The Model Forest Ensemble Kalman Filter Andrey A. PopovyzandAdrian Sanduz Abstract. Traditional data assimilation uses information obtained from the propagation of one physics-driven

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The Model Forest Ensemble Kalman Filter∗

Andrey A. Popov,† ‡ and Adrian Sandu‡

Abstract. Traditional data assimilation uses information obtained from the propagation of one physics-driven

model and combines it with information derived from real-world observations in order to obtain a

better estimate of the truth of some natural process. However, in many situations multiple simulation

models that describe the same physical phenomenon are available. Such models can have diﬀerent

sources. On one hand there are theory-guided models are constructed from ﬁrst physical principles,

while on the other there are data-driven models that are constructed from snapshots of high ﬁdelity

information. In this work we provide a possible way to make use of this collection of models in

data assimilation by generalizing the idea of model hierarchies into model forests—collections of

high ﬁdelity and low ﬁdelity models organized in a groping of model trees such as to capture various

relationships between diﬀerent models. We generalize the multiﬁdelity ensemble Kalman ﬁlter that

previously operated on model hierarchies into the model forest ensemble Kalman ﬁlter through a

generalized theory of linear control variates. This new ﬁlter allows for much more freedom when

treading the line between accuracy and speed. Numerical experiments with a high ﬁdelity quasi-

geostrophic model and two of its low ﬁdelity reduced order models validate the accuracy of our

approach.

Key words. Bayesian inference, control variates, data assimilation, multiﬁdelity, ensemble Kalman ﬁlter, re-

duced order modeling

MSC codes. 62F15, 62M20, 65C05, 65M60, 76F70, 86A22, 93E11

1. Introduction. In many situations the availability of multiple models that describe the

same physical system is a valuable asset for obtaining accurate forecasts. For example the

Coupled Model Intercomparison Project [10] used by the International Panel on Climate

Change is an eﬀort to utilize an aggregate of a wide array of climate models for the purposes

of increasingly accurate predictions. It is a recognition by the climate community that a

collection of models is greater than the sum of its parts.

The idea of leveraging a collection of models to improve data assimilation [1,20,28] has

seen an explosion of research over the last several years. Multilevel data assimilation was

ﬁrst developed in the context of Monte Carlo methods [13,14], wherein a hierarchy of models,

through successive coarsening in the time dimension, was used to perform inference with

the accuracy of the ﬁnest level coarsening with a larger and larger amount of samples from

the coarser levels. The ideas of multilevel Monte Carlo were transferred to the ensemble

Kalman ﬁlter (EnKF) in a series of works developing the multilevel ensemble Kalman ﬁlter

(MLEnKF) [4,5,17–19] aiming to provide more operationally viable methods.

∗Submitted to the ArXiv October 24, 2022.

Funding: The work of Popov and Sandu was supported by DOE through award ASCR DE-SC0021313, by NSF

through award CDS&E–MSS 1953113, and by the Computational Science Laboratory at Virginia Tech.

†Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX

(apopov@vt.edu)

‡Computational Science Laboratory, Department of Computer Science, Virginia Tech, Blacksburg, VA

(sandu@cs.vt.edu).

arXiv:2210.11971v1 [cs.CE] 21 Oct 2022

2 A. A. POPOV AND A. SANDU

The multiﬁdelity ensemble Kalman ﬁlter (MFEnKF) [7,23,25,26] circumvents numerical

diﬃculties present in the MLEnKF through a robust use of linear control variate theory. The

MFEnKF also extends the idea of model coarseness to arbitrary non-linear couplings between

high ﬁdelity (ﬁne level) and low ﬁdelity (coarse level) model states, allowing the use of various

types of reduced order models (ROMs) to form a model hierarchy.

This work further extends the EnKF ideas and brings two novel contributions. (i) First,

it extends model hierarchies to model trees and model forests, covering the situation were the

collection of models cannot neatly form a model hierarchy. (ii) Second, it extends the multiﬁ-

delity ensemble Kalman ﬁlter to the model forest Kalman ﬁlter allowing data assimilation to

make use of model forests in a rigorous way.

Given one high ﬁdelity model and a collection of low ﬁdelity models, it is not always

possible to organize them in a strict model hierarchy. Following this observation we introduce

the ﬁrst key contribution of the this work (i); we generalize the idea of model hierarchies to

model trees, where one model is allowed to have multiple low ﬁdelity models on the same level

below it; the low ﬁdelity models are surrogates for the high ﬁdelity one, but they may not

have a direct relationship with each other. This results in a tree structure of models with the

high ﬁdelity model acting as the root. We further extend model trees by leveraging the idea of

model averaging [8]. Assuming that we have a collection of model trees, each with their own

high ﬁdelity model at the root, we organize them in a “model forest” and build an averaging

procedure over all the trees in the forest.

By bringing together the ideas of the MFEnKF with that of model forests, we make the

second key contribution (ii) of this work; we replace the MFEnKF with the model forest

ensemble Kalman ﬁlter, which also has the acronym MFEnKF as we show that the former is

a special case of the latter.

Numerical tests on the Quasi-Geostrophic equations with a quadratic reduced order model

and an autoencoder-based surrogate show that our proposed extension signiﬁcantly decreases

the number of high ﬁdelity model runs required to achieve a certain level of analysis accuracy.

This paper is organized as follows. Relevant background information including the se-

quential data-assimilation problem, model hierarchies, model averages, and the multiﬁdelity

ensemble Kalman ﬁlter are presented in Section 2. The extension of model hierarchies to

model trees, and the extension of model averages to model forests is described in Section 3.

Next the extension of the multiﬁdeity ensemble Kalman ﬁlter to the model forest Kalman

ﬁlter is explained in Section 4. The quasi-geostrophic equations and two surrogate models are

detailed in Section 5. Numerical experiments on various model trees and model forests are

presented in Section 6. Finally, some closing remarks are stated in Section 7.

2. Background. We review relevant background on data assimilation, including model

hierarchies, linear control variates, model averaging, and the multiﬁdelity ensemble Kalman

ﬁlter.

2.1. Data Assimilation. Let Xt

idenote the state of some natural process at time ti, where

the superscript t represents ground-truth. Assume that we have some prior information about

this state represented by the distribution of the random variable Xb

i. Assume also that we

MODEL FOREST ENKF 3

have access to some sparse noisy observations of the truth represented by,

(2.1) Yi=H(Xt

i) + εi,

where His a non-linear observation operator and iis a random variable representing obser-

vation error. For the remainder of this paper we assume that the observation error is normal

with distribution

(2.2) εi∼ N(0,ΣYi,Yi).

Finally, assume we have some inexact numerical model Mthat approximates the dynamics

of the natural process, i.e., evolution of the truth,

(2.3) Xt

i=M(Xt

i−1) + ξi,

where the random variable ξirepresents the model error.

Data assimilation [1,9,28] seeks to combine the prior information Xb

iwith the sparse

noisy observations Yiinto a posterior representation Xa

iof the information, commonly through

Bayesian inference,

(2.4) π(Xa

i) = π(Xb

i|Yi)∝π(Yi|Xb

i)π(Xb

i),

where the distribution π(Xa

i) represents our full knowledge about the state of the system at

time ti.

The model (2.3) also forecasts the posterior information at time index i−ito prior

information at time i, through the relation,

(2.5) Xb

i=M(Xa

i−1) + ξi.

2.2. Notation. In this work, the mean of the random variance Xis denoted by, µX, and

the covariance between the random variable Xand the random variable Yis denoted by, ΣX,Y .

An ensemble of Nsamples from the random variable Xis denoted by, EX= [X1,X2,...,XN],

with the ensemble mean denoted by,

µX=

i=1

NXi,

the scaled ensemble anomalies denoted by,

AX=1

√N−1EX−e

µX1T

N,

where 1Nis a column vector of Nones, and the unbiased sample covariance between Xand

Ydenoted by, e

ΣX,Y =AXAT

4 A. A. POPOV AND A. SANDU

M(1)

M(1,1)

M(1,1,1)

Figure 2.1. A visual representation of a model hierarchy with two surrogate models. The principal model

,M(1), has a surrogate model , M(1,1), which in turn has its own surrogate model , M(1,1,1).

2.3. Model Hierarchies and Order Reduction. Assume there exists a model which is

expensive to compute from which we are attempting to glean some information through a

sampling procedure. Call this model the principal model. Assume that there exists a surrogate

model with which we can bootstrap our knowledge about the principal model. We can then

assume that the previously mentioned surrogate model is its own principal model in its own

model hierarchy that has its own surrogate model. This process can be repeated ad inﬁnitum

to obtain a model hierarchy of a desired size. Figure 2.1 provides an illustration of a model

hierarchy for one principal model which has a surrogate that itself has a surrogate.

Let the tuple Irepresent the index of a model in the model hierarchy, such that the model

MIhas a surrogate model MI·1, with ‘ ·’ representing tuple concatenation, e.g., (1,2) ·3 =

(1,2,3). This particular notation helps with deﬁning model trees and model forests later.

We make the following assumptions:

•The dynamics of the high ﬁdelity ‘principal’ model MIis embedded into the space

XI, i.e., MI:XI→XI.

•The dynamics of the low ﬁdelity ‘surrogate’ model MI·1is embedded into the reduced

space XI·1, i.e., MI·1:XI·1→XI·1.

•There exists a (possibly non-linear) projection operator that maps the states of the

principal model to its surrogate:

(2.6) θI·1:XI→XI·1.

•There exists an interpolation operator that reconstructs an approximation of the state

of the principal model from that of the surrogate model:

(2.7) φI·1:XI·1→XI,

•The two operators obey the right-invertible consistency property [26],

(2.8) θI·1◦φI·1= id,

MODEL FOREST ENKF 5

ensuring that reconstruction has the same representation of the full order information

in the reduced space.

2.4. Linear Control Variates for Model Hierarchies. We discuss the speciﬁc case of a

biﬁdelity model hierarchy, , with the high ﬁdelity model having one surrogate. Assume that

the information about our high ﬁdelity model run is represented by the distribution of the

random variable Xknown as the principal variate. Assume also that there exist two random

variables whose distributions describe the information about the surrogate model: the control

variate b

Uwhich is highly correlated to X, and the ancillary variate Uwhich is uncorrelated

with the other variates, but shares its mean with b

U. The variates X,b

Uand Uare known as

the constituent variates.

Given some (possibly non-linear) functions hand g, the total variate which describes the

total information of the hierarchy in the linear control variate framework is given by,

(2.9) Zh=h(X)−Shg(b

U)−g(U)i

where Sis known as the gain operator. The choice of hand glargely depends on, and deﬁnes,

the information that is encapsulated by the diﬀerent variates, and has to be carefully chosen

for each given problem.

Theorem 2.1. The optimal gain matrix Sthat minimizes the trace generalized variance of

Zin (2.9)is given by,

(2.10) S=Σh(X),g(b

U)Σg(b

U),g(b

U)+Σg(U),g(U)−1.

Proof. By [22], the derivative with respect to Sof the trace generalized variance of Zis

∂

∂Str(ΣZ,Z ) = −2Σh(X),g(b

U)+ 2SΣg(b

U),g(b

U)+Σg(U),g(U),

and as the Hessian is always symmetric positive deﬁnite,

∂2

∂S2tr(ΣZ,Z )=2Σg(b

U),g(b

U)+Σg(U),g(U)⊗I≥0,

the global minimum is attained when,

(2.11) ∂

∂Str(ΣZ,Z ) = 0,

which is satisﬁed by (2.10), as required.

We now describe the generalization to a model hierarchy. Assume that the ancillary variate

with indexing tuple Iis the total variate estimator for

(2.12) UI=hIXI−SI·1hgI·1b

UI·1−gI·1UI·1i,

with Zh:=U(1) representing the total variate

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TheModelForestEnsembleKalmanFilterAndreyA.Popov,yzandAdrianSanduzAbstract.Traditionaldataassimilationusesinformationobtainedfromthepropagationofonephysics-drivenmodelandcombinesitwithinformationderivedfromreal-worldobservationsinordertoobtainabetterestimateofthetruthofsomenaturalprocess.However,inm...

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The Model Forest Ensemble Kalman Filter Andrey A. PopovyzandAdrian Sanduz Abstract. Traditional data assimilation uses information obtained from the propagation of one physics-driven

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