COMBINED DATA AND DEEPLEARNING MODEL UNCERTAINTIES ANAPPLICATION TO THE MEASUREMENT OF SOLID FUEL REGRESSION RATE

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COMBINED DATA AND DEEP LEARNING MODEL

UNCERTAINTIES: ANAPPLICATION TO THE MEASUREMENT OF

SOLID FUEL REGRESSION RATE

Georgios Georgalis

Data Intensive Studies Center

Tufts University

Medford, MA 02155, USA

georgios.georgalis@tufts.edu

Kolos Retfalvi, Paul E. DesJardin

Department of Mechanical and Aerospace Engineering

University at Buffalo, The State University of New York,

Buffalo, NY 14260, USA

{kretfalvi, ped3}@buffalo.edu

Abani Patra

Data Intensive Studies Center and Department of Mathematics

Tufts University

Medford, MA 02155, USA

abani.patra@tufts.edu

ABSTRACT

In complex physical process characterization, such as the measurement of the regression rate for

solid hybrid rocket fuels, where both the observation data and the model used have uncertainties

originating from multiple sources, combining these in a systematic way for quantities of interest (QoI)

remains a challenge. In this paper, we present a forward propagation uncertainty quantiﬁcation (UQ)

process to produce a probabilistic distribution for the observed regression rate

˙r

. We characterized

two input data uncertainty sources from the experiment (the distortion from the camera

and the

non-zero angle fuel placement

Uγ

), the prediction and model form uncertainty from the deep neural

network (

), as well as the variability from the manually segmented images used for training it (

We conducted seven case studies on combinations of these uncertainty sources with the model form

uncertainty. The main contribution of this paper is the investigation and inclusion of the experimental

image data uncertainties involved, and how to include them in a workﬂow when the QoI is the result

of multiple sequential processes.

Keywords

Uncertainty characterization, Deep learning model uncertainty, Image data uncertainty, Combustion

Experiments, Regression rate density estimation

1 Introduction

When analyzing a physical system, we use the “input data” (knowledge we have) to run the “model” and evaluate its

outputs (knowledge we want to obtain). All of the components involved in the process carry their own assumptions,

limitations, and uncertainties. When the outputs are part of a larger computational framework or are used in decision

making, reporting the associated uncertainty is required, because the modeling process and the model evaluations

are approximations of the underlying true physical process. Probability models are common representations of such

uncertainty in both the inputs and outputs. Forward-propagation of the uncertainty of input data and estimation of the

probability distribution of the simulation output via sampling based Monte-Carlo methods or functional approximations

is abundant in literature with recent applications in many ﬁelds (e.g., see [

], [

]). Similarly, observation data-driven

calibration procedures using Markov Chain Monte Carlo or variants are common [4].

However, in cases where the “input data” are results of a complex physical process that is inherently variable (e.g., from

a complex experiment), and the “modeling” includes multiple sequential models and dependencies, propagating and

Manuscript published at International Journal for Uncertainty Quantiﬁcation:

http: // dx. doi. org/ 10. 1615/ Int. J. UncertaintyQuantification. 2023046610

arXiv:2210.14287v2 [cs.LG] 18 Mar 2023

Figure 1: Overview of the workﬂow and uncertainties at each step to estimate the fuel regression rate

˙r

from a slab

burner experiment. Hybrid rocket image (left) is NASA’s Peregrine rocket motor.

combining all uncertainty sources in a systematic way for a required quantity of interest (QoI) remains a challenge

[

]. This is especially true when using operations like image segmentation models [

] for identiﬁcation of surfaces

and interfaces which implicitly and explicitly use modeling and calibration. In particular, we focus here on the

characterization of the “input data" uncertainty when it is acquired in a complex experiment and used within a

model-based interpretation that transforms the raw observation into a QoI.

In hybrid rocket motor combustion, one QoI is the rate at which the fuel surface recedes during the burn, deﬁned as the

regression rate

˙r

. The regression rate has direct impact on the geometrical design of the rocket motor and its performance

([

], [

]). Due to the high cost of building a full-scale hybrid motor, it is common practice to estimate the regression

rate a priori via a smaller-scale experiment of a slab burner [

]. Tracking the surface of the fuel from experimental

images during the burn with time data is one way to estimate

˙r

[

]. In this case, “inputs” are all of the assumptions

and limitations of the experiment, the equipment, and any processing to get the fuel surface images. “Modeling” is

the process of translating the images to the regression rate, which is practically a sequence of multiple models: a deep

learning convolutional network model to segment the fuel masks of an entire experiment (e.g., Monte-Carlo Dropout

(MCD) U-net from [

]), a process to detect the boundaries of the segmented fuel masks, and a process to estimate

˙r

from the changing boundaries with time data (Fig. 1).

The regression rate and its measure of uncertainty are necessary to either validate multi-scale computational models

(e.g., simulations of turbulent combustion chemistry coupled with ﬂow dynamics [

]), or for decision making on the

rocket motor design and performance. In these coupled systems, there is the additional requirement that the uncertainty

information of the QoI be expressed in a simple way (e.g., as a probability distribution) because simulations can be

sequential: the distribution of the regression rate represented by an ensemble is an input to another simulation (e.g., see

[13] for an application of sequential uncertainty quantiﬁcation in combustion systems).

iterature has focused on separating the different types of uncertainty within a model as aleatoric (AU) or epistemic

uncertainty (EU) and how to measure them in physical models such as hypersonic ﬂows [

], in machine learning [

in Bayesian convolutional networks [

], and in safety decision making [

] [

] among others. In the context of a

complex “input data” and “modeling” process as in the regression rate case (Fig. 1), these two distinct uncertainty

deﬁnitions are often challenging to separate or measure. For example, the inherent camera structure and how it represents

an image captured in the experimental environment and whether the researcher misplaced the fuel specimen by a few

degrees include some inherent randomness, but we have the capability to estimate some parts of that experimental

randomness by analyzing the setup and the optical errors. Therefore, in this work, we do not classify uncertainty as

aleatoric or epistemic, but rather characterize and quantify the individual uncertainty measures based on their source

(i.e., experiment or modeling).

In this paper, we present an uncertainty quantiﬁcation (UQ) process to produce a probabilistic distribution for the

regression rate

˙r

from the input experimental images, by combining the input data uncertainty with inference model

uncertainty. We characterize the input data uncertainties from the experiment

Udata ={Uc, Uγ}

, the model form

uncertainty from the MCD U-net (

), and the variance in the model prediction from the variability of manually

tracing the masks used to train the MCD U-net (

). The process is shown in Fig. 1:

(a)

We introduce the input data

uncertainty

Udata

to create sequenced ensembles of experimental images,

(b)

pass the ensembles through the U-net to

get an estimate of the mean predicted masks

µmask

and corresponding uncertainty maps

Umask

which carry both the

data and model uncertainties, and lastly (c) process each sequenced ensemble of predicted fuel masks and uncertainty

maps to get an estimate for the regression rate ˙rand its bounds.

The sources of input data uncertainty are most relevant for this problem because optical distortion from the camera (

)

or non-zero angle placement of the fuel (

Uγ

) may misrepresent the fuel boundaries in the images that are tracked to

accurately measure the regression rate. The model form uncertainty (

) is computed as the entropy of the probability

prediction vector of the MCD U-net. The possible variation in the manually segmented masks used for training the

MCD U-net is added as a prediction variance Usdirectly to the ﬁnal resulting uncertainty map Umask.

Manuscript published at International Journal for Uncertainty Quantiﬁcation:

http: // dx. doi. org/ 10. 1615/ Int. J. UncertaintyQuantification. 2023046610

Figure 2: Slab burner test chamber with parafﬁn wax sample.

The paper is organized as follows. Section 2 includes details about the experimental setup and the characterization for

each of the studied uncertainty measures. Section 3 presents a series of cases that investigate how the different types of

uncertainties impact the overall ﬂow of uncertainty measure in the forward-propagation process (i.e., how the input data

uncertainties individually and together combine with the model form uncertainty and the manual segmentation variance

to form the output uncertainty map used in the measuring

˙r

). Section 4 is an application of the entire workﬂow shown

in Fig. 1, and includes the resulting distribution for

˙r

via forward-propagation after including all the uncertainty sources

in sequence. Section 5 summarizes the conclusions of the paper with directions on future work.

2 Formulation of Uncertainty Sources and Regression Rate Measurement

In this section, we describe the experimental setup and equipment used, the characterization of the input data uncer-

tainties

Udata ={Uc, Uγ}

and how they are introduced to the experimental images.

corresponds to error from

optical distortion that is inherent to the camera placement in respect to the fuel specimen.

Uγ

corresponds to the

non-zero angle fuel placement error, which may result from the researcher not perfectly placing the fuel orthogonal to

the camera axis, thus having the camera misrepresent the distance between the fuel boundaries correctly. We also show

the formulation for the model form uncertainty

and the manual segmentation variance

, as well as how the model

outputs µmask, Umask are used to measure the regression rate ˙rand its bounds.

2.1 Experimental setup and its uncertainties

The experimental setup follows closely the one developed by Dunn et al. [

] and can be seen in Fig. 2. The fuel

specimen is placed in a chamber consisting of two stainless steel plates on the top and bottom and high temperature

borosilicate glasses on the side of the experiment for optical access. The chamber is 15.24cm long, 2.54cm tall and

2.54cm wide, with a oxidizer inlet pipe of 1.83cm long and 2.54cm in diameter. Based on the entrance length, the

inlet ﬂow is assumed to be fully developed as it enters the chamber. We used lab-grade parafﬁn wax from the Carolina

Biological Company as the fuel during the experiments and they were cast in a stainless steel mold. The temperature of

the mold was monitored during the solidiﬁcation process to avoid impurities in the samples. The average dimensions

for the fuel specimens used in the experiments were 9.4mm wide, 80.7mm long and 11.2mm in height. Each sample

had a 45

◦

slant in the front to guide the ﬂow. The oxidizer used for the experiment was 100

gaseous oxygen which

was regulated with solenoids and measured with an Omega FMA 1744a mass ﬂow meter. The measurement range of

the ﬂowmeter was 5 - 500 SLMs with ±1.5% accuracy.

To obtain the image dataset used in this paper, we conducted two experiments with measured oxidizer mass ﬂuxes of

G1= 6.96 kg

m2s

and

G2= 10.96 kg

m2s

respectively. During the automated experimental sequence the oxidizer was ﬁrst

introduced and the slab ignited using a Bosch diesel glow plug which was lowered to make contact with the slab and

later retracted using a stepper motor. The experiment continued until the ﬂame was extinguished. The experiment was

recorded with a Chronos 2.1 high speed camera during the experimental runs, with a frame rate of 1000 frames per

second. The lens used was a NIKON AF Nikkor 50mm lens with apertures ranging from 1.8 to 8 with an exposure

of 1

µs

. The full burn of the specimen took between 8-12 seconds. Some example images from the experiment with

oxidizer ﬂux G2= 10.96 kg

mssat different times can be seen in Fig. 3.

2.1.1 Optical distortion error Uc

The image data from the experiment are not exactly accurate representations of the real phenomena during the burn. One

reason is that despite best efforts to set the camera correctly for best image quality, the fuel surface moves with respect

to the camera during the experiment, adding a possibility of optical distortions. To account for the possible camera

distortions as part of the input data uncertainty, we followed a calibration process by using checkerboard patterns.

Assuming the pinhole model for the camera, i.e. the real points of the captured space are imaged by rays that pass

Manuscript published at International Journal for Uncertainty Quantiﬁcation:

http: // dx. doi. org/ 10. 1615/ Int. J. UncertaintyQuantification. 2023046610

Figure 3: Example images from the experiment with oxidizer ﬂux

G2= 10.96 kg

mss

at early (a), middle (b), and later (c)

stages.

through a single origin of the camera lens [

], we estimated the intrinsic and extrinsic parameters of the camera and

the distortion coefﬁcients as outlined in [21].

Given the global coordinate system

(x, y, z)

and the camera coordinate system

(i, j, k)

, a point in the real world is

represented by the position vector

p= [px, py, pz]T

. The same point in the camera coordinate system is represented

by the position vector

q= [qi, qj, qk]T=RT(p−τ)

, where

is the rotation matrix and

the translation vector

(extrinsic parameters). The point represented in the camera coordinates as

is then mapped into the image plane

using the intrinsic parameters matrix

and represented as

r=Kq="fx0 0

s fy0

cxcy1#[qi, qj, qk]T

where

fx, fy

are the

focal lengths,

is the skew coefﬁcient, and

cx, cy

are the coordinates of the optical center. The rotation matrix

, the

translation vector

, and the intrinsic matrix

are all estimated during the camera calibration process. In addition, we

also estimate the camera distortion coefﬁcients

k1, k2

that characterize radial distortion in the image coordinates [

rx,distorted =rx

fx(1 + k1d2+k2d4)and ry,distorted =ry

fy(1 + k1d2+k2d4), where d= ( rx

fx)2+ ( ry

fy)2.

To calibrate the camera, we took ten images of a square checkerboard pattern with a side of 2 cm under different

angles (Fig. 4), and processed them using MATLAB’s camera calibrator app [

]. The resulting estimated parameters

and camera resolution are shown in Table 1. The rotation matrices and translation vectors are not shown because

they are different for each calibration image. During the calibration process, the true locations of the checkerboard

pattern are detected and compared with their locations on the reconstructed images from the camera model. The

reconstruction error is then

ei,j =si,j −ˆsi,j

, where

si,j

are the true detected points on the calibration patterns and

ˆsi,j

the re-constructed points. To arrive at a measure of distortion uncertainty from the camera that can be used to introduce

uncertainty to the slab burner images, we count how many of the calibration points have a total error that is greater

than 1 pixel in distance, for each of the calibration images. The threshold is set at 1 pixel because when we represent

the experimental images as tensors for the deep learning model, an error greater than 1 pixel means the image point

is distorted enough to not populate that ﬁeld in the tensor anymore. The total number of distorted pixels from each

calibration image are expressed as a percentage of the total pixels in the image. The result of this process is shown in

Fig. 5: the expected maximum number of distorted points with an error greater than 1 pixel corresponds to 0.7

of the

image and the minimum number to

0.133%

of the image. Therefore, based on the calibration results, we expect that

an experimental image taken with our camera has a number of distorted pixels

Uc∼ U(0.133,0.7)[%]

, expressed as

a percentage of the image. Since there are only two options for the pixels, distorted or not, for a given experimental

image

of resolution 512 by 64, there is a corresponding distortion map

Dmap,k

. The distortion map follows the

binomial distribution

Dmap,k ∼B(n= 64 ∗512, p =Uc)

. The pixels in

that have a success trial on the distortion

map

Dmap,k

, are distorted, and therefore have their intensity reduced to zero when we introduce distortion uncertainty

Ucto the original image (Alg. 1).

Table 1: Camera resolution and parameter estimation from calibration

Camera Parameter

Resolution 1920x1080 [pix]

fx4386.76 ±27.41 [pix]

fy4374.34 ±26.68[pix]

s0.0 (perpendicular axes)

cx971.65 ±1.25 [pix]

cy503.64 ±1.22 [pix]

k14.30 ±0.24 [–]

k228.74 ±13.40 [–]

Manuscript published at International Journal for Uncertainty Quantiﬁcation:

http: // dx. doi. org/ 10. 1615/ Int. J. UncertaintyQuantification. 2023046610

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COMBINEDDATAANDDEEPLEARNINGMODELUNCERTAINTIES:ANAPPLICATIONTOTHEMEASUREMENTOFSOLIDFUELREGRESSIONRATEGeorgiosGeorgalisDataIntensiveStudiesCenterTuftsUniversityMedford,MA02155,USAgeorgios.georgalis@tufts.eduKolosRetfalvi,PaulE.DesJardinDepartmentofMechanicalandAerospaceEngineeringUniversityatBuffalo,T...

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COMBINED DATA AND DEEPLEARNING MODEL UNCERTAINTIES ANAPPLICATION TO THE MEASUREMENT OF SOLID FUEL REGRESSION RATE

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