=Paper=
{{Paper
|id=Vol-2587/article_2
|storemode=property
|title=A Weighted Sparse-Input Neural Network Technique Applied to Identify Important Features
for Vortex-Induced Vibration
|pdfUrl=https://ceur-ws.org/Vol-2587/article_2.pdf
|volume=Vol-2587
|authors=Leixin Ma,Themistocles Resvanis,Kim Vandiver
|dblpUrl=https://dblp.org/rec/conf/aaaiss/MaRV20
}}
==A Weighted Sparse-Input Neural Network Technique Applied to Identify Important Features
for Vortex-Induced Vibration==
https://ceur-ws.org/Vol-2587/article_2.pdf
A weighted sparse-input neural network technique applied to identify
important features for vortex-induced vibration
Leixin Ma, Themistocles L. Resvanis, J. Kim Vandiver
Department of Mechanical Engineering, Massachusetts Institute of Technology
leixinma@mit.edu
Abstract To identify the key mechanisms and the governing di-
Flow-induced vibration depends on a large number of pa- mensionless parameters behind the complicated fluid-
rameters or features. On the one hand, the number of candi- structure interaction process, extensive investigations have
date physical features may be too big to construct an inter- been made through structural response measurement, flow
pretable and transferrable model. On the other hand, failure
visualization and various force modeling techniques
to account for key dependence among features may over-
simplify the model. Feature selection is found to be able to (Sarpkaya 2004). VIV research over the past decades has
reduce the dimension of the physical problem by identifying revealed that Strouhal number, Reynolds number, mass
the most important features for a certain prediction task. In ratio, damping parameter etc. are all relevant VIV features
this paper, a weighted sparse-input neural network (Vandiver 1993, Govardhan and Williamson 2006, Van-
(WSPINN) is proposed, where the prior physical knowledge
diver et.al. 2018). However, if one is only interested in
is leveraged to constrain the neural network optimization.
The effectiveness of this approach is evaluated when ap- predicting a certain quantity of interest, such as cylinder’s
plied to the vortex-induced vibration of a long flexible cyl- vibration amplitude in the crossflow direction, some of
inder with Reynolds number from 104 to 105. The important these candidate features may be redundant or unimportant.
physical features affecting the flexible cylinders’ crossflow Feature selection algorithms are intended to extract the
vibration amplitude are identified.
most important features out of the full set of candidate fea-
tures with the goal of keeping prediction accuracy at a de-
Introduction sirable level, but with a reduced set of features. A pre-
analysis of a features’ importance can be conducted by
Vortex-induced vibration (VIV) is a multi-physics problem examining the statistical correlations among the features.
associated with a number of features (or variables) that However, the statistical analysis often fails to consider the
characterize either the structure or the flow individually or complicated interactions among the physical input parame-
their interaction. As flow passes around a cylinder, the ters (features). To solve this problem, the importance of
wake becomes unstable. The periodically shed vortices each feature subset can be assessed according to their pre-
induce unsteady forces on the cylinder which lead to VIV. diction accuracy using a learning machine, such as deep
Moreover, the VIV of long cylinders in ocean currents may neural network (DNN). Several learning machine-based
vary from single mode dominated, narrow-band random feature selection approaches have already been developed
vibration to multi-mode response, characterized by broad- to iteratively search the optimal feature subset that gives
band random vibration. Different current profiles may similar prediction accuracy as the full feature set, but they
cause structural vibration with standing waves or travelling can be computationally expensive especially when the
wave patterns (Bourguet et.al. 2011; Vandiver et.al. 2018). number of input features becomes very large (Guyon
The complexity of the nonlinear fluid-structure interaction 2003).
process, especially for the VIV of long, flexible cylinders To efficiently identify important features in a learning
in high Reynolds numbers fluid flows, precludes exact machine, several regularization techniques are introduced
analytical solutions and CFD simulations are not yet up to to the machine learning process. Rudy et. al. (2017) devel-
the task. oped a sequential threshold ridge regression, which helped
discover governing partial differential equations of a sys-
tem from measured time series. Inspired by the effective-
Copyright © 2020, for this paper by its authors. Use permitted under
ness of the group lasso regularization in linear regression,
Creative Commons License Attribution 4.0 International (CC BY 4.0). Feng (2017) and Scardapane (2017) developed a sparse-
input neural network by imposing group lasso regulariza- the input feature and the neuron in the first hidden layer,
tion on the weight groups connecting each input neuron. respectively. The magnitude of weight group for feature p
The effectiveness of the approach was demonstrated is measured by W p . Since the l0 norm is non-convex and
through theoretical derivations and empirical evidence. non-differentiable, l1 norm, which calculates the sum of
However, for physical problems, some of the system’s absolute values of the vectors, is often used as a convex
properties may be known in advance or can be obtained proxy (Tibshirani, 1996). It can be shown geometrically
from the governing physical laws and dimensional analysis that l1 norm is the closest convex approximation for l0
(Sonin, 2001). Studies have shown that incorporating the norm (Rosasco, 2010). Following the convex approxima-
prior physical knowledge can help build more interpretable tion, we obtain Equation (1),
machine learning models (Ye et. al 2018). 1 N 2
( )
M
In this paper, the sparse-input neural network proposed min ∑ L y ( ) , yˆ ( ) + λ ∑ ∑ wpm
w N
n n
( ) (2)
by Feng (2017) and Scardapane (2017) is modified to effi- = n 1 = p∈P m 1
ciently identify the important features on top of prior phys-
The second term in Equation (2) introduces bias term for
ical information. Comparison with searching all combina-
prediction. The hyperparameter λ is known as the group
tions of additional features shows its effectiveness in build-
lasso penalty, which adjusts the sparsity of the input fea-
ing compact predictive models, while maintaining prior
tures versus the prediction accuracy. When λ grows, the
physical information. The method was applied to the VIV
neural network will try to minimize the sum of the weight
response amplitude prediction problem at dominant vibra-
groups, and therefore more weight groups are likely to
tion frequencies. On top of the Reynolds number and
shrink to near 0. The input features with nonzero weight
damping parameter, the in-line-cross-flow coupling and
groups are the remaining features that contribute to the
modal participation are found to be important global VIV
prediction. In this way, the model can be built out of fewer
features.
input features, but the prediction accuracy may decrease
due to the loss of information.
Weighted sparse-input neural network
(WSPINN) incorporating prior physical
knowledge
We consider a fully connected DNN with P input features
x ∈ R P in the input layer and M neurons in the first hidden
layer that predict a certain target output y ∈ R1 . The weight Figure 1: A DNN example with 3 inputs and 4 neurons in the first
hidden layer
connecting the pth input feature and mth neuron in the first
hidden layer is denoted as wpm. Figure 1 shows an example
of the DNN with P=3 and M=4. The sparse-input neural However, for many physical problems, some of the fea-
network (Feng 2017; Scardapane 2017) aims at accom- tures are known to be important in advance, which are
plishing two tasks simultaneously: On the one hand, it min- termed as prior knowledge. In this case, the objective is to
imizes L ( y, yˆ ) , which is the prediction error (or loss) select a small number of additional features that will com-
between the predicted ŷ and the measured y. Meanwhile, plement the input features that are considered prior
it tries to constrain the number of input features to the knowledge and lead to predictions of acceptable accuracy.
DNN to be no greater than k. To implement this constraint, Since the conventional sparse-input neural network cannot
we need to group the weights outgoing from the same input tell the difference between prior knowledge and additional
feature together, and then limit the number of non-zero features, the optimization objective in Equation (2) needs
weight groups to be no larger than k. Hence, the mathemat- to be modified as follows,
ical expression for the optimization objective can be ex- M M
2
min L ( y, yˆ ) + s p λ ∑ ∑ ( wpm ) + sa λ ∑ ∑ ( wpm )
2
(3)
pressed as, w
= p∈Pp m 1 = p∈Pa m 1
min L ( y, yˆ )
w Where Pp denotes the feature set representing the prior
subject to knowledge, and Pa denotes the set of all the additional fea-
(1) tures. The parameters sp and sa are the weights assigned to
M
the prior knowledge and additional features, respectively.
w 0 { p : W p ≠ 0}= ∑(w )
2
= ≤ k p: ≠ 0 ≤ k
pm
These weights represent the level of confidence on the fea-
m =1
ture’ importance for prediction (Lian, 2018). The conven-
Where w 0 is the l0 norm of weight vector w. |…| is the tional sparse-input neural network in Equation (2) is a spe-
cardinality of the weight groups; p and m are the index for cial case for the weighted formulation in Equation (3),
where sa=sp=1, which assumes equal confidence for all the at every instant Hence, the parameterization for VIV force
features’ importance. For the prior knowledge (i.e., feature in the “power-in” region may involve,
set known to be important), we’d like to prevent the algo- Fcf = f (U ( z ) , x ( z, t ) , y ( z, t ) , ρ , µ , L, L in , D, m ( z ) ,
rithm from minimizing their weight groups to near 0, hence (6)
cs ,cf ( z ) , ch,cf ( z ) , P ( z, t ) , EI ( z ))
sp/sa should be set close to 0.
Fil = f (U ( z ) , x ( z , t ) , y ( z , t ) , ρ , µ , L, L in , D, m ( z ) ,
(7)
Relevant features for long flexible cylinders cs ,il ( z ) , ch ,il ( z ) , P ( z , t ) , EI ( z ))
subjected to vortex-induced vibrations Where ρ , µ , L and D are fluid density and dynamic
viscosity, cylinder’s length and diameter, respectively.
If the spatiotemporal root-mean-square (rms) amplitude
Flexible cylinder VIV modeling
of crossflow vibration Arms,cf in the power-in region is the
Figure 2 is a sketch of a tensioned elastic cylinder under a target output, then from Equations (4)-(7), the predictive
linearly sheared current profile U(z) distributed along axis model can be expressed as,
z, which causes the cylinders’ vibration in both the inline Arms ,cf = f (U ( z ) , x ( z , t ) , y ( z , t ) , ρ , µ , L, L in , D, m ( z ) ,
(IL) and crossflow (CF) directions with respect to the in- (8)
coming current. The vibration of the elastic cylinder can be cs ,cf ( z ) , ch ,cf ( z ) , P ( z , t ) , EI ( z ))
approximated as a tensioned Euler-Bernoulli beam. The It can be observed that Equations (6)-(8) involves spa-
equation of motion in crossflow direction and inline direc- tial-temporal distribution of structural response and system
tion can be expressed as, properties, which will be further simplified and represented
∂2 y ∂y ∂2 y ∂4 y by some global VIV features.
m ( z ) 2 + cs ( z ) − P ( z , t ) 2 + EI ( z ) 4 = Fcf (4)
∂t ∂t ∂z ∂z
Spatial-temporal analysis for typical VIV
∂2 x ∂x ∂2 x ∂4 x
m ( z ) 2 + cs ( z ) − P ( z , t ) 2 + EI ( z ) 4 = Fil (5) VIV measurements from the 2011 Shell experiments on a
∂t ∂t ∂z ∂z
38-m-long cylinder (Lie et al 2013) were studied in this
investigation.
Where x and y are the displacement in inline and cross-
The measured crossflow displacements in a linearly
flow direction. m(z) is the cylinder’s mass per unit length,
sheared current are presented in Figure 3. The top figures
P(z,t) is tension of the vibrating cylinder, EI(z) represents
are the CF response time series at two locations within the
bending stiffness. cs is the structural damping coefficient
“power-in” region, while their corresponding wavelet
per unit length, Fcf and Fil are the vortex induced forces on
analysis is shown at the bottom. The vibration is found to
the cylinder. The loading transfers energy from fluid to the
be narrow-banded with the dominant frequency 𝜔𝜔𝑐𝑐𝑐𝑐 drift-
structure in a well-defined region with length Lin, which is
ing in time. Given the dominant vibration frequency and
the “power-in” region. Outside this region, the vortex load-
structural properties, the corresponding wavenumber k 𝑐𝑐𝑐𝑐
ing dissipates energy by transferring energy from the struc-
can be estimated by the dispersion relationship.
ture to the fluid through hydrodynamic damping coeffi-
cient ch(z). The location of the “power-in” region can be
identified from structural vibration measurements in exper-
iments or simulation (Rao 2015). Under steady-state, nar-
row-banded vibration, the total power dissipation in the
flexible pipe can be normalized to an equivalent damping
coefficient ce (Vandiver et.al. 2018).
Figure 2: Side and front view of a cylinder under VIV
Figure 3: Top: Time series of crossflow displacement at two loca-
tions in the power-in region; Bottom: Wavelet analysis on the
The VIV loading is the result of nonlinear interaction
measured time series. (2011 Shell experiment, D=30 mm, linearly
between vortex shedding and structural vibration via com-
sheared flow, Umax=1.6 m/s)
plicated feedback mechanisms that depend on the structur-
al properties, the current profile and the structure’s motion
Meanwhile, Figure 4 shows the corresponding spatial- Arms ,cf f (U rms , ∆U , Arms ,il , ωcf , ωil , k cf , k il , α cf , α il ,
=
temporal distribution of crossflow displacement for the (9)
κ cf , κ il , ρ , µ , L, L in , D, m, cs , ce,cf , ce,il , P0 , P, EI )
same test condition. The response is nonstationary, with a
mixture of standing wave and travelling wave components. Where Urms and ∆U / L are the spatial root-mean-square
To better capture the temporal variation of the vibration and the shear gradient of the current profile, respectively
signal, a moving window analysis is conducted. The vibra- within the power-in region. Arms,cf and Arms,il are the spatio-
tion signal is windowed into overlapping time frames over temporal rms for the crossflow and inline VIV amplitude
each 3 vibration cycles, with 75% overlap. in the power-in region. cs and ce are respectively, the struc-
Complex proper orthogonal decomposition (POD) is tural damping coefficient and the equivalent rigid cylinder
conducted on the crossflow displacement in each spatial- damping coefficients that will lead to the same power dis-
temporal window in the “power-in” region to decompose sipation as discussed by (Vandiver et.al. 2018). P0 and P
the displacement in each window into several orthogonal are the initial tension before VIV and the mean tension
complex modes (Feeny 2008). The ratio between the mod- during the VIV process, respectively.
al energy of the dominant POD mode and the total energy Non-dimensionalizing Equation (9) gives,
is defined as κ , which suggests the dominance of the prin- Acf* = f (Re, β , Ail* ,Vrcf ,Vril , Lk cf , α cf , α il , κ cf , κ il ,
cipal mode. Additionally, by comparing the real and imag- P L2 (10)
L / L in , L / D, m*ζ , ccf* , cil* , 0 , P )
inary component of the dominant complex mode, the trav- EI ( EIk cf2 )
elling wave index α can be defined, with α = 1 for travel- Where Acf* = Arms , cf / D , Ail* = Arms ,il / D are the dimension-
ling waves, and α = 0 for standing waves (Feeny 2008). less crossflow and inline response amplitude, respectively.
The middle and bottom of Figure 4 shows the temporal Re = ρU rms D / µ is Reynolds number;
= β ( D / U rms )( ∆U / L )
variation of the travelling wave index and the modal domi-
nance factor analyzed in the power-in region, which sug-
is known as the shear parameter; Vrcf = 2π U / ωcf D , ( )
Vril = 2π U / (ωil D ) are the crossflow and inline reduced
gests that the VIV process is single POD mode dominated,
but the mode may vary from standing to travelling waves.
velocities, respectively; m*ζ = 4mζ / πρ D 2 is known as ( )
the mass damping parameter in the VIV literature, which
Analysis from inline vibration also shows similar spatial- historically has been thought to be important in controlling
temporal distribution. rigid cylinder’s VIV amplitude. ccf* = 2ce , cf ωcf / ρU rms 2
and
cil = 2ce,il ωil / ρU rms are the dimensionless forms of the
* 2
equivalent damping parameter in the crossflow and inline
directions, respectively (Vandiver et.al. 2018).
Although Equation (10) suggests that crossflow response
prediction in the “power-in” region may require consider-
ing the effect of all the 17 dimensionless variables, it is
likely that the dimension of the input features can be fur-
ther reduced due to redundancy or correlation between
features or irrelevance to the prediction target. We are in-
terested in finding a smaller and more manageable subset
of parameters that are ultimately the most important out of
the full set when it comes to determining the CF response
Figure 4: Contour plot of CF vibration around the dominant fre- amplitude. The motivation behind this is our interest to
quency, its corresponding travelling wave index α cf , and mode understand what causes the CF response variability that is
dominance factor κ cf in the estimated “power-in” region observed in the temporal domain. At the very least, we
(z/L=0~0.3) would like to start associating changes to certain parame-
ters with that variability that is often observed but is too
Dimensional analysis for narrow-banded VIV complicated to understand.
process
For a homogeneous, tensioned cylinder in uniform or line-
Feature selection for flexible cylinder VIV
arly sheared current undergoing narrow-banded VIV,
Equation (8) can be approximated by the following rele-
vant global quantities,
Dataset description
The dataset is from a set of experiments conducted by
Shell Oil Co. in 2011 at Marintek. The vibration of 38-
meter-long cylinders under various current profiles were
measured. The test matrix included two pipes with differ-
ent diameters (30-mm and 80-mm) but of the same bend- two parameters are designated to be used as prior
ing stiffness. The cylinders were tested in uniform and lin- knowledge. The shear parameter β which is ideally suited
early sheared current profiles with the maximum flow to differentiating between uniform or sheared flows was
speed, Umax, ranging from 0.5 m/s to 2.5 m/s. This resulted the third parameter that was chosen as prior knowledge
in the Reynolds number Re ranging from 1.0 ×104 – before starting the feature selections process.
20×104. The dataset also included cases where the 80-mm
pipe was covered with strakes over 50% of its length. The Feature selection on top of prior physical
pipe tests were conducted in uniform flows with Umax vary- knowledge
ing from 0.5 m/s to 1.5 m/s. The strakes dissipated vibra-
The feature selection procedure was conducted by increas-
tion energy and limited the power-in region to Lin = 0.5L ,
ing the hyperparameter λ from 0.01 until all the input fea-
50% of the cylinder’s length. Detailed descriptions of the
tures except the prior knowledge shrank to 0. Figure 5
experiments can be found in Lie (2013) and Rao (2015).
demonstrates how varying the value of λ determines the
The structural damping ratio ζ in the experiment was
number of features chosen by the proposed algorithm. In
around 0.5% (Vandiver et.al. 2018). The cross flow re-
the figure the retained features are indicated by the pres-
duced velocity Vrcf varies in a narrow range from 6 to 9
ence of a black bar at each λ value tested.
and Vrcf / Vril ≈ 2 .
Deep neural network setup
The deep neural network was constructed using two hidden
layers. Each hidden layer had twenty neurons using a sig-
moid activation function. The total number of data points
was around 6000. 70% of the experimental data were used
as the training data, while the rest was used as the test data.
The input variables x were standardized to keep the fea-
tures at the same scale, while the output variables y were
normalized to values between 0 to 1. The mean absolute
percentage error (MAPE) was chosen as the loss function
between prediction and measurement L ( y, yˆ ) . The neural
network optimization was conducted via FTRL algorithm
(McMahan 2013). During neural network training, the
batch size was 128 and learning rate was 0.01. The sa and
sp in Equation (3) were fixed to be 1 and 0.02, respectively.
After the optimization, we remove the input features whose Figure 5: Top: The variation of remaining features with λ. The
magnitude of the weight groups have shrunk to near 0 from black bars represent the features selected by the neural network.
prediction model. In this paper, the magnitude of a weight Bottom: Comparison of the prediction error (MAPE) between
group is considered to be near 0 when it’s value is less than weighted sparse-neural network (WSPINN) prediction and the
5% of the maximum magnitude among all of the input fea- DNN prediction using combinatorically searched features under a
tures. given number of features
Prior physical knowledge for VIV The prediction error varies with the retained features in
the prediction model, which is presented in the bottom part
Experimental studies on small spring-mounted rigid cylin-
of Figure 5. At each number of features, a brute force ap-
ders show that the response amplitude increases with in-
proach that searches all the possible combinations of the
creasing Reynolds number in the range 103 to 104 and de-
additional features is also carried out. The error obtained
creases as the dimensionless damping increases
from the WSPINN is compared with hundreds runs of
(Govardhan & Williamson 2006, Vandiver 2012).
DNN predictions using combinatorically searched features
Similarly, studies on long flexible cylinders have shown
in addition to the 3 features representing prior knowledge.
that the Reynolds number and the dimensionless damping
The comparison suggests that the WSPINN is able to find
continue to play important roles on the VIV response am-
the feature subsets that gives smallest prediction error
plitude (Resvanis 2012, Rao 2015) but as discussed earlier,
among all the feature combinations. Besides, it can be ob-
the large number of potentially relevant parameters and the
served that there could be multiple combinations of fea-
response variability result in scatter in the data.
tures that give similar prediction accuracy. For example,
Because it is known that Reynolds number Re and di-
both the additional features Ail* , κ cf and Ail* , κ il gives predic-
mensionless damping parameter ccf* are important, these
tion error around 13%. This suggests the correlations and
interactions among some of the VIV features.
After balancing the prediction accuracy with the sparsity
of input features, we find that the feature subset containing
5 features: Re, β , ccf* , Ail* , κ cf gives 13% MAPE, which is
close to 10.6% MAPE using all 17 features.
We have also applied the WSPINN algorithm to other
VIV related problems, such as the prediction for rigid cyl-
inder’s VIV amplitude and flexible cylinders’ VIV ampli-
tude at higher harmonics etc. Because of space limitations
we cannot demonstrate this here. Moreover, since this pa-
per only studied the important parameters for VIV sheared Figure 6: Contours of CF RMS amplitude as a function of ccf*
and uniform current profiles, the importance of the features and Re (in uniform flow)
may be different for more complicated current profiles.
Physical insight interpretation
The importance of the identified features for flexible cylin-
der VIV can be examined by systematically varying the
ranges of input features to the constructed neural network
models. Figure 6 and Figure 7 show the effect of varying
Re, ccf* while constraining the other variables in the predic-
tion model to characteristic values most often observed in
the Shell experiments. The black dots are the experimental
measurements within 20% from the referenced values and
are included to demonstrate that the prediction model (con- Figure 7: Contours of CF RMS amplitude as a function of ccf*
tours) did in fact have data in that vicinity. and Re (in sheared flow)
The results demonstrate that increasing Reynolds num-
ber tends to increase the spatiotemporal CF RMS ampli-
tude. This Reynolds number effect is obvious in the uni-
form flow data which typically has small dimensionless
damping values ( ccf* <0.3-0.4). While the Reynolds number
effect is virtually non-existent when looking at the sheared
flow cases with damping parameters ( ccf* >0.4).
Figure 8 shows the effect of varying Ail* and κ cf while
constraining the other variables. It can be found that the
crossflow response tends to increase with inline response.
Such a relationship has also been observed in spring-
mounted rigid cylinder’s VIV experiments (Dahl 2008),
where the fluctuating inline force increased with crossflow Figure 8: Contours of CF RMS amplitude as a function of Ail*
motion. Finally, the prediction model suggests that as the and κ cf (in sheared flow)
mode-participation factor increases so does the CF re-
sponse amplitude. Note that both standing wave and travel-
ling wave response can result in high mode-participation Special properties of the approach compared
factors and in this situation, the factor primarily character- to other machine learning
izes whether all points on the flexible cylinder are respond-
ing in a similar manner (spanwise coherence). 1. Direct and learning task dependent dimension reduction
in the original feature space while retaining the prior in-
formation in the model.
The WSPINN is one of the dimension reduction ap-
proaches. However, different from widely used PCA or
auto-encoders, WSPINN seeks to reduce the dimension
directly in the original input feature space. Moreover,
through machine learning prediction, WSPINN is able to
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