictions,
as compared to sliding EDMD (red line). Note that while a zoomed-in section is
shown here, the improvement was computed over a $1000$ time-step simulation.
(C) Because there are a finite (and small) number of possible values
$\lambda_{t}$ can take, there is repetition of transitions (examples
highlighted in green and orange) rectangles. (D) These repetitions can be
identified by characteristic Koopman spectral objects [colored the same as in
(C)].

As can be seen, the predictions from sliding EDMD does a poor job matching the
true evolution of $x_{t}$ when the sliding window includes data from two
different values of $\lambda_{t}$ (Fig. 3A, B). However, while the Koopman
models from these windows having information that is not directly helpful for
prediction, they do have identifiable Koopman eigenvalues and modes across
repeats of the same transitions (e.g.,
$\lambda_{t}=-0.99\rightarrow\lambda_{t}=1.0101$) (Fig. 3C, D). This suggests
that utilizing the episodic memory mechanism may lead to improvements in
prediction.

Because $x_{t}$ can grow and shrink with time (depending on whether
$|\lambda_{t}|>1$ or $|\lambda_{t}|<1$), time windows with matching dynamics
(and thus, similar Koopman eigenvalues and modes) can have different
magnitudes of $x_{t}$. This can complicate directly using previously observed
data as predictions. We correct for this by computing the ratio between the
current time point’s magnitude and that of the first time point in the
recalled time window, and using this ratio as a re-scaling factor for our
prediction. Additionally, we adaptively set the variance of the Gaussian
radial basis dictionary functions to be the maximum value in the current
sliding window, instead of the maximum value over the entire data set. This
helps mitigate slight differences in Koopman spectra that artificially arise
due to differences in scale.

We find that the episodic memory mechanism can indeed boost the predictive
capability of sliding EDMD on this data set (Fig. 3B). While not perfect, EDMD
with memory is more capable at accurately predicting the future state of
$x_{t}$, even when the data in its sliding window comes from two different
dynamical regimes (for instance, it can predict periods of oscillation,
$\Delta=5$ time steps ahead, from a sliding window that includes both
exponentially growing and oscillating data). This suggests that the episodic
memory mechanism is mitigating the repetition of the same mistakes, once a
given transition has been observed. To quantify the improvement, we compute
the median relative prediction error as
$100\%\times\left[\frac{\text{median}_{t}(|x_{t+\Delta}-\hat{x}_{t+\Delta}^{\text{EDMD}}|)}{\text{median}_{t}(|x_{t+\Delta}-\hat{x}_{t+\Delta}^{\text{mem}}|)}-1\right]$,
where $\hat{x}_{t+\Delta}^{\text{EDMD}}$ is sliding EDMD’s prediction $\Delta$
time steps ahead at time $t$, $\hat{x}_{t+\Delta}^{\text{mem}}$ is EDMD with
memory’s prediction $\Delta$ time steps at time $t$, and
$\text{median}_{t}(\cdot)$ is the median value taken over all values of time.
We find that EDMD with memory leads to an improvement of $650\%$ in median
relative prediction error, as compared to sliding EDMD (Fig. 3B).

###  4.2 U.S. Flu Cases

We now turn our attention to applying EDMD with episodic memory on real-world,
non-stationary data. As already mentioned in the Introduction, the number of
U.S. flu cases per week333Data from
https://gis.cdc.gov/grasp/fluview/fluportaldashboard.html. has repeated
structure (Fig. 1) that our episodic memory mechanism should be able to take
advantage of.

We find that adding the episodic memory mechanism to sliding EDMD yields an
improvement of almost $55\%$ on one-week ahead forecasting (Fig. 4A),
decreasing the median relative error from $22.6\%$ to $14.6\%$. Examples of
EDMD with memory’s predictions of individual yearly peaks (Fig. 4C) show
clearly better matching with the ground truth. Increasing the forecast horizon
to 5 weeks ahead, we again find that EDMD with memory can lead to large
improvements over sliding EDMD (Fig. 4B, D), decreasing the error by over
$28\%$.

Figure 4: Episodic memory improves the ability of Koopman learning to predict
the number of U.S. flu cases. (A)–(B) Predicted weekly number of U.S. flu
cases one and five weeks ahead, using sliding EDMD with (blue line) and
without (red line) episodic memory. Black line denotes true data. (C)–(D)
Zoomed-in windows of (A) and (B).

###  4.3 Washington D.C. Bike Share

Lastly, we apply EDMD with memory to predict the number of bikes in use, per
hour, from the Washington D.C. bike share program during January 1, 2011 –
December 31, 2012 444Data from https://www.kaggle.com/datasets/marklvl/bike-
sharing-dataset.. This data set is especially challenging, given that it
contains $17,000+$ time points and exhibits highly transient dynamics (Fig.
5A, B).

We find that EDMD with memory can provide improvements in median relative
prediction error of more than $110\%$ over sliding EDMD, in the case of one
hour ahead forecasting, and over $62\%$, in the case of four hour ahead
forecasting (Fig. 5). We believe this improvement in performance, relative to
the U.S. flu cases, comes from the bike share data having considerably more
time points. This enables the episodic memory mechanism to learn and leverage
more repeated patterns. Additionally, the variability in the data makes the
application of sliding EDMD more challenging, allowing information “recalled”
from the past, via the episodic memory mechanism, to have a larger impact.

Figure 5: Episodic memory improves the ability of Koopman learning to predict
the number of bikes in use, per hour, from the Washington D.C. bike share
program. (A) Predicted number of bikes used one hour ahead, from the
Washington D.C. bike share program using sliding EDMD with (blue line) and
without (red line) episodic memory. Due to the large number of time points
($17,000+$), the data has been zoomed-in around distinct five day (120 hour)
periods (top and bottom). Note that the statistics presented in the top box is
computed across the entirety of the data set. (B) Same as (A), but for four
hour ahead predictions.

##  5 Discussion

In this paper, we present an episodic memory mechanism that can be easily
integrated into the framework of Koopman learning (Fig. 2) and can improve the
predictive capabilities of sliding EDMD on non-stationary data (Figs. 3–5) by
enabling the “recall” of what occurred directly after episodes in the past
that had similar dynamical features. A simple consequence of this is that EDMD
with memory is better able to learn that transient increases are followed by
transient decreases in real-world data (Figs. 4 and 5). While our simple
implementation of Koopman learning with episodic memory (which we call “EDMD
with memory”) has limitations, it opens up a number of directions for
expanding the ideas developed here that we believe will push forward the state
of Koopman learning.

###  5.1 Limitations

While the presence of repeating structure in time-series data (e.g., Fig. 1B)
motivates the development of approaches that can identify and leverage them
for future prediction, the success of such methods are necessarily limited in
several ways.

First, the repeated structure that EDMD with memory identifies is related to
constraints and response properties of the system under study. For instance,
in the case of U.S. weekly flu cases, the repeated structure is related
healthcare infrastructure and supply chains, weather, and flu variants. When
these are relatively consistent over the span of several years, EDMD with
memory can be used to identify past data that is directly relevant. However,
in the case where healthcare infrastructure and supply chains, as well as
disease variants, rapidly evolve with time (as in the case of COVID-19), we
expect EDMD with memory to be less useful.

A second limitation has to do with the fact that EDMD with memory requires
having seen a given dynamical feature in the past to predict it. While it can,
in principle, learn the pattern in a single shot (making it considerably more
efficient than some standard ML methods), during the first exposure it does
not contribute to the prediction. Because sliding EDMD has been found to be a
powerful tool for data-driven prediction [8, 10], this should not be severely
limiting to the predictive capability of our approach. However, there may be
other methods, such as physics (or domain) informed techniques, that would be
better to deploy in the case where first identification is critical. However,
we note that the absence of a good match between the current set of spectral
objects and the library of saved spectral objects, provides information.
Namely, that a new regime, unlike what has been seen previously, is being
entered. If the library is created from a sufficiently large set of time
points, this could provide confidence that novel dynamics are being
encountered, and appropriate measures could be taken.

Finally, in our current implementation of EDMD with memory, all saved Koopman
eigenvalues and modes are given equal weighting, and only one, of possibly
several close (with respect to the Wasserstein distance) previously saved
memories, are used for prediction. Because this approach does not take into
account whether this past data is close in time to the current data and/or
whether this estimate came from data with high noise or uncertainty, we
imagine that this approach is sub-optimal and can be improved upon.

###  5.2 Expansions

There are several ways in which this preliminary development of Koopman
learning with episodic memory can be expanded. First, recent developments in
Koopman operator theory for reduced order modeling allows for the noisy
components of the Koopman spectra to be removed and the uncertainty in the
estimate to be quantified [45]. Integrating this into EDMD with memory will
enable better estimates of the true Koopman spectra, as well as better
matching between saved spectral objects. Second, a powerful capability of
Koopman operator theory is its natural connection to observables beyond the
state-space variables. As many state-space variables of interest (e.g., flu
cases) are naturally functions of other variables (e.g., weather), these
additional observables can be added to better approximate the dynamics, in
ways that existing predictive methods may struggle with. We imagine including
more observables into estimates of the Koopman operator will make EDMD with
memory more accurate. Finally, we believe that research into human memory and
decision making [46] can be connected with our work and used to inspire
further advances in Koopman learning.

### Acknowledgements

We thank the members of AIMdyn Inc. for useful discussion on our results and
Thomas Redman for pointing out [46]. This material is based upon work
supported by the DARPA Small Business Innovation Research Program (SBIR)
Program Office under Contract No. W31P4Q-21-C-0007 to AIMdyn, Inc. Any
opinions, findings and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of
the DARPA SBIR Program Office. Distribution Statement A: Approved for Public
Release, Distribution Unlimited.

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