ENCODING NONLINEAR AND UNSTEADY AERODYNAMICS OF LIMIT CYCLE OSCILLATIONS USING NONLINEAR SPARSE BAYESIAN LEARNING

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ENCODING NONLINEAR AND UNSTEADY AERODYNAMICS OF

LIMIT CYCLE OSCILLATIONS USING NONLINEAR SPARSE

BAYESIAN LEARNING

Rimple Sandhu ∗

Department of Civil and Environmental Engineering

Carleton University

Ottawa, ON, Canada

Brandon Robinson

Department of Civil and Environmental Engineering

Carleton University

Ottawa, ON, Canada

Mohammad Khalil †

Quantitative Modeling & Analysis Department

Sandia National Laboratories

Livermore, CA, United States

Chris L. Pettit

Aerospace Engineering Department

US Naval Academy

Annapolis, MD, United States

Dominique Poirel

Department of Mechanical and Aerospace Engineering

Royal Military College of Canada

Kingston, ON, Canada

Abhijit Sarkar

Department of Civil and Environmental Engineering

Carleton University

Ottawa, ON, Canada

ABSTRACT

This paper investigates the applicability of a recently-proposed nonlinear sparse Bayesian learning

(NSBL) algorithm to identify and estimate the complex aerodynamics of limit cycle oscillations.

NSBL provides a semi-analytical framework for determining the data-optimal sparse model nested

within a (potentially) over-parameterized model. This is particularly relevant to nonlinear dynamical

systems where modelling approaches involve the use of physics-based and data-driven components.

In such cases, the data-driven components, where analytical descriptions of the physical processes

are not readily available, are often prone to overﬁtting, meaning that the empirical aspects of these

models will often involve the calibration of an unnecessarily large number of parameters. While it

may be possible to ﬁt the data well, this can become an issue when using these models for predictions

in regimes that are different from those where the data was recorded. In view of this, it is desirable

to not only calibrate the model parameters, but also to identify the optimal compromise between

data-ﬁt and model complexity. In this paper, this is achieved for an aeroelastic system where the

structural dynamics are well-known and described by a differential equation model, coupled with a

semi-empirical aerodynamic model for laminar separation ﬂutter resulting in low-amplitude limit

cycle oscillations. For the purpose of illustrating the beneﬁt of the algorithm, in this paper, we use

synthetic data to demonstrate the ability of the algorithm to correctly identify the optimal model and

model parameters, given a known data-generating model. The synthetic data are generated from a

forward simulation of a known differential equation model with parameters selected so as to mimic

the dynamics observed in wind-tunnel experiments.

Keywords

Aeroelasticity, unsteady aerodynamics, inverse problems, sparse learning, Bayesian inference, nonlinear

dynamics

∗Currently at National Renewable Energy Laboratory, USA

†

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering

Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National

Nuclear Security Administration under contract DE-NA-0003525.

arXiv:2210.11476v1 [cs.CE] 18 Oct 2022

1 Introduction

In this paper we demonstrate the applicability of the recently proposed nonlinear sparse Bayesian learning (NSBL)

algorithm [

] to a single degree of freedom (SDOF) aeroelastic oscillator that is undergoing low amplitude limit

cycle oscillations. This experimental setup has been studied extensively through experimentation [

], numerical

modelling of laminar separation ﬂutter using high-ﬁdelity large eddy simulations (LES) [

] and unsteady Reynolds

averaged Navier-Stokes (URANS) model [

]. Furthermore, the wind tunnel experiments have provided a reliable

test-bed for developing Bayesian techniques for system identiﬁcation and model selection for nonlinear dynamical

systems [

]. The work in [

] use standard methods of evidence-based Bayesian model selection, which

allows for the systematic comparison of a set of candidate models with varying degrees of complexity. The model

evidence as a criterion for model selection ensures a balance of favouring models with superior average data-ﬁt, while

penalizing models that are overly complex and thus prone to overﬁtting [

]. In this context, model complexity is

quantiﬁed by the KL-divergence of the parameter posterior probability density function (pdf) from the parameter prior

pdf. For parameters where there exists little prior knowledge, it is typical to assign non-informative priors, however the

width of the distribution used for the non-informative prior will inﬂuence the optimal complexity of the model. The

issue of sensitivity to prior width is addressed in [

], whereby the problem is reposed as a sparse learning problem.

Rather than non-informative priors, parameters with little prior information are assigned Gaussian automatic relevance

determination (ARD) priors. The precision (inverse of the variance) of these ARD priors are determined through

evidence optimization. In this re-framing of the inference problem, the optimal model is still quantiﬁed as such based

on the model evidence. In contrast to the previous approach, rather than proposing an entire set of nested candidate

models to determine the optimal model complexity, the current paper approaches the problem as an automatic discovery

of the optimal sparse model nested within a single (potentially) over-parameterized model. Herein lies an additional

beneﬁt of approaching the problem as a sparse learning task; it is only necessary to obtain the parameter posterior for a

single model, whereas standard methods require the calibration of each model in the candidate in order to then obtain an

estimate of the model evidence. The shortcoming of this approach lies in the fact that the optimization process involves

the use of Markov Chain Monte Carlo (MCMC) sampling at each iteration. This is addressed in the current NSBL

framework, which removes the use of MCMC from within the optimization loop, resulting in signiﬁcantly improved

computational efﬁciency.

The NSBL framework presented here is an extension of the sparse Bayesian learning (SBL) also known as the relevance

vector machine (RVM) algorithm [

]. Both methods are motivated by the desire to avoid overﬁtting during

Bayesian inversion. SBL/RVM and the similar Bayesian compressive sensing (BCS) algorithm [

] provide analytical

expressions for a sparse parameter posterior distribution owing to the analytical conveniences of the semi-conjugacy

that exists between the Gaussian likelihood functions, and Gaussian ARD priors that are conditioned on hyperpriors

that are Gamma distributions. The SBL methodology is extended to be applicable to nonlinear-in-parameter models

and for non-Gaussian prior distributions, as these both commonly arise in engineering applications. We provide the

minimum required mathematical details to understand the objectives of the algorithm and to provide a complete account

of all terms shown in the equations used in this paper. For the full detailed derivation and additional details, we refer the

reader to [1, 2].

2 Methodology: Nonlinear sparse Bayesian learning

The NSBL methodology is applicable to general nonlinear mappings,

f:φ7→ y

where the model operator

maps the

unknown model parameter vector

φ∈RNφ

to the observable model output

y∈RNy

. In this speciﬁc application,

represents the aeroelastic model,

are the deterministic system parameters and the stochastic parameters (relating to

the model error), and

are the system output. Sensor measurements of the system output

at discrete points in time

are denoted as

. The likelihood function

p(D|φ)

can be computed for any

, using the observations

, and these

observation may be noisy, sparse, and incomplete measurements of the system state. The purpose of the algorithm is to

obtain a data-optimal sparse representation of φusing Bayesian inversion, while removing redundant parameters.

NSBL operates within the following Bayesian framework; we seek the posterior distribution of the unknown model

parameters φconditioned on the data Dand hyperparameters α,

p(φ|D,α) = p(D|φ)p(φ|α)

p(D|α)=p(D|φ)p(φ|α)

Rp(D|φ)p(φ|α)dφ(1)

for given data and hyperparameters, the denominator, which represents the model evidence (or marginal likelihood or

type-II likelihood), is just a normalization constant. The parameter prior

p(φ|α)

is also conditional on the hyperparam-

eter. Though the objective is not to perform full hierarchical Bayesian inference, we nevertheless deﬁne a prior on

p(α)

(which is notably absent in the expression above); this hyperparameter prior (or hyperprior) becomes relevant during

the optimization of α.

The following sections outline the three principal tasks involved in the NSBL framework, as depicted in Figure 1.

Namely:

(i)

in section 2.1 we discuss the assignment of a hybrid prior, wherein we distinguish between a priori relevant

parameters and questionable parameters, assigning known priors and ARD priors, respectively,

(ii)

in section 2.2, we detail the incorporation of data and the physics-based model through the construction of a

Gaussian mixture model (GMM) over samples generated from the product of the likelihood function and the

known prior, and

(iii)

in section 2.3 we discuss the optimization of the hyperparameters. The derivation of various semi-analytical

entities that enable the NSBL methodology is outlined in A.

(i) (ii) (iii)

Identify a priori relevant

parameters Assign known prior pdf

GMM of the likelihood

times known prior

Computation of

semi-analytical

entities

Assign ARD prior pdf

& Gamma hyperprior pdf

Identify questionable

parameters

Likelihood function

computation

Optimization of

hyperparameters

Figure 1: Summary of the main steps involved in the NSBL algorithm.

2.1 Hybrid prior pdf

The model parameter vector

is ﬁrst decomposed as

φ={φα,φ-α}

, distringuishing between the set of parameters

that are known to be relevant a priori, denoted as

φα∈RNα

, and the parameters that the modeller has deemed to

be questionable, denoted

φ-α∈RNφ−Nα

. This classiﬁcation as questionable encompasses any parameter for which

little or no prior knowledge exists, where a non-informative prior with large support would usually be used. The

vector of questionable parameters are the set of parameters among which we will induce sparsity, as a subset of these

parameters may be redundant. The mechanism for inducing sparsity follows SBL, where

φα

is assigned a Gaussian

ARD prior

p(φα|α) = N(φα|0,A−1)

. This prior is a normal distribution, whose mean vector is an

Nα×1

zero

vector, with a covariance matrix of

A−1

, where

is the precision matrix. Following SBL [

], prior independence

of the questionable parameters

φα

is assumed, hence, the precision matrix is diagonal,

A=diag(α)

. Furthermore,

each parameter

φi∈φα

has a unique variable precision

αi

, such that we can write

p(φi|αi) = N(φi|0, α−1

. The

hyperparameter

αi

dictates the prior precision of parameter

φi

; where low precision (or high variance) reduces to a

non-informative prior, and conversely, a high precision (or low variance) results in an informative prior with a mean of

zero. In the limit where the precision tends to inﬁnity, the ARD prior becomes a Dirac delta function centered at zero,

effectively pruning the parameter. Hence, the motivation behind NSBL is that optimally selecting

, can allow us to

discover the model having the optimal complexity given the available data. The optimization criteria and methodology

are presented later in section 2.3.

The joint prior pdf of φis summarized as [2]

p(φ|α) = p(φ-α)p(φα|α) = p(φ-α)N(φα|0,A−1).(2)

This hybrid prior pdf enables sparse learning of questionable parameters in

φα

through the use of an ARD prior

p(φα|α)

while incorporating prior knowledge about parameters

φ-α

through an informative prior

p(φ-α)

. The ARD

prior is a conditional Gaussian distribution, whose precision depends on the hyperparameter

. The marginal hyperprior

pdf

p(αi)

is chosen to be a Gamma distribution. Given the assumption of prior independence, the joint hyperprior

p(α)

is written as

p(α) =

Nα

i=1

p(αi) =

Nα

i=1

Gamma(αi|ri, si) =

Nα

i=1

sri

Γ(ri)αri−1

ie−siαi,(3)

where

Gamma(αi|ri, si)

denotes a univariate Gamma distribution parameterized by shape parameter

ri>0

and rate

parameter

si>0

. The use of a Gamma distribution as the hyperprior allows us to enforce the requirement that the

precision parameters

be positive. Furthermore, for speciﬁc combinations of shape and rate parameters,

and

the Gamma function can assume many forms of informative and non-informative priors. For instance, using values

si≈1

and

ri≈0

gives a ﬂat prior over

αi

, or values of

si≈0

and

ri≈0

gives Jeffreys prior, which is a ﬂat

over

log αi

. In fact, for reasons discussed in later sections, the NSBL algorithm operates on the natural logarithm of

the hyperparameters rather than the hyperparameters directly. For this reason, we choose to use Jeffreys prior for the

numerical results section.

Using a univariate transformation of random variables [16], the hyperprior in Eq. (3) becomes

p(log α) =

Nα

i=1

p(log αi) =

Nα

i=1

p(αi)

d

dαilog αi

Nα

i=1

sri

Γ(ri)αri

ie−siαi.(4)

2.2 Gaussian mixture-model approximation

After deﬁning the joint parameter prior distribution as in Eq. (2), we substitute the resulting expression into the

conditional posterior distribution from Eq. (1), yielding [2]

p(φ|D,α) = p(D|φ)p(φ-α)N(φα|0,A−1)

p(D|α).(5)

Given that the ARD priors are normally distributed, NSBL constructs a GMM approximation of the remaining terms

in the numerator,

p(D|φ)p(φ-α)

, which enables the derivation of semi-analytical expressions for many entities of

interest. Obtaining expressions for the model evidence and objective function (A.1), the parameter posterior (A.2), and

the gradient and Hessian of the objective function (A.3) makes the optimization of the hyperparameters analytically

tractable. Moreover,the use of a GMM enables the preservation of non-Gaussianity in both the likelihood function and

the known prior. The construction involves the estimation of kernel parameters a(k),µ(k), and Σ(k),

p(D|φ)p(φ-α)≈

k=1

a(k)N(φ|µ(k),Σ(k)),(6)

where

a(k)

, and

N(φ|µ(k),Σ(k))

are the total number of kernels, the kernel coefﬁcient (

a(k)>

0), and a Gaussian

pdf with mean vector

µ(k)∈RNφ

and covariance matrix

Σ(k)∈RNφ×Nφ

[

]. For a Gaussian likelihood function and a

Gaussian known prior, this reduces to the case of SBL or RVM [

], and a single kernel is sufﬁcient. Otherwise, except

in the case of a Laplace approximation [

] of the likelihood function times the known prior, multiple kernels will

generally be required. The construction of the GMM typically involves the use of MCMC in order to generate samples

from the arbitrary distribution, which requires repeated function evaluations for different samples of the unknown

parameter vector. The model itself operates as a black-box, thus, the analytical form of the model does not need to

be known; the model is only needed in order to compute the likelihood function. Once samples have been generated

from the posterior distribution, the estimation of the kernel parameters in Eq. (6) can be performed numerically using

methods such as kernel density estimation (KDE) or expectation maximization (EM) [

]. Since the construction of the

GMM involves numerous forward solves of the model, this step is the most computationally demanding component

of the algorithm. Notably, the GMM itself is independent of the hyperparameters, thus this process only needs to be

performed once at the onset, and does not need to be repeated during the optimization of the hyperparameters.

2.3 Sparse learning optimization problem

The critical step in the NSBL algorithm is the optimization of the hyperparameter,

. Within a hierarchical Bayesian

framework, we seek a point estimate for the hyperparameters, rather than obtaining posterior estimates thereof. As in

SBL, we perform type-II maximum likelihood, seeking the values

, which maximize the hypeperparameter posterior,

p(α|D) = p(D|α)p(α)

p(D)∝p(D|α)p(α).(7)

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ENCODINGNONLINEARANDUNSTEADYAERODYNAMICSOFLIMITCYCLEOSCILLATIONSUSINGNONLINEARSPARSEBAYESIANLEARNINGRimpleSandhuDepartmentofCivilandEnvironmentalEngineeringCarletonUniversityOttawa,ON,CanadaBrandonRobinsonDepartmentofCivilandEnvironmentalEngineeringCarletonUniversityOttawa,ON,CanadaMohammadKhalilyQ...

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ENCODING NONLINEAR AND UNSTEADY AERODYNAMICS OF LIMIT CYCLE OSCILLATIONS USING NONLINEAR SPARSE BAYESIAN LEARNING

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