A Random Batch Method for Eﬃcient Ensemble Forecasts of Multiscale Turbulent Systems Di Qiaand Jian-Guo Liub

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A Random Batch Method for Eﬃcient Ensemble Forecasts of

Multiscale Turbulent Systems

Di Qiaand Jian-Guo Liub

aDepartment of Mathematics, Purdue University, 150 North University Street, West Lafayette,

IN 47907, USA

bDepartment of Mathematics and Department of Physics, Duke University, Durham, NC 27708,

USA

Abstract

A new eﬃcient ensemble prediction strategy is developed for a general turbulent model framework with

emphasis on the nonlinear interactions between large and small scale variables. The high computational cost in

running large ensemble simulations of high dimensional equations is eﬀectively avoided by adopting a random

batch decomposition of the wide spectrum of the ﬂuctuation states which is a characteristic feature of the

multiscale turbulent systems. The time update of each ensemble sample is then only subject to a small portion

of the small-scale ﬂuctuation modes in one batch, while the true model dynamics with multiscale coupling

is respected by frequent random resampling of the batches at each time updating step. We investigate both

theoretical and numerical properties of the proposed method. First, the convergence of statistical errors in the

random batch model approximation is shown rigorously independent of the sample size and full dimension of

the system. Then, the forecast skill of the computational algorithm is tested on two representative models of

turbulent ﬂows exhibiting many key statistical phenomena with direct link to realistic turbulent systems. The

random batch method displays robust performance in capturing a series of crucial statistical features of general

interests including highly non-Gaussian fat-tailed probability distributions and intermittent bursts of instability,

while requires a much lower computational cost than the direct ensemble approach. The eﬃcient random batch

method also facilitates the development of new strategies in uncertainty quantiﬁcation and data assimilation for

a wide variety of complex turbulent systems in science and engineering.

1 Introduction and background

Turbulent dynamical systems appearing in many natural and engineering ﬁelds [27, 7, 36, 18, 38] are characterized

by a wide range of spatiotemporal scales in a high dimensional phase space. Small uncertainties in the multiscale

high-dimensional states can be rapidly ampliﬁed through the nonlinearly coupled dynamics and inherent instability

possessed by the turbulent ﬂow. These distinctive features give rise to a wide variety of complex phenomena such as

intermittent bursts of extreme ﬂow structures and strongly non-Gaussian probability density functions (PDFs) in

the key state variables [43, 26, 23, 42]. A probability representation for the evolution of the major ﬂow states is thus

essential to accurately quantify the uncertainty in the practical prediction of such turbulent systems. The ensemble

forecast through a Monte Carlo (MC) type approach estimates the evolution of the PDFs by tracking an ensemble

of trajectories solved independently from an initial distribution [15, 35, 16]. The empirical statistics of the ensemble

solutions are used to approximate the model uncertainty due to randomness from various internal and external

sources. A particular issue with large societal impacts is to accurately capture the non-Gaussian PDFs related to

the extreme event outliers [32, 8, 2] using the ﬁnite size ensemble. However, the ‘curse-of-dimensionality’ forbids

direct MC simulations of such high-dimensional systems especially in cases including strongly coupled multiscale

nonlinear interactions [35, 18]. A very large ensemble size is usually needed to suﬃciently sample the entire coupled

ﬂuctuation modes in a wide energy spectrum, while only a small number of solutions are aﬀordable in many practical

situations such as climate forecast [36, 10]. It remains a grand challenge to obtain accurate statistical estimates for

the key physical quantities from the multiscale interaction between the large-scale mean ﬂow and the interacting

high-dimensional small-scale ﬂuctuations.

arXiv:2210.00373v1 [physics.flu-dyn] 1 Oct 2022

In this paper, we propose an eﬃcient method for the understanding and ensemble forecast of a general group

of complex turbulent systems accepting coupled multiscale dynamics [23, 24]. We use the ideas in the Random

Batch Method (RBM) originally developed for interacting particle systems [12, 13, 9], and apply it to the very

diﬀerent problem of multiscale turbulent systems. Inspired by the stochastic gradient descent [34, 1, 11] in machine

learning, the RBM randomly divides and constrains the large number of interacting particles into small batches

in each time interval. The RBM can greatly reduce the computational cost in large particle systems and has got

many successful applications such as on manifold learning [8, 9] and quantum simulations [14]. In the ensemble

prediction of turbulent systems, we focus on the dominant ﬂow structure in the largest scale thus the required

ensemble size to sample a low-dimensional subspace can be controlled. This reduced modeling strategy usually

suﬀers diﬃculties in practice since the large scale is closely coupled with all the unresolved small-scale ﬂuctuations,

thus it is impossible to only perform ensemble simulation inside the large-scale subspace. Using random batches,

this diﬃculty is eﬀectively overcome by regrouping the large number of small-scale ﬂuctuating modes into small

batches each containing only a few modes. Then the batches from a single simulation of the ﬂuctuation modes are

used for updating diﬀerent ensemble members of the large-scale state separately during a short time step update.

This approximation is based on the important observation that the small-scale ﬂuctuations often decorrelate much

faster in time and contain less energy than the mean-ﬂow state on the largest scale. The batches of diﬀerent modes

are randomly resampled before each time updating step thus contributions from all scales are well characterized

during the evolution in time. In this way, we achieve an eﬃcient algorithm that gains high skill to capture the fully

non-Gaussian statistical feature in the most important large-scale mean ﬂow state, while greatly reduce the high

computational cost independent of the dimensionality of the full system.

In order to achieve a detailed analysis of the method for the complex turbulent system which usually combines

various complex eﬀects from diﬀerent sources, we develop the the new RBM algorithm on a uniﬁed class of turbulent

systems with emphasis on the explicit coupling between large and small scales, while the unresolved nonlinear

coupling among small scales is modeled by damping and white noise forcing. The simpliﬁed formulation reveals the

most important key physical processes on the nonlinear interaction between the large-scale mean ﬂow component

and the smaller scale ﬂuctuation components. On the other hand, the extra complexity due to the nonlinear

self-interactions among the less important small scales are avoided to provide a cleaner model setup. The model

framework is shown to have many representative applications in physical and engineering problems [17, 27, 39, 29,

31]. Precise error estimations are derived based on this general framework for the convergence of the RBM approach

using a ﬁnite number of samples. The convergence of the statistical quantiﬁes is proved based on the semigroups

generated by the backward Kolmogorov equations [40] of the RBM model. The RBM is then veriﬁed numerically

on two representative prototype turbulent models. Diﬀerent non-Gaussian statistics are observed in the two models

inferring strong intermittency and extreme events induced from distict physical mechanisms. The numerical tests

show accurate prediction of both transient and equilibrium PDFs recovering various non-Gaussian features under a

much lower computational cost requiring a much smaller ensemble size.

First in the following, we display the general structures that are representative in the turbulent dynamics and

illustrate the central issues in achieving an accurate probability solution.

General formulation for turbulent systems and challenges in eﬃcient statistical fore-

cast

The complex turbulent systems discussed above can be written as the following canonical equation about the state

variable uin a high-dimensional phase space

dt= Λu+B(u,u) + F(t) + σ(t)˙

W(t;ω).(1.1)

On the right hand side of the equation (1.1), the ﬁrst component, Λ = −D+L, represents linear dissipation and

dispersion eﬀects (with a negative-deﬁnite dissipation operator D < 0and a skew-symmetric dispersion operator

LT=−Las in [23]). One representative feature of such complex systems is the nonlinear energy conserving

interaction that transports energy across scales. The nonlinear eﬀect is introduced through a bilinear quadratic

form, B(u,u), that satisﬁes the conservation law u·B(u,u)≡0. External forcing eﬀects are decomposed into a

deterministic component, F(t), and a stochastic component represented by a Gaussian random process, σ˙

The evolution of the model state udepends on the sensitivity to the randomness in initial conditions and

external stochastic eﬀects. Combined with the inherent internal instability due to the nonlinear coupling term in

(1.1), small perturbations are ampliﬁed in time thus requiring a probabilistic description to completely characterize

the development of uncertainty in the model state u. The time evolution of the PDF p(u, t)can be found directly

from the solution of the associated Fokker-Planck equation (FPE) [40]

∂p

∂t =LFPp:=−divu[Λu+B(u,u) + F]p+1

2divu∇σσTp,(1.2)

with an initial condition p|t=0=µ0. However, it is still a challenging task for directly solving the FPE (1.2) as a high

dimensional PDE system. As an alternative approach, ensemble forecast by tracking the Monte-Carlo solutions [35]

estimates the essential statistics through empirical averages among a group of independently sampled trajectories

of (1.1). In particular, the ensemble members u(i)are sampled at the initial time t= 0 according to the initial

distribution µ0. The PDF solution p(u, t)at each time instant t > 0is then approximated by evolving each sample

independently in time according to the same dynamical equation.

Though simple to implement, a direct ensemble forecast running the original model (1.1) suﬀers several diﬃculties

in accurately recovering the key model statistics and PDFs in a high dimensional space. First, the ensemble size

required to achieve desirable accuracy will grow exponentially in direct ensemble simulation of the full model as

the dimension of the system increases. Second, the turbulent systems often contain strong internal instability and

multiscale coupling along the entire spectrum. Thus reduced models by simply truncating the stabilizing small-scale

modes [23] are not feasible to correctly represent the true model dynamics. Besides, diﬀerent orders of statistical

characteristics are fully coupled in the general formulation (1.1) so it is diﬃculty to identify the contributions of

diﬀerent scales especially when highly non-Gaussian statistics appear. These are the central diﬃculties we will

address in this paper using ideas in the RBM approximation.

In the following part of the paper, we ﬁrst propose a more tractable model framework with emphasis on the

explicit coupling between large and small scales in Section 2. Based on this model framework, the RBM algorithm

for ensemble prediction is developed in Section 3. Detailed error estimation and convergence analysis of the new

RBM method are obtained in Section 4. The performance of the method is veriﬁed on two prototype turbulence

models with practical importance in Section 5. A summary of this paper is given in Section 6.

2 A turbulent model framework with explicit multiscale coupling mech-

anism

We ﬁrst introduce a simpliﬁed modeling framework that enables us to focus on the key nonlinear large and small

scale coupling mechanism in the general system (1.1). One major diﬃculty in complex turbulent systems is the

strong nonlinear coupling across scales where the large-scale state can destabilize the smaller scales with small

variance, while the increased ﬂuctuation energy contained in the large number of small-scale states can inversely

impact the development of the coherent largest scale structure. To address this central issue of coupling with mixed

scales in modeling turbulent systems, we propose a mean-ﬂuctuation decomposition of the model state u, so that

the multiscale interactions can be identiﬁed in a natural way. To achieve this, we view uas a random ﬁeld and

separate it into the composition of a large-scale random mean state and stochastic ﬂuctuations in a ﬁnite-dimensional

representation under a pre-determined basis {vk}K

k=1 (for example, the Fourier expansion oﬀers a natural basis for

periodic boundary)

u(t;ω) = ¯

u+u0:=¯

u(t;ω) + 1

√K

k=1

Zk(t;ω)vk,(2.1)

where we use the overbar ‘•’ to denote a proper average operator. In this way, ¯

u(t;ω)represents the random mean

ﬁeld of the dominant largest scale structure (for example, the zonal jets in geophysical turbulence [17] or the coherent

radial ﬂow in fusion plasmas [4]); and Z(t;ω) = {Zk(t;ω)}K

k=1 are stochastic coeﬃcients measuring the uncertainty

in multiscale ﬂuctuation processes u0on the eigenmodes vk(with a zero averaged mean u0= 0). Usually, ¯

ucan be

represented in a much lower dimension dthan the dimension Kof full stochastic modes Zrepresenting ﬂuctuations

along a wide spectrum of scales. The state decomposition (2.1) enables us to analyze the individual contributions

from diﬀerent scale modes to the large and small scale dynamics. Therefore, we derive the governing equations for

the mean state and ﬂuctuation modes separately according to the decomposition (2.1).

First, by averaging over the original equation (1.1) and applying the mean-ﬂuctuation decomposition (2.1), the

evolution equation of the large-scale mean state ¯

uis given by the following dynamics

d¯

dt= Λ¯

u+B(¯

u,¯

u) + 1

k,l=1

ZkZl¯

B(vk,vl) + F.(2.2a)

Above, the small-scale nonlinear ﬂuctuating feedback to the large-scale mean dynamics is represented by the

quadratic coupling ZkZlwith the coupling coeﬃcients ¯

B(vk,vl). The term B(¯

u,¯

u)represents the self-interaction

within the mean state. Next, by projecting the ﬂuctuation equation to each orthogonal basis element viwe obtain

the evolution equation for the stochastic ﬂuctuation coeﬃcients

dZk

dt=1

l=1

γkl (¯

u)Zl+1

K3/2

m,n=1

ZmZnB0(vm,vn)·vk+σ(t)

K1/2˙

W(t;ω)·vk,(2.2b)

where γkl (¯

u) = [Λvl+B0(¯

u,vl) + B0(vl,¯

u)] ·vkcharacterizes the quasilinear coupling from the mean state ¯

uin

the ﬂuctuation modes Z. The interactions between the ﬂuctuation modes in diﬀerent scales are summarized in the

second term on the right hand side of (2.2b) with the ﬂuctuation coeﬃcients B0=B−¯

B. In addition, without loss

of generality, we assume that the deterministic forcing Fexerts on the large-scale mean state, while the ﬂuctuation

modes are subject to the coupled white noise forcing.

Still, the fully coupled mean-ﬂuctuation model (2.2) contains multiple linear and nonlinear interaction compo-

nents involving both large-scale mean ¯

uand small-scale ﬂuctuations u0, thus it may not be a desirable starting

model for identifying the central dynamics in multiscale interactions. Rather, we would like to propose a further

simpliﬁed model based on mean-ﬂuctuation interaction mechanism which only maintains the key large and small

scale interaction explicitly while eliminates the large number of small-scale self-interaction terms. Considering this,

we assume that the combined nonlinear feedback among diﬀerent small-scale modes (2.2b) can be parameterized

by independent damping and stochastic forcing. This leads to the simpliﬁed multiscale model with large-small scale

interaction

d¯

dt=V(¯

u) + 1

k,l=1

ZkZl¯

B(vk,vl) + F,

dZk

dt=1

l=1

γkl (¯

u)Zl−dkZk+σk˙

Wk, k = 1,··· , K.

(2.3)

Above, the linear and nonlinear eﬀects within the mean state, Λ¯

uand B(¯

u,¯

u), are summarized in a single term

V(¯

u). The model simpliﬁcation occurs in the small-scale dynamics (2.2b) for Zkwhich replaces the combined

nonlinear small-scale coupling ZmZnB0(vm,vn)·vkby equivalent damping and noise with parameters dk, σk. In

fact, this replaced term represents the higher-order moment feedback to the covariance dynamics from a detailed

statistical analysis of the moment equations [23]. The simpliﬁcation is derived from the important observation that

these small-scale modes are fast mixing (thus decorrelate fast in time) so the average of the large number of modes

plays an equivalent role as the linear damping and white noise as in the homogenization theory. The introduced

model parameters dk, σkare usually picked according to the equilibrium statistical spectrum Ek=σ2

2dk=E0|k|−s

[17] with sdetermining the energy decaying rate in the ﬂuctuation modes.

The resulting model (2.3) recovers the most essential coupling mechanism between the large-scale mean state

uand the small-scale ﬂuctuation modes Zkwhich is explicitly modeled through the quasi-linear operator γkl (¯

in the ﬂuctuation equations and quadratic feedback term in the mean equation. Notice that the strong internal

instability which is the key feature of turbulent systems is maintained in both the mean and ﬂuctuation modes

by coupling terms Vand γkl in (2.2a) and (2.2b). The only model approximation comes from parameterization of

the complicated self-interactions of small-scale ﬂuctuations. Using this simpliﬁed model avoids the various sources

of uncertainties from the ﬂuctuation scales so that the dominant large-small scale interaction is identiﬁed. The

thorough analysis of the fully coupled nonlinearly ﬂuctuation model (2.2b) will be left for the future study.

In addition, the multiscale model formulation (2.2b) enjoys the advantage of more ﬂexibility to run ensemble

simulations for statistical forecast, uncertainty quantiﬁcation and data assimilation in practical applications. A

wide variety of turbulent systems [29, 19] can be categorized into this framework so that it has wide validity in

developing the eﬃcient ensemble forecast methods. Two typical prototype models accepting the dynamical structure

(2.3) with a wide multiscale spectrum will be discussed in Section 5 displaying a wide variety of diﬀerent turbulent

features. In the following sections, we will develop eﬃcient ensemble forecast strategies based on this representative

turbulent model formulation (2.3).

3 Random batch method for ensemble forecast of turbulent models

Next, we propose an eﬃcient ensemble forecast method to describe the time evolution of the probability distribution

of the state u. The direct Monte-Carlo approach runs an ensemble simulation using Nindependent samples

u(i)=¯

u(i),Z(i), i = 1,··· , N, with ¯

u∈Rdthe large-scale mean state and Z={Zk}K

k=1 ∈RKthe entire small-

scale ﬂuctuation modes in a high dimensional space Kd. The samples are drawn from the initial distribution

u(i)(0) ∼µ0(u)at the starting time t= 0, and the time-dependent solution of each sample u(i)(t)is achieved by

solving the equation (2.3) independently in time. The resulting PDF at each time instant tis approximated by the

empirical ensemble representation,

p(u, t)'pMC (u, t):=1

i=1

δu−u(i)(t),u∈Rd+K.(3.1)

Associated with the PDF, the statistical expectation of any function ϕ(u)can be estimated by the empirical average

of the samples according to (3.1)

Epϕt(u)'EMCϕt(u) = 1

i=1

ϕu(i)(t).

In particular for the model (2.3), all the small-scale modes contribute to the mean state equation as a combined

feedback. As a result, even though we are mostly interested in the statistics from the mean state samples ¯

u(i)

in the relatively low dimensional subspace, the solution of entire Ksmall-scale modes Z(i)must be computed.

The K-dimensional equations for ﬂuctuation modes also need to be solved repeatedly Ntimes for all the samples

i= 1,··· , N. Thus, the direct ensemble method reaches a high computational cost of ONK2(d+ 1)for one

time step update. Furthermore, the required number of samples Nto maintain accuracy in the empirical PDF

(3.1) is also dependent on the system dimension (d+K)and will grow exponentially as Kincreases (known as the

curse-of-dimensionality [3, 6]). Therefore, this direct MC approach will quickly become computational unaﬀordable

as a larger Nis needed to resolve all the detailed small scale ﬂuctuations.

Here, we propose to reduce the computational cost in ensemble simulations of the turbulent models using the

idea in the eﬀective random batch method (RBM) [12, 13]. We focus on the ensemble sampling of the most dominant

mean state statistics ¯

u∈Rdin a much lower dimensional subspace dK. Thus accurate empirical estimation of

the marginal probability distribution of ¯

uin (2.3) can be reached using a much smaller ensemble size N1N

pRBM (¯

u) = 1

i=1

δ¯

u−¯

u(i),¯

u∈Rd.(3.2)

Accordingly, the expectation in the resolved mean state is computed by the empirical estimation, ERBMϕt(¯

u) =

N1Piϕ¯

u(i)(t). In the main idea of the RBM model, we no longer run the associated large ensemble simulation

of the full ﬂuctuation modes Z(i)∈RK×Nassociated with each mean state sample ¯

u(i). Instead, only one

stochastic trajectory of Z(t)is solved in time while the Kspectral modes are divided into smaller subgroups

(batches) for updating diﬀerent ensemble members ¯

u(i). The RBM approximation is introduced considering the

typical property of the turbulent system where the energy inside the single small-scale mode E|Zk|2, k 1decays

fast and decorrelates rapidly in time (see examples in Figure 5.1 and 5.4 of Section 5). On the other hand, ergodicity

in the stochastic ﬂuctuation modes [41, 25] implies that updating the mean state ¯

uusing fractional ﬂuctuation modes

at each time step with consistent time-averaged feedback can provide an equivalent total contribution without

altering the original statistical equations.

Next, we describe the detailed RBM approach for modeling turbulent systems. To accurately quantify the small-

scale feedback in the mean state dynamics, we introduce a partition In={In

i}of the mode index k= 1,··· , K at

the start of each time updating step t=tn−1. Thus, the full spectrum of modes is divided into N1small batches

of size p=|In

i|according to the total number of samples i= 1,··· , N1of ¯

u(i)

∪N1

i=1 In

i={k: 1 ≤k≤K}.(3.3)

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摘要：

ARandomBatchMethodforEcientEnsembleForecastsofMultiscaleTurbulentSystemsDiQiaandJian-GuoLiubaDepartmentofMathematics,PurdueUniversity,150NorthUniversityStreet,WestLafayette,IN47907,USAbDepartmentofMathematicsandDepartmentofPhysics,DukeUniversity,Durham,NC27708,USAAbstractAnewecientensemblepredicti...

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A Random Batch Method for Eﬃcient Ensemble Forecasts of Multiscale Turbulent Systems Di Qiaand Jian-Guo Liub

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