# A Machine Learning and Computer Vision Approach to Rapidly Optimize
Multiscale Droplet Generation

Alexander E. Siemenn  Department of Mechanical Engineering, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA<EMAIL_ADDRESS>Evyatar
Shaulsky  Department of Chemical Engineering, Northeastern University, Boston,
MA 02115, USA  Matthew Beveridge  Department of Electrical Engineering and
Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  Tonio Buonassisi  Department of Mechanical Engineering, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Sara M. Hashmi  Department
of Chemical Engineering, Northeastern University, Boston, MA 02115, USA
<EMAIL_ADDRESS>Iddo Drori  Department of Electrical Engineering and
Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA<EMAIL_ADDRESS>
###### Abstract

Generating droplets from a continuous stream of fluid requires precise tuning
of a device to find optimized control parameter conditions. It is analytically
intractable to compute the necessary control parameter values of a droplet-
generating device that produces optimized droplets. Furthermore, as the length
scale of the fluid flow changes, the formation physics and optimized
conditions that induce flow decomposition into droplets also change. Hence, a
single proportional integral derivative controller is too inflexible to
optimize devices of different length scales or different control parameters,
while classification machine learning techniques take days to train and
require millions of droplet images. Therefore, the question is posed, can a
single method be created that universally optimizes multiple length-scale
droplets using only a few data points and is faster than previous approaches?
In this paper, a Bayesian optimization and computer vision feedback loop is
designed to quickly and reliably discover the control parameter values that
generate optimized droplets within different length-scale devices. This method
is demonstrated to converge on optimum parameter values using 60 images in
only 2.3 hours, $30\times$ faster than previous approaches. Model
implementation is demonstrated for two different length-scale devices: a
milliscale inkjet device and a microfluidics device.

###### keywords:

Bayesian optimization, droplet generation, computer vision control,
microfluidic devices, inkjet printing, Rayleigh instability, capillary
instability

##  1 Introduction

Figure 1: Fluid flow before and after control parameter optimization to form
droplets. (A) Rayleigh instability and (B) capillary instability forming after
optimization for the inkjet and microfluidics devices, respectively. (C)
Deposited droplet patterns from the inkjet device. The images show a case from
before optimization, using Latin hypercube sampled control parameter values,
and a case from after optimization. Our model learns how to optimize droplet,
which indirectly optimizes (D) Rayleigh and (E) capillary instabilities within
the fluid stream. Rayleigh instability forms within the inkjet device by
applying piezoelectric-driven perturbations, $\varepsilon_{i}$, to a water
flow of radius $R_{0_{i}}$ being acted on by gravity in ambient air. Capillary
instability forms within the microfluidic device by applying viscous stress-
driven perturbations, $\varepsilon_{m}$, from the oil medium to a water flow
of radius $R_{0_{m}}$. (F) Length scale comparison of an inkjet-generated
droplet (left in mm) and a microfluidic-generated droplet (right in $\mu$m).

Generating discrete and uniform droplets from a fluid stream requires the
fine-tuning of experimental conditions to transition the flow from a
continuous, stable jet to a separated, unstable stream of droplets 1.
Rayleigh-Plateau instability governs the separation of a fluid stream into
discrete droplets through gravity-driven perturbations and capillary
instability governs the separation of a fluid stream into discrete droplets
through pressure-driven perturbations 2. Analytical relationships between
governing parameters and droplet characteristics simply do not exist due to
the complex, non-linear physical relationships between all forces acting on
the fluid at various length scales 3. Thus, it is analytically intractable to
determine the parameter values that produce optimized droplets, hence,
requiring experimentation. Typically, researchers tune experimental controls
until they reach the desired droplet characteristics 4, 1, 5, 6. Our method
provides an efficient way to find the appropriate values of experimental
control parameters that achieve the desired outcome without developing a
device-specific control model. Thus, we are no longer restricted to using our
method only on devices with similar governing physics. We demonstrate that the
method successfully optimizes droplets generated by devices that have both
dissimilar control parameters and droplet length-scales, with few
modifications. Furthermore, the novelty of our approach lies not only in its
flexible application to devices with different governing physics and control
parameters, but also in it requiring no upfront time to train, enabling
researchers to quickly apply our method to their data and get results within
hours rather than days 7.

Prior works develop proportional integral derivative (PID) controllers or
classification machine learning to predict or control droplet states 8, 7.
However, both of these approaches are inflexible as they either require
specific models to simulate the physics of a specific device, in the case of
PID, or require training on large datasets of data specific to a single
device, in the case of classification. Additionally, it has been shown that an
estimated three days of upfront training is necessary to effectively classify
and predict the droplet states of a microfluidic device 7. Our method requires
no upfront training, instead, our algorithm learns the dynamics between input
control parameters iteratively as the researcher performs experiments. We
demonstrate that after initialization, our method requires only four rounds of
experimentation with a batch size of 10 before converging on an optimum.
Resulting in an average total optimization time of 2 hours, compared to
previously reported $>70$ hours via supervised classification 7.

Flexible machine learning (ML) processes, such as Bayesian optimization (BO),
have the promise to greatly improve the efficiency of exploring parameter
spaces by saving expensive lab time and resources as they are generalizable to
parameter spaces of varying dimensionality and require no upfront model
training since they use a Gaussian Process (GP) backbone 9, 10, 8, 11, 12. In
this paper, we develop a universal BO and computer vision feedback loop to
quickly and reliably discover Rayleigh and capillary unstable conditions in
different droplet-generating devices with different governing control
parameters and physics 13, 14. We demonstrate the functionality of our method
on two devices: (1) an milliscale inkjet droplet-generator with the
controllable parameters of (a) pressure, (b) actuation frequency, and (c)
translation speed and (2) a microfluidic droplet-generator with the
controllable parameters of (a) water pressure and (b) oil pressure, shown in
Fig. 1 and Fig. 3.

The challenge faced in optimizing droplets at different length scales is in
designing a tool robust enough to predict the physics of each device parameter
space to make meaningful and reliable predictions of optimized printing
conditions. The challenge is addressed in this paper through our physical
model-agnostic approach of optimization, which implements computer vision to
detect the droplet flows and converts them into numerical representations of
the parameter space for the Bayesian inference algorithm to learn. The
significance of developing a tool that rapidly optimizes droplets at multiple
length scales is in providing researchers with a universal tool for tuning
devices that requires little customization or ML expertise. Generating these
optimized droplets has importance across several application fields depending
on the length scale of the droplet.

###  1.1 Prior Work on Millimeter-scale Droplets

Depositing millimeter-scale droplets onto a substrate is a process useful for
high-throughput characterization of material, most notably for finding
optimized semiconductor materials that maximize efficiency or stability 15,
16. Semiconductors such as perovskites have vast and complex composition
spaces which makes it challenging to discover optimum compositions 17, 18, 19.
Thus, using discrete inkjet-deposited droplets of varying semiconductor
compositions for high-throughput experimental characterization elicits an
accelerated search of this vast composition space for an optimum composition.
However, studies such as Bash et al., (2020) rely on a domain expert to fine-
tune experimental conditions that establish flow instability control prior to
semiconductor characterization and, hence, require and understanding of system
physics prior to optimization. Rather, in this study, the non-linear physical
relationships between control parameters are iteratively learned by the
probabilistic ML surrogate model as data is collected.

###  1.2 Prior Work on Micrometer-scale Droplets

Creating microspheres with a high surface area to volume ratio is beneficial
for a large variety of applications such as materials generation,
bio(chemical) analysis, polymeric microcapsules, and more 20, 21, 22.
Industries like food science, biomedical, and cosmetics use microcapsules as
materials delivery vehicles with the ability to tune the capsule wall
chemistry for slow or triggered release of the capsulated material 23, 24, 25.
For each of these unique applications, a different microdroplet volume is
required. However, these droplet volumes and structures are a function of
several physical parameters like the interfacial tension between the
immiscible fluids, their viscosities, the microfluidic device geometry, and
fluid flow rates 26, 27.

Existing literature uses supervised neural networks to inform device geometry
or to classify different control states for droplet generation. This method
requires a large quantity of training data and requires an estimated 70 hours
for model training. However, a gap still exists on how ML can be applied to
optimize the device control parameters using GP techniques that do not require
a priori training and can, in turn, operate much faster 28, 7.. Thus, in this
study, we use Bayesian inference methods to optimize the microspheres using
the control parameters of water and oil pressures directly, rather than
developing a PID model or supervised neural network to simulate the physical
parameters and their non-linear relationships.

###  1.3 Main Findings

We demonstrate accurate and repeatable discovery of Rayleigh and capillary
unstable regions to generate droplets of high yield and high circularity
within two fluid devices at different length scales using Bayesian
optimization and computer vision. For each of these fluid devices, three
trials of Bayesian optimization were run using three different decision policy
acquisition functions in which all trials converged to learning similar bounds
for generating high circularity and high yield droplet structures. Convergence
on optimized control parameters is attained for both devices using the same
method within 2 hours while synthesizing 60 samples only.

##  2 Methods

Figure 2: Control feedback loop of the hardware-algorithm interface used for
droplet optimization. (A) The BO experiment is initialized using LHS, which
samples control parameter values from the device parameter space. (B) For the
first loop, LHS parameter values are used to print and image droplets. For
proceeding rounds, learned parameter values by BO are used to print and image
droplets. (C) Computer vision is used to score each droplet image and assign a
label that quantifies droplet circularity and yield. The hardware (B) and
computer vision (C) subsystems interface with each other by simply providing
an image of the output droplets to the computer vision code. (D) The label and
parameter values from the current loop and all prior loops are input into the
BO decision policy directly from the computer vision algorithm, which informs
the next round of parameter values to print. The iterative retraining of BO
from processes (B)–(D) autonomously drives the control of the Rayleigh and
capillary instability phenomena without a priori domain knowledge of a
physical model. (E) Convergence on optimized control parameter values is
assessed after four loops of iteratively retraining BO with new experimentally
generated data.

Fig. 2 illustrates the workflow presented in this paper for multiscale droplet
optimization. The user collects a set of initialization images of droplets
generated from the device over a range of experimental control parameters
(e.g., fluid pressures, nozzle frequency). The optimization software analyzes
the data and generates a new set of control parameters to be used in the
device, and the process is then repeated. Within only a few iterations, the
efficient algorithm converges to a set of control parameters to produce the
desired drops.

###  2.1 Device Hardware

(a) Inkjet Device Architecture

(b) Microfluidics Chip Architecture

Figure 3: Droplet-generating device hardware used in this study. (a) Inkjet
device with inset image showing the top view of the deposition plate and the
(b) microfluidics device with inset image showing the droplet generating
interface. For the inkjet device, the positive pressure motor controls the
fluid pressure in the vials; the frequency of the actuated nozzle controls the
rate of droplet deposition; the speed of the laterally translating plate
controls the droplet deposition placement. For the microfluidics device, the
channel is flowed with mineral oil from the oil line as the suspending fluid;
oil pinches off water from the water line to form droplets; all fluid exits
through the outlet. The entire microfluidics device has a height of 30 $\mu$m.

####  2.1.1 Inkjet

The inkjet device has three possible control parameters: (1) fluid pressure,
(2) piezoelectric actuation frequency, and (3) nozzle translation speed, of
which we know the discrete values of at all times, but do not know the
combination of these values that produce optimized droplets. These three
parameters influence the shape and yield of droplets deposited on a plate 6.
Pressurizing the fluid within the pipes drives both the radius and the flow
rate of the stream for a fixed orifice size; our inkjet device operates at
pressures limits of 0.03–0.15 MPa. Applying an alternating current electrical
signal to the piezoelectric material within the pipe actuates a membrane at a
given frequency to induce perturbations, $\varepsilon$, within the fluid
stream 13, 14; our inkjet device operates at frequency limits of 1.0–600.0 Hz
using. The fluid is ejected from a nozzle which translates in 1D above a
deposition site, which also translates in 1D perpendicular to the nozzle,
driving 2D deposition of the fluid onto the deposition plate; our inkjet
device operates at translation speed limits of 10–360 mm/s. None of these
three control parameters are artificially constrained, the limits are defined
by the physical limits of the hardware only. We keep both the fluid (dyed
water) and nozzle diameter (70 $\mu$m) constant for all experiments, shown in
Fig. 3(a). The intrinsic fluid properties are constant and we operate at flow
velocities between 0.8–4 m/s, given the range of operating pressure. For every
printed sample, these known values of the control parameters are appended to
the image and normalized to a range of $[0,1]$ before being fed into the BO
algorithm.

The printable fluid parameter space is governed by the fluid density, surface
tension, viscosity, as well as the driving pressure and nozzle geometry 29,
30, 31. While analytical relationships between these variables and the output
droplet circularity and yield are intractable, the droplet properties can be
described by dimensionless relationships between forces: (1) Ohnesorge number
(Oh), which compares viscous forces to inertia and surface tension, (2) Weber
number (We), which compares inertial forces to surface tension, and (3)
Reynolds number (Re), which compares inertial to viscous forces 29, 30.

$\textrm{Oh}=\frac{\mu}{\sqrt{\rho\sigma d}},\qquad\textrm{We}=\frac{\rho
v^{2}a}{\sigma},\qquad\textrm{Re}=\frac{\rho vd}{\mu},$ (1)

where $\mu$ is the fluid dynamic viscosity, $\rho$ is the fluid density,
$\sigma$ is the surface tension, $d$ is the jet diameter, $a$ is the droplet
diameter, and $v$ is the characteristic flow velocity. Oh depends on fluid
properties and nozzle diameter only, and so is fixed in our system at
$\textrm{Oh}\approx 0.01$. This value falls in the range typically considered
to facilitate satellite-drop formation; we observe satellite drops in
approximately 10% of the inkjet samples 29. Moreover, we determine the ranges
of the variable dimensionless numbers to be $\textrm{We}\approx$ 1–200 and
$\textrm{Re}\approx$ 10–50, which are further discussed in the results.
Gravity is negligible in our system because the Bond number $=\frac{\rho
gd^{2}}{\sigma}\approx 0.001$, where $g$ is gravitational acceleration,
implying that capillary forces dominate the gravitational forces.
Interestingly, despite operating in the regime of satellite drops, the data-
driven BO approach efficiently explores and optimizes the parameter space for
desired drop properties.

####  2.1.2 Microfluidic

The microfluidic device has two possible control parameters: (1) pressure to
drive the inner droplet fluid (water) and (2) pressure to drive the outer
suspending fluid (mineral oil), of which we know the discrete values of at all
times, but do not know the combination of these values that produce optimized
droplets. Both the water and oil pressure are controlled by a flow controller
(Flow-EZ, Fluigent); our microfluidic device operates at pressure limits of
0-2000 mbar. Neither of these two control parameters are artificially
constrained, the limits are defined by the physical limits of the hardware
only. The outer mineral oil pinches off droplets of the inner DI water at a
junction that is 40 $\mu$m wide, the droplets then travel into a channel 200
$\mu$m wide, shown in Fig. 3(b); the entire device has a height of 30 $\mu$m.
Microfluidic droplet makers operate in one of five regimes at any given time:
no inner fluid flow, unbroken inner fluid flow, dripping, transition, and
jetting 32, 33, 34. Our experiment explores two out of the three drop
generation regimes: (1) dripping, where the drop forms while touching the
constriction walls, and (2) transition, where the drop forms in the
constriction without touching the walls. Jetting does not occur in our
experiments due to the device structure, fluid viscosities, and experimental
parameter space 32. Similar to the inkjet device, for every printed sample,
the control parameters, normalized to the range $[0,1]$, are appended to the
image and fed into the BO algorithm.

The droplet properties in the microfluidic system are primarily described by
two dimensionless numbers: (1) Capillary number (Ca), which compares viscous
forces to surface tension and (2) Weber number (We), which compares inertial
forces to surface tension 35, 5, 36, 32.

$\textrm{Ca}=\frac{\mu\dot{\gamma}a}{\sigma},\qquad\textrm{We}=\frac{\rho
v^{2}a}{\sigma},$ (2)

where $\mu$ is the viscosity of the outer fluid or continuous phase,
$\dot{\gamma}$ is the shear rate, $a$ is the droplet diameter, $\sigma$ is the
surface tension between the two fluids, $\rho$ is the density of the
continuous phase, and $v$ is the characteristic flow velocity. The ranges of
these dimensionless numbers within our system are found to be
$\textrm{Ca}\approx$ 0.005–0.05 and $\textrm{We}\approx$ 0.0005–0.003, which
are further discussed in the results. Viscous stresses dominate the inertial
effects in the microfluidics system because $\textrm{Re}\approx$ 0.04–0.4.

###  2.2 Parameter Space Initialization

Without a priori knowledge of the parameter space dynamics from a domain
expert, we look towards using machine learning methods, in particular,
Bayesian optimization (BO), to learn the topology of the parameter space. As
more experimental data is acquired, BO refines its search of the parameter
space for optimized printing conditions. The user collects training data over
the range of parameter space in which optimization is desired. For the purpose
of this demonstration, data was collected over the entire range of
experimentally accessible parameters within our system.

To give BO the best chance at finding a global optimum condition, we provide
the algorithm with our best estimation of the parameter space as a whole in
the form of an initialization dataset. Latin hypercube sampling (LHS) is a
sampling tool that generates an initialization sample set
$\mathbf{x}\in\mathcal{X}^{(N)}$ to capture the variability of an
$N$-dimensional parameter space with low bias 37. LHS stratifies the range of
each control parameter into $K$ strata ($K=20$ in this paper) of equal
marginal probability such that each stratum is randomly sampled once to
generate $\mathcal{X}^{(N)}$. The low variance demonstrated by LHS in
literature, relative to random and stratified sampling, illustrates the
goodness of LHS as an unbiased estimator for selecting initialization
conditions 37.

###  2.3 Computer Vision Characterization

Figure 4: Computer vision characterization of droplets based on circularity
and yield targets. The watershed process segments the droplets from the image
background such that $\mathcal{L}_{\textrm{geom}}$ and
$\mathcal{L}_{\textrm{yield}}$ can be computed. The developed computer vision
code is robust to variations in droplet size, allowing for optimization of
droplets at several scales, without adjustment – in this paper, milliscale and
microscale droplets are explored. The developed computer vision algorithm is
robust to variations in image artifacts by using pre-processing thresholds
such that uniform image normalization is achieved.

The user can define a target. We defined our targets to be the geometric
circularity and yield of droplets, as shown in Fig. 4. An optimization
objective label $\ell$ is computed for each droplet sample via computer
vision, which quantifies these properties of the droplet flow. A simple
scalarized linear combination of loss functions is implemented to support
flexibility of the user to implement their own loss functions.

After generating each sample, the flow is imaged and fed into the computer
vision software. The computer vision software utilizes a watershed
segmentation process flow 38. The watershed process defines a dynamic
threshold to segment each droplet in the sample from the background, as shown
in Fig. 4, such that we generate a set of indexed droplet pixels
$P_{\textrm{drop}}$. We characterize all indexed droplets within a sample by
the optimization objective label, $\ell\in[0,1]$, which we aim to minimize.

Droplet circularity is the first component of $\ell$ and it is calculated by
computing the loss between all droplets in the image $P_{\textrm{drop}}$ and
perfect circles mapped onto the centroids of each droplet
$P_{\textrm{circle}}$.

$\mathcal{L}_{\textrm{geom}}=\frac{\sum\overline{P_{\textrm{drop}}\cap
P_{\textrm{circle}}}}{\sum P_{\textrm{drop}}}.$ (3)

By computing the major axis chord $r_{\textrm{major}}$ and minor axis chord
$r_{\textrm{minor}}$ of each indexed droplet, a perfect circle is mapped to
each droplet such that the circle’s centroid is the intersection point of
$r_{\textrm{major}}$ and $r_{\textrm{minor}}$ and the radius is the average of
these chords. Fig. 4 illustrates the process of mapping a circle to each
watershed indexed droplet in the sample. We aim to minimize
$\mathcal{L}_{\textrm{geom}}\in[0,1]$ to achieve droplets of high circularity.

Droplet yield is the second component of the optimization objective and it is
computed by taking the loss of all droplet pixels and all non-droplet pixels
to minimize the number of non-droplet pixels.

$\mathcal{L}_{\textrm{yield}}=\frac{1}{2}\frac{\sum\overline{P_{\textrm{drop}}}}{\sum
P_{\textrm{drop}}\cup\overline{P_{\textrm{drop}}}}+\frac{1}{2}\frac{\max(\textbf{count})-\textbf{{count}}(P_{\textrm{drop}})}{\max(\textbf{count})},$
(4)

where $\max(\textbf{count})$ denotes the maximum count over all
$P_{\textrm{drop}}$, which keeps $\ell$ as a minimization problem. We aim to
minimize $\mathcal{L}_{\textrm{yield}}\in[0,1]$ to achieve droplets of high
area and high count.

The total loss function $\ell(\mathbf{x})$ is constructed by taking the
average of $\mathcal{L}_{\textrm{geom}}$ and $\mathcal{L}_{\textrm{yield}}$,
as is common practice in computer vision studies 39. We give equal importance
to the geometric circularity and yield objectives during optimization which is
suitable for our purposes. Every experimentally generated sample is labeled
with a value of $\ell$ such that we aim to minimize $\ell\in[0,1]$ to achieve
droplets of both high geometric circularity and high yield. The objective
function in this study is user-defined, meaning that other important
measurable droplet properties can be explicitly included by the user and
tailored on the basis of the application.

###  2.4 Bayesian Optimization

Algorithm 1 Droplet Optimization by BO in Loop

Inputs: (1) LHS-initialized set $\mathbf{x}\in\mathcal{X}^{(N)}$, the
$N$-dimensional control parameter space. (2) Batch size, $b$.

Outputs: (1) Vector of optimized control parameter values,
$\mathbf{x}^{*}=\underset{\mathbf{x}\in\mathcal{X}}{\operatorname{arg\
min}}\;\ell(\mathbf{x})$. (2) Running minimum loss across all samples,
$\ell^{*}=\textbf{{min}}(\ell(\mathbf{x}))$.

Let
$\ell(\mathbf{x})=\frac{\mathcal{L}_{\textrm{geom}}(\mathbf{x})+\mathcal{L}_{\textrm{yield}}(\mathbf{x})}{2}$,
the scoring function from computer vision.

Let $\ell(\mathbf{x})\in\mathcal{S}\;\forall\;\mathbf{x}\in\mathcal{X}^{(N)}$,
the set of losses of the training samples.

while no convergence do

Learn: Estimate parameter space with the GP surrogate,
$\ell_{\textrm{pred}}(\mathbf{x})\sim\mathcal{GP}(\cdot)$.

Predict: Using an acquisition function $a\in\\{EI,MPI,LCB\\}$, predict control
parameter values, $\mathbf{x}_{i}$ for sample $i$, which minimize
$\ell_{\textrm{pred}}(\mathbf{x}_{i})$.

Synthesize: Synthesize $b$ samples using $\mathbf{x}_{i}$ parameters and image
the sample.

Update: Update $\ell_{\textrm{pred}}(\mathbf{x}_{i})$ with the computer vision
image measurement $\ell(\mathbf{x}_{i})$. Concatenate $\mathbf{x}_{i}$ to
$\mathcal{X}^{(N)}$ and $\ell(\mathbf{x}_{i})$ to $\mathcal{S}$.

end while

return $\mathbf{x}^{*}$, $\ell^{*}$.

We drive the efficient discovery of unstable fluid conditions using BO in
loop, shown in Algorithm 1, where previous batches of generated samples inform
the next batch of generated samples. The control parameter values
$\mathbf{x}\in\mathcal{X}^{(N)}$ and labels $\ell\in\mathcal{S}$ of previous
batches serve as a likelihood to a probabilistic surrogate model, in this case
a Guassian Process (GP) 40. The GP utilizes a decision policy, called an
acquisition function, to acquire new control parameter values $\mathbf{x}$
that better optimize $\ell(\mathbf{x})$ or explore new regions of the
parameter space that may contain more optimized values of $\ell$. We aim to
minimize $\ell(\mathbf{x})$ to find the control parameter conditions that
produce highly circular and high yield droplets 12, 41:

$\mathbf{x}^{*}=\underset{\mathbf{x}\in\mathcal{X}}{\operatorname{arg\
min}}\;\ell(\mathbf{x}),$ (5)

where $\mathbf{x}$ is an $N$-dimensional vector of printing condition in the
set $\mathcal{X}^{(N)}$.

GP is the surrogate model used to predict how the droplet flow
$\ell(\mathbf{x})\in\mathcal{S}$ changes as the control parameter values
$\mathbf{x}\in\mathcal{X}^{(N)}$ change:

$\ell_{\textrm{pred}}(\textbf{x})\sim\mathcal{GP}(\mu(\mathbf{x}),\kappa(\mathbf{x},\mathbf{x}^{\prime})).$
(6)

A Gaussian prior, $\mathcal{N}$, is assigned to the initial GP likelihood of
the objective function $\ell(\mathbf{x})$ to estimate its posterior mean
$\mu(\mathbf{x})$ and (co)variance $\kappa(\mathbf{x},\mathbf{x}^{\prime})$
12:

$\displaystyle\mu(\mathbf{x})$ $\displaystyle=\mathbb{E}[\ell(\mathbf{x})],$
(7) $\displaystyle\kappa(\mathbf{x},\mathbf{x}^{\prime})$
$\displaystyle=\mathbb{E}[(\ell(\mathbf{x})-\mu(\mathbf{x}))(\ell(\mathbf{x}^{\prime})-\mu(\mathbf{x}^{\prime}))].$
(8)

In this work, three common acquisition functions are used to guide the
iterative selection of $\mathbf{x}$ 12, 41: (1) expected improvement (EI), (2)
maximum probability of improvement (MPI), and (3) lower confidence bound
(LCB). We introduce these three distinct acquisition functions to illustrate
the robustness of the computer vision-BO in loop method for discovering
control parameter values that produce unstable fluid flow in both milliscale
and microscale droplet-generating devices.

We begin an experiment by initializing the algorithm using LHS. Then, we
iterate interleaving sampling and update steps. For every iteration of the BO
in loop, 10 new $\mathbf{x}$ are acquired using the acquisition function’s
decision policy. The decisions of the acquisition function are based on the GP
parameter space estimation of $\ell$, computed from the labeled batch set
$\mathcal{X}^{(N)}$, which contains all previous batch results. These new
$\mathbf{x}$ are concatenated to $\mathcal{X}^{(N)}$ and $\ell$ are
concatenated to $\mathcal{S}$ before acquiring the next batch of $\mathbf{x}$.
The decision policies for each acquisition function used in this study are
detailed next.

####  2.4.1 Expected Improvement Acquisition

Expected improvement (EI) is a decision policy that samples optima
$\mathbf{x}$ from the GP by evaluating the function $\ell(\mathbf{x})$ and
comparing the value to the current running minimum function evaluation
$\ell^{*}=\textbf{{min}}(\ell(\mathbf{x}))$. Since our batch size is 10
samples per iteration, the EI acquisition function outputs 10 samples that
have the highest improvement from previous $\ell^{*}$ 11, 41:

$a_{\textrm{EI}}(\mathbf{x})=\int^{\ell^{*}}_{-\infty}(\ell^{*}-\ell(\mathbf{x}))\mathcal{N}(\ell;\mu(\mathbf{x}),\kappa(\mathbf{x},\mathbf{x}^{\prime}))\textrm{d}\ell.$
(9)

A local penalization evaluator is used which penalizes points closely sampled
to each other within a batch, thus, the best point is selected first, then the
second best (within some distance), and so on.

####  2.4.2 Maximum Probability of Improvement Acquisition

Maximum probability of improvement (MPI) is a decision policy that samples
optima $\mathbf{x}$ similarly to EI, however, MPI does not scale the function
values proportional to the magnitude of improvement. This means that MPI
computes the location where $\ell^{*}=\textbf{{min}}(\ell(\mathbf{x}))$ but
does consider how much the function value $\ell$ improves from the current
running minimum $\ell^{*}$. Thus, the acquisition of MPI is the same as EI but
without the $(\ell^{*}-\ell)$ term 11, 41:

$a_{\textrm{MPI}}(\mathbf{x})=\int^{\ell^{*}}_{-\infty}\mathcal{N}(\ell;\mu(\mathbf{x}),\kappa(\mathbf{x},\mathbf{x}^{\prime}))\textrm{d}\ell.$
(10)

####  2.4.3 Lower Confidence Bound Acquisition

Lower confidence bound (LCB) is a decision policy that samples optima
$\mathbf{x}$ through an explicit trade-off scheme between exploitation and
exploration. A pure exploitation acquisition function outputs optima
$\mathbf{x}$ exactly at evaluated GP function mean values $\mu(\mathbf{x})$
that minimize $\ell$, i.e., where the standard deviation
$\sqrt{\kappa(\mathbf{x},\mathbf{x}^{\prime})}$ about the mean is zero. A pure
exploration acquisition function outputs optima $\mathbf{x}$ within some large
number of standard deviations about the mean. The LCB acquisition function
balances exploiting the GP mean values with exploring the GP variance by
varying a hyperparameter $\beta>0$ 41:

$a_{\textrm{LCB}}(\mathbf{x};\beta)=\mu(\mathbf{x})-\beta\sqrt{\kappa(\mathbf{x},\mathbf{x}^{\prime})}.$
(11)

###  2.5 Evaluating Parameter Convergence

To evaluate the performance of the presented BO and computer vision method, we
visualize the topology of the objective $\ell$ within the parameter space of
each trial for both devices. First, density estimations of each experimentally
sampled parameter are shown to illustrate the regions of the parameter space
most searched by the tool for each device and each acquisition function. Next,
a subset of the imaged droplet flows are shown. Finally, the measured $\ell$
are plotted within the parameter space to demonstrate the learning of the
method. Parameters with low $\ell$ generate the most circular and highest
yield droplet flows for both devices at both milliscale and microscale. We
compute the sample efficiency of the method by determining how many sampled
points fall within the feasibility bounds. We identify that $\ell<0.75$
demarcates conditions that can feasibly generate droplets.

##  3 Results

###  3.1 Experimentally Sampled Conditions

(a) Inkjet Control Parameter Sampling

(b) Microfluidic Control Parameter Sampling

Figure 5: id=AES, comment=Adjust scale of this figure to improve
readability.Density of control parameter values explored by each acquisition
function over four batches of samples for the (a) inkjet device and (b)
microfluidic device. As the number of batches increases, BO becomes more
certain of what control parameter values produce the most optimized droplets.

The computer vision-driven BO in loop approach presented in this paper is
demonstrated to consistently discover optimum control parameter conditions
within both milliscale and microscale droplet-generating devices.
Demonstrating the optimization capabilities of this approach at two different
length scales illustrates its robustness and utility as a universal tool for
experimentalists to tune their droplet devices without physics-based models,
regardless of length scale. Furthermore, even though the phenomena of droplet
breakup is governed by non-linear differential equations, our approach
efficiently determines an appropriate set of experimental parameters to obtain
the desired drops.

Fig. 5(a) illustrates the regions of the inkjet parameter space explored by
each acquisition function for each batch of conditions acquired by BO in loop.
It should be noted that these curves are density estimations of sampling
frequency for each parameter and, thus, interpolate the shape of the curve
between points. The inkjet device is driven by control parameters of pressure,
frequency, and speed such that BO predicts parameters $\mathbf{x}\in X^{(3)}$
that minimize $\ell(\mathbf{x})$. For the pressure condition, it is shown that
BO discovers samples across the whole range of pressures, indicating that the
droplet flows are not sensitive to changes in pressure. Conversely, BO
sampling is more concentrated for the frequency and speed conditions,
indicating that high circularity and high yield droplet flows form using low
frequency and high speed conditions.

(a) Inkjet Droplets

(b) Microfluidic Droplets

Figure 6: Experimentally generated droplet samples labeled with the computer
vision-computer optimization objective, $\ell$ for the (a) inkjet device and
(b) microfluidic device. Control parameters that generate a sample with
$\ell<0.75$ are shown to produce discrete droplets, hence, exhibiting the
feasibility of Rayleigh and capillary instability occurring at those
conditions. Conversely, control parameters that generate a sample with
$\ell\geq 0.75$ are shown to have continuous or connected fluid flows,
implying it is not feasible for Rayleigh or capillary instability to occur at
these conditions.

Fig. 5(b) illustrates the regions of the microfluidic parameter space explored
by each acquisition function. The microfluidic device is driven by two control
parameters of water pressure and oil pressure such that BO predicts parameters
$\mathbf{x}\in X^{(2)}$ that minimize $\ell(\mathbf{x})$. Unlike the inkjet
device, the droplet flows are shown to be sensitive to both the control
parameters in the microfluidics device. BO acquires points from low water
pressure conditions and high oil pressure conditions.

The results from Fig. 5 illustrate the sampling mechanics of the GP-EI, GP-
MPI, and GP-LCB BO models when all functions are initialized using the same
dataset for each respective device. The regions of the parameter space where
$\ell$ is minimized are learned using the sampling mechanics of BO. It will be
shown that for each experiment, all of the tested BO models converge to
similar optimum control parameters.

###  3.2 Learned Parameter Space Topology

Table 1: Time required to perform each process step in Bayesian optimization of droplets. Process steps are separated into whether it occurs on the software- or hardware-end of the optimization procedure. Averages and standard deviations of 12 model runs are shown, each run generating 10 droplet samples. The total time for convergence for 4 rounds of updates plus the initialization dataset is $total\ time\ per\ sample\ \textrm{[s]}*60$, since 60 samples are generated and analyzed for an entire optimization procedure. Process Step | Time per Sample [s] | Type
---|---|---
Read Images | 0.10 $\;\;\;\pm$ 0.01 | Software
Computer Vision | 2.7 $\;\;\;\;\;\pm$ 0.2 | Software
Retrain Surrogate | 0.40 $\;\;\;\pm$ 0.1 | Software
Acquisition | 0.50 $\;\;\;\pm$ 0.1 | Software
Device Configuration | 70.0 $\;\;\;\pm$ 10 | Hardware
Print Droplets | 30.0 $\;\;\;\pm$ 5 | Hardware
Image Droplets | 35.0 $\;\;\;\pm$ 10 | Hardware
Total | 3.7 $\;\;\;\;\;\pm$ 0.2 | Software
Total | 135.0 $\;\,\pm$ 15 | Hardware
Total | 138.7 $\,\pm$ 15 | Both
| Time for Convergence [hr]
Total | 0.062 $\;\,\pm$ 0.0004 | Software
Total | 2.25 $\;\;\;\,\pm$ 0.03 | Hardware
Total | 2.31 $\;\,\,\,\pm$ 0.03 | Both

Our method discovers the control parameter conditions that produce droplets
due to fluid instabilities at both the millimeter and micrometer scales using
the same computer vision-driven BO in loop method. Table 1 notes the time
required to perform each process step. Our method takes $138.7$ seconds per
sample to run, and $2.3$ hours to run the full optimization procedure,
compared to $>70$ hours via prior methods 7. The time for convergence is shown
to be dominated by hardware calibration and synthesis processes (97.3% of
total convergence time), rather than software processing or model training
(2.7% of total convergence time). This is true since the learning process
occurs in the loop and only $O(10^{1})$ data points are used in our GP
approach, compared to the $O(10^{6})$ data points used for upfront training in
prior neural network approaches 7. Fig. 6 illustrates a subset of droplet
flows from stable flows, i.e., outside of the feasibility bounds, $\ell\geq
0.75$, and from unstable flows, i.e., inside the feasibility bounds
$\ell<0.75$. Fig. 7 shows the parameter space topology of the objective values
$\ell$ obtained from each experiment. It should be noted that topologies are
Gaussian interpolated from the raw data for clarity; no extrapolation is done.
The sample efficiency of each BO model is determined based on the number of
sampled points that fall within the feasibility bounds – meaning that a model
with higher sample efficiency requires fewer samples to achieve an optimum
solution.

(a) Learned Rayleigh Unstable Conditions

(b) Learned Capillary Unstable Conditions

Figure 7: id=AES, comment=Increased figure font size.Control parameter values
learned by BO to form (a) Rayleigh and (b) capillary instability to generate
optimized droplets within the inkjet device and microfluidics device,
respectively. Three rows of figures are illustrated, one row for each
independent experiment run with a different acquisition function (EI, MPI, and
LCB). The first column illustrates the values of all control parameter values
sampled by all batches of BO over the course of the experiment. The second
column illustrates the where BO has discovered that Rayleigh/capillary
instability can occur within the parameter space ($\ell<0.75$). The third
column shows the measured value of $\ell$ within the feasibility bounds. Lower
values of $\ell$ denote more optimized droplet flows.

(a) Inkjet Drop Diameter

(b) Inkjet Objective Value

(c) Microfluidic Drop Diameter

(d) Microfluidic Objective Value

Figure 8: Reynolds number versus Weber number for (a) inkjet drop diameter,
(b) inkjet objective value, $\ell$; and capillary number versus Weber number
for (c) microfluidic drop diameter, (d) microfluidic objective value, $\ell$.
Each subfigure includes all $n=120$ synthesized droplet patterns across the
three acquisition policy experiments, not including the initialization set.
The dashed curves represent the envelop of the parameter space explored. For
the inkjet device, the diameter of the in-air droplets are computed using a
spherical end-cap and the measured contact angle 42. For the microfluidics
device, the velocity and fluid density are referring to the continues phase
(mineral oil) and the interfacial tensions for water in oil. The location of
the most optimal droplet pattern, i.e., lowest objective value, is indicated
as a white star in (b) and (d).

For the inkjet device, BO discovers the feasible control parameter values for
Rayleigh instability to be in low-mid range frequencies and mid-high range
translation speeds using different decision policy acquisition functions.
Moreover, Fig. 7(a) illustrates that this instability may exist for all
pressure values within the inkjet device. The importance ranking of each
parameter-objective relationship is indicated using SHAP in Fig. 5(a) for the
inkjet device 43. The model sample efficiencies when integrated into the
optimization of the inkjet device are: (1) 65.0% using GP-EI, (2) 45.0% using
GP-MPI, and (3) 40.0% using GP-LCB. Fig. 8(b) shows a scatter plot of Re
versus We for every inkjet-printed sample output by the BO model; the colorbar
indicates the objective value. These experimental results illustrate the
highly non-linear relationship between Re, We, and the objective value.
Despite the complexity of this parameter space, our method is still shown to
discover the correct optimized region of the parameter space as three
independent experimental trials converge to the same optimum conditions, as
shown in Fig. 7. These optimized conditions fall within the regions of the
Re–We space that have optimal drop-on-demand printing conditions, as suggested
in the literature 29, 44. Our results in Fig. 8(b) validate the expectation
that droplets do not reliably form due to Rayeligh instability at
$\textrm{We}\lesssim 3$ 44. The computed We may differ slightly from the
expected value since the jet diameter contracts and expands as a function of
velocity 45. Comparing these results to Fig. 7(a) and Fig. S-2, the optimized
parameter space is shown to correspond to low actuation frequency and fast
translation speed throughout the pressure range. Additional information
regarding the impact of control parameters on droplet optimization can be
found in the Supplementary Information.

For the microfluidic device, BO discovers the feasible control parameter
values for capillary instability to be in low-mid range water pressure values
for the whole range of oil pressure values using different decision policy
acquisition functions, as shown in Fig. 7(b). The importance ranking of each
parameter-objective relationship is indicated using SHAP in Fig. 5(b) for the
microfluidic device 43. The model sample efficiencies when integrated into the
optimization of the microfluidic device are: (1) 67.5% using GP-EI, (2) 97.5%
using GP-MPI, and (3) 75.0% using GP-LCB. Predictions of droplet shape and
size as a function of the device control parameters remain intractable to
predict 14. Fig. 8(c)–8(d) illustrate the non-linear relationship between We,
Ca, and the objective value, further demonstrating the significance of
exploring the experimental configuration space via BO. The optimized objective
value is found in the region $\textrm{Ca}\sim$ 0.03 and $\textrm{We}\sim$
0.002. Comparing these results to the optimization shown in Fig. 7(b) and Fig.
S-3, a roughly linear band, in which the oil and water pressures increase
proportionally to each other, can be seen. However, the non-linear bounds on
the learned optimized region in the parameter space, shown again in Fig. 7(b),
are a reminder of the non-linear physics governing droplet formation and
characteristics.

##  4 Discussion

Our work addresses the challenge of developing a universal tool for optimizing
droplet-generating devices at multiple length scales. It is often the case
where no analytical function exists to map device control parameters to a
target material property, thus, resulting in wasteful trial-and-error
experimentation to discover those optimized conditions. We demonstrate the
ability of a computer vision-integrated Bayesian optimization framework to
accurately and rapidly discover control parameter values with repeatability
that optimize the generated droplets with no a prioi domain knowledge of the
device. The significance of this paper is in providing a scale-invariant and
physical model-agnostic method for droplet-generating devices optimization
that can be used to flexibly optimize many different droplet-generating
systems in $2.3$ hours.

We demonstrate that for these milliscale and microscale droplet-generating
devices, a machine learning algorithm can learn which control parmeter values
produce Rayleigh/capillary instabilities and, in turn, transition a continuous
fluid stream to a decomposed flow of droplets. The performance of this method
is demonstrated using three independent experiments for each device, each with
a different acquisition function but with the same initialization dataset and
surrogate model: (1) GP-EI, (2) GP-MPI, and (3) GP-LCB. After four batches of
data sampling and model updates, all three experiments converge to similar
optimized regions of the parameter space for both length scale devices. We
illustrate the significance of our method in an application space where
droplets optimized for high circularity and high yield do not always
correspond with droplets of a certain size, see Fig. 8(a)–8(b), hence,
demonstrating the power of a flexible ML method to predict the values of
control parameters that optimize droplets. Convergence occurs within four
rounds of sampling, resulting in a total average optimization time of $2.3$
hours, compared to $>70$ hours previously reported using supervised learning,
hence, accelerating throughput by $30\times$ 7. Unlike previous literature
where the majority of the optimization time is dominated by model training,
the majority of optimization time using our method is dominated by running
experiments on the device hardware. Therefore, this method provides an
extremely fast way to optimize droplets within complex parameter spaces and
has the potential to be accelerated further by an increase in experimental
throughput.

For the inkjet device, converged conditions are the full pressure range, low
piezoelectric frequencies, and high axial translation speeds. For the
microfluidic device, converged conditions are low-mid water pressures and high
oil pressures. However, each of these acquisition functions has a different
sampling efficiency, indicating that the acquisition function affects the rate
of learning differently depending on the length scale of the droplets
generated by a device. The highest efficiency acquisition function for the
inkjet device is GP-EI where 65.0% of sampled points are within the feasiblity
bounds, whereas the highest efficiency acquisition function for the
microfluidic device is GP-MPI where 97.5% of sampled points are within the
feasibility bounds. The differences in these results can be explained by each
device having different governing physics and a different number of control
parameters that comprise the parameter space. Although GP-LCB results in a low
sampling efficiency, it provides a powerful mode of exploration rather than
exploitation, which could be used to improve our understanding of the
governing physics of droplet formation across length scales. It should be
noted that even though the acquisition function was varied in the computer
vision-Bayesian optimization algorithm, all experiments results in convergence
to optimized droplet flows.

##  5 Conclusion

We present a machine learning-based method for rapid optimization of several
droplet-generating devices at different length scales. Tuning an experimental
device to produce an optimized product is dynamic; the optimized conditions
change based on the operating length scale of a device due to the governing
physics of the device changing. Thus, there is the challenge to design a
universal method for scale-invariant droplet optimization. In this study, we
develop a method of Bayesian optimization in loop with computer vision
characterization to discover the optimum conditions for generating droplets at
two different length scale experimental devices in $2$ hours, without upfront
model training.

Through several trials of experimentation, we demonstrate that our machine
learning and computer vision approach discovers optimized droplet-generating
device control parameter values at both the milliscale and microscale.
Furthermore, our method does not require knowledge of droplet size dependence
on control parameters, thus, this optimization approach is demonstrated to be
scale-invariant. As no physical or mathematical model is required to perform
optimization, this optimization approach is also demonstrated to be physical
model-agnostic; meaning that this approach can be extended to explore more
complicated devices like those using non-Newtonian fluids, polymeric fluids
that crosslink in flow, or phase-change materials with few adjustments.
Moreover, our method implements probabilistic surrogate model training and
retraining in parallel with data acquisition, significantly reducing the time
required to run the optimization procedure compared to methods of upfront
model training. Developing this multiscale droplet optimization method
provides researchers with a universal tool for accelerating smart device
conditioning and reduces wasteful trial-and-error experimentation.

We thank James Serdy (MIT) for knowledge contributions and engineering
expertise in hardware development and running physical experiments. We thank
Dr. Armi Tiihonen for guidance and assistance with Bayesian optimization of
physical systems. We thank Dr. Shijing Sun for contributions to methodology
development and framing of this paper. This material is based upon work
primarily supported by the Center to Center (C2C) International Collaboration
on Advanced Photovoltaics as part of the Engineering Research Center Program
of the National Science Foundation and the Office of Energy Efficiency and
Renewable Energy of the Department of Energy under NSF Cooperative Agreement
No. EEC-1041895. Iddo Drori would like to thank Google for a cloud research
grant.

Data Availability. All data and code are available at https://github.com/PV-
Lab/ML-Multiscale-Droplets

Conflict of Interest Statement. Author T.B. holds equity in a start-up company
(Xinterra) focused on commercializing machine learning technologies for
accelerated materials development.

Supporting Information. Additional information provided for the parameter
space search of each decision policy, importance ranking of control
parameters, relevant dimensionless numbers, and images of the experimental
setups, including the data collected for all experiments.

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GitHub: https://github.com/PV-Lab/ML-Multiscale-Droplets

##  6 For Table of Contents only

##  S-1 Supporting Information:
A Machine Learning and Computer Vision Approach to Rapidly Optimize Multiscale
Droplet Generation

Alexander E. Siemenn,∗,† Evyatar Shaulsky,‡ Matthew Beveridge,¶ Tonio
Buonassisi,† Sara M. Hashmi,∗,‡ and Iddo Drori∗,¶

${\dagger}$Department of Mechanical Engineering, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA

${\ddagger}$Department of Chemical Engineering, Northeastern University,
Boston, MA 02115, USA

${\mathparagraph}$Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
For each loop of optimization, the regions of the parameter space where new
optima are acquired by each decision policy are highlighted in dark blue in
Figure S-2 for the inkjet device and in Figure S-3 for the microfluidic
device. For each of these sampled batches, all of the acquired points are
plotted as box plots to illustrate the ranges of objective values. This
information is useful when designing a fluid device such that a researcher may
engineer the device to have experimental parameter values within the regions
of high acquisition value.

Figure S-4 illustrates the running minimum objective value across all sampled
points for the inkjet and microfluidic devices. As the autonomous optimization
procedure runs, more highly-optimized droplet structures are discovered.

The importance impact of each experimental parameter on the objective value
$\ell$ is illustrated in Figure S-5. For the inkjet device, frequency is
illustrated to have a much larger impact on the objective value compared to
pressure and translation speed. For the microfluidic device, water pressure is
illustrated to have only a marginally larger impact on the objective valued
compared to the water pressure. These results are useful for designing new
fluid devices since understanding the impact of a device’s input on the output
product provides insight into which input parameters should receive more
attention during tuning.

The parameter space envelope for each BO acquisition function shown in Figures
7 and 8. For a microfluidics device, the slope of the linear ratio correlation
will change depending on the properties of the device 32, 5. The width of the
instability drop formation region will change as a result of the fluids
characteristics such as viscosity ratio. In the high-valued regions of Ca and
We numbers, the oil pressure exceeds the viscosity ratio and, in turn, results
in no water flow or water backflow. Moreover, in the low-valued regions of Ca
and We numbers, the water pressure exceeds the viscosity ratio and forms a
continuous water stream instead of droplets. In the specific case of flow-
focusing microsystem, the drop formation region can be divided into three
distinguished sections. The first regime, "squeezing" or "geometric
restriction", appears at the low flows/pressures and low capillary number
area. The inner fluid is touching the restriction wall before breaking into
individual drops 36. In the squeezing regime, there is a good correlation
between the drop size and the fluid flow fraction. The second regime,
"dripping" or "transition" appearing at the mid-range flows/pressures and
capillary number. In this regime, the formation of the droplets is due to an
increased flow velocity inside the restrictions that leads to pinched off
droplets. In this case, smaller droplets can be formed, and as we are
increasing the fraction ratio between the inner and continuous liquids, a high
volume of small drops is generated. The third regime is the "jetting" that
appears at the highest pressures and capillary numbers, where the droplets
form after the restriction.

(a) Inkjet Device Hardware

(b) Microfluidics Device Hardware

Figure S-1: Images of the (a) inkjet droplet-generator and the (b)
microfluidics droplet-generator.

(a) Inkjet EI Acquisition Value per Batch

(b) Inkjet EI Objective Value per Batch

(c) Inkjet MPI Acquisition Value per Batch

(d) Inkjet MPI Objective Value per Batch

(e) Inkjet LCB Acquisition Value per Batch

(f) Inkjet LCB Objective Value per Batch

Figure S-2: Acquisition function search spaces by batch in 3-dimensions. The
regions where each acquisition function is likely to draw points from within
the inkjet parameter space are illustrated in blue for each acquisition
function: (a) EI, (c) MPI, (e) LCB – these denote regions of high acquisition
value. The initialization acquisition space is a uniform color because points
are picked uniformly during initialization via LHS. Based on the mechanics of
each acquisition function, the acquisition value is computed by balancing the
exploitation of the GP posterior mean and exploring the GP (co)variance. The
acquisition search space evolves every loop, which updates the sampling
locations of data points, shown for each acquisition function: (b) EI, (d)
MPI, (f) LCB. The reason why the same data point that maximizes the
acquisition value is not sampled 10 times for every batch in the loop is
because of evaluator used is local penalization – points sampled close to each
other are penalized, thus, the best point is selected first, then the second
best (within some distance), and so on.

(a) Microfluidic EI Acquisition Value per Batch

(b) Microfluidic EI Objective Value per Batch

(c) Microfluidic MPI Acquisition Value per Batch

(d) Microfluidic MPI Objective Value per Batch

(e) Microfluidic LCB Acquisition Value per Batch

(f) Microfluidic LCB Objective Value per Batch

Figure S-3: Acquisition function search spaces by batch in 2-dimensions. The
regions where each acquisition function is likely to draw points from within
the microfluidic parameter space are illustrated in blue for each acquisition
function: (a) EI, (c) MPI, (e) LCB – these denote regions of high acquisition
value. The initialization acquisition space is a uniform color because points
are picked uniformly during initialization via LHS. Based on the mechanics of
each acquisition function, the acquisition value is computed by balancing the
exploitation of the GP posterior mean and exploring the GP (co)variance. The
acquisition search space evolves every loop, which updates the sampling
locations of data points, shown for each acquisition function: (b) EI, (d)
MPI, (f) LCB. The reason why the same data point that maximizes the
acquisition value is not sampled 10 times for every batch in the loop is
because of evaluator used is local penalization – points sampled close to each
other are penalized, thus, the best point is selected first, then the second
best (within some distance), and so on.

(a) Inkjet Running Best Objective Value, $\ell^{*}$

(b) Microfluidic Running Best Objective Value, $\ell^{*}$

Figure S-4: The running minimum objective function evaluation $\ell^{*}$ for
(a) the inkjet device and (b) the microfluidic device. The optimum unstable
fluid conditions that generate high yield and high uniformity droplets are
labeled by $\ell^{*}\in[0,1]$ – the running minimum objective value.
$\ell^{*}$ as a function of sample number is illustrated for each acquisition
function after the 20 LHS initialization data points. The vertical axes are
scaled differently for the inkjet and microfluidics devices, this scaling is
arbitrary and is a result of the different droplet generating dynamics within
each parameter space.

(a) Inkjet SHAP Feature Importance

(b) Microfluidic SHAP Feature Importance

Figure S-5: The SHAP feature importance of the (a) inkjet and (b) microfluidic devices 43. Random forest regression with an 80/20 training/test split is used to compute the SHAP feature importance of all acquisition function-sampled data. Each printing condition is shown to have a certain SHAP value – the magnitude of the SHAP value describes how much that condition impacts the objective values $\ell$. For example, low frequency values (blue) are shown to generate low $\ell$ values (high yield, uniform droplets), i.e., illustrated by high negative SHAP values; high frequency values (red) are shown to generate high $\ell$ values (low yield, not uniform droplets), i.e., illustrated by high positive SHAP values. High speed values have little impact on $\ell$, whereas low speed values generate high $\ell$ values. Pressure, in general, has little impact on $\ell$, shown by the clustering of all values of pressure conditions centered on a 0.0 SHAP value. Table S-1: Inkjet conditions for three independent Bayesian experiments of droplet structures using three different decision policies: EI, MPI, LCB. The initialization dataset is obtained via LHS. This table details all of the experiments conducted for the Bayesian experimentation and optimization of droplet structures using the inkjet droplet deposition system. The synthesis conditions of $N=3$ control parameters: pressure, frequency, and speed, all reported as normalized values. The last column is the computer vision-computed loss, $\ell$, for each measurement. Sample | Batch | Method | Pressure | Frequency | Speed | Loss
---|---|---|---|---|---|---
1 | Initialization | LHS | 0.583 | 0.719 | 0.002 | 1.000
2 | Initialization | LHS | 0.917 | 0.294 | 0.059 | 1.000
3 | Initialization | LHS | 0.667 | 0.552 | 0.118 | 1.000
4 | Initialization | LHS | 0.333 | 0.927 | 0.165 | 1.000
5 | Initialization | LHS | 0.083 | 0.169 | 0.226 | 1.000
6 | Initialization | LHS | 0.083 | 0.491 | 0.272 | 1.000
7 | Initialization | LHS | 0.167 | 0.660 | 0.312 | 1.000
8 | Initialization | LHS | 0.750 | 0.106 | 0.373 | 1.000
9 | Initialization | LHS | 0.250 | 0.303 | 0.443 | 0.717
10 | Initialization | LHS | 0.000 | 0.958 | 0.483 | 1.000
11 | Initialization | LHS | 0.500 | 0.500 | 0.520 | 0.720
12 | Initialization | LHS | 0.000 | 0.050 | 0.581 | 0.564
13 | Initialization | LHS | 0.750 | 0.036 | 0.608 | 0.630
14 | Initialization | LHS | 0.917 | 0.817 | 0.691 | 1.000
15 | Initialization | LHS | 0.417 | 0.377 | 0.704 | 0.728
16 | Initialization | LHS | 0.250 | 0.207 | 0.779 | 0.760
17 | Initialization | LHS | 0.667 | 0.423 | 0.811 | 0.678
18 | Initialization | LHS | 1.000 | 0.770 | 0.861 | 1.000
19 | Initialization | LHS | 0.000 | 0.638 | 0.930 | 1.000
20 | Initialization | LHS | 0.417 | 0.850 | 0.981 | 0.739
21 | Batch 1 | EI | 0.000 | 0.000 | 1.000 | 1.000
22 | Batch 1 | EI | 1.000 | 0.000 | 1.000 | 0.831
23 | Batch 1 | EI | 0.333 | 0.065 | 0.996 | 0.562
24 | Batch 1 | EI | 0.833 | 0.032 | 0.989 | 0.665
25 | Batch 1 | EI | 0.000 | 0.090 | 0.992 | 0.567
26 | Batch 1 | EI | 0.333 | 0.007 | 0.992 | 0.850
27 | Batch 1 | EI | 0.833 | 0.109 | 0.972 | 0.393
28 | Batch 1 | EI | 0.000 | 0.029 | 0.867 | 0.715
29 | Batch 1 | EI | 0.833 | 0.013 | 0.985 | 0.715
30 | Batch 1 | EI | 0.000 | 0.013 | 0.995 | 0.840
31 | Batch 2 | EI | 0.833 | 0.000 | 1.000 | 0.812
32 | Batch 2 | EI | 0.083 | 0.027 | 0.958 | 0.736
33 | Batch 2 | EI | 1.000 | 0.076 | 0.969 | 0.603
34 | Batch 2 | EI | 0.000 | 0.197 | 0.979 | 0.411
35 | Batch 2 | EI | 0.083 | 0.009 | 1.000 | 0.833
36 | Batch 2 | EI | 1.000 | 1.000 | 1.000 | 1.000
37 | Batch 2 | EI | 1.000 | 0.000 | 1.000 | 0.829
38 | Batch 2 | EI | 0.917 | 0.007 | 0.007 | 1.000
39 | Batch 2 | EI | 0.000 | 0.009 | 0.119 | 0.521
40 | Batch 2 | EI | 0.083 | 0.998 | 0.998 | 0.790
41 | Batch 3 | EI | 1.000 | 0.201 | 1.000 | 0.685
42 | Batch 3 | EI | 0.833 | 0.176 | 0.999 | 0.686
43 | Batch 3 | EI | 1.000 | 0.223 | 0.970 | 0.673
44 | Batch 3 | EI | 1.000 | 0.332 | 0.996 | 0.663
45 | Batch 3 | EI | 0.833 | 0.192 | 0.990 | 0.685
46 | Batch 3 | EI | 0.333 | 0.165 | 0.999 | 0.366
47 | Batch 3 | EI | 0.583 | 0.147 | 1.000 | 0.361
48 | Batch 3 | EI | 0.583 | 0.228 | 0.986 | 0.702
49 | Batch 3 | EI | 0.750 | 0.149 | 0.995 | 0.368
50 | Batch 3 | EI | 0.833 | 0.298 | 0.987 | 0.682
51 | Batch 4 | EI | 0.000 | 0.147 | 1.000 | 0.460
52 | Batch 4 | EI | 1.000 | 0.514 | 1.000 | 1.000
53 | Batch 4 | EI | 0.167 | 0.143 | 1.000 | 0.449
54 | Batch 4 | EI | 0.250 | 0.136 | 1.000 | 0.432
55 | Batch 4 | EI | 0.000 | 0.447 | 1.000 | 0.809
56 | Batch 4 | EI | 0.417 | 0.131 | 1.000 | 0.431
57 | Batch 4 | EI | 0.250 | 0.482 | 1.000 | 0.767
58 | Batch 4 | EI | 0.000 | 0.137 | 0.890 | 0.449
59 | Batch 4 | EI | 0.250 | 0.145 | 0.954 | 0.434
60 | Batch 4 | EI | 0.917 | 0.492 | 0.994 | 1.000
21 | Batch 1 | MPI | 0.417 | 0.000 | 1.000 | 0.885
22 | Batch 1 | MPI | 1.000 | 0.027 | 1.000 | 0.680
23 | Batch 1 | MPI | 0.083 | 0.036 | 0.907 | 0.704
24 | Batch 1 | MPI | 1.000 | 0.028 | 0.794 | 0.634
25 | Batch 1 | MPI | 0.000 | 0.066 | 0.986 | 0.640
26 | Batch 1 | MPI | 1.000 | 0.518 | 1.000 | 1.000
27 | Batch 1 | MPI | 0.000 | 0.058 | 0.523 | 0.516
28 | Batch 1 | MPI | 1.000 | 0.014 | 0.402 | 0.629
29 | Batch 1 | MPI | 0.000 | 0.476 | 0.999 | 0.809
30 | Batch 1 | MPI | 0.083 | 0.004 | 0.002 | 1.000
31 | Batch 2 | MPI | 0.000 | 0.000 | 0.644 | 0.877
32 | Batch 2 | MPI | 1.000 | 0.000 | 0.669 | 0.811
33 | Batch 2 | MPI | 0.000 | 0.000 | 0.879 | 0.915
34 | Batch 2 | MPI | 0.000 | 0.055 | 0.491 | 0.539
35 | Batch 2 | MPI | 0.000 | 0.001 | 0.748 | 0.915
36 | Batch 2 | MPI | 0.750 | 0.060 | 0.826 | 0.510
37 | Batch 2 | MPI | 0.500 | 1.000 | 0.717 | 1.000
38 | Batch 2 | MPI | 0.000 | 0.048 | 0.597 | 0.474
39 | Batch 2 | MPI | 0.917 | 0.027 | 0.715 | 0.617
40 | Batch 2 | MPI | 0.000 | 0.009 | 0.514 | 0.766
41 | Batch 3 | MPI | 0.917 | 0.000 | 0.658 | 0.811
42 | Batch 3 | MPI | 0.000 | 0.037 | 0.556 | 0.689
43 | Batch 3 | MPI | 0.417 | 0.010 | 0.660 | 0.686
44 | Batch 3 | MPI | 1.000 | 0.022 | 0.948 | 0.681
45 | Batch 3 | MPI | 0.000 | 0.029 | 0.799 | 1.000
46 | Batch 3 | MPI | 1.000 | 0.141 | 0.414 | 1.000
47 | Batch 3 | MPI | 0.000 | 0.672 | 0.584 | 1.000
48 | Batch 3 | MPI | 1.000 | 1.000 | 1.000 | 1.000
49 | Batch 3 | MPI | 0.083 | 0.016 | 0.304 | 0.605
50 | Batch 3 | MPI | 1.000 | 0.079 | 0.981 | 0.511
51 | Batch 4 | MPI | 0.833 | 0.000 | 1.000 | 0.885
52 | Batch 4 | MPI | 0.083 | 0.019 | 0.905 | 0.759
53 | Batch 4 | MPI | 0.917 | 0.001 | 0.013 | 1.000
54 | Batch 4 | MPI | 0.000 | 0.086 | 0.962 | 0.636
55 | Batch 4 | MPI | 1.000 | 0.097 | 0.996 | 0.430
56 | Batch 4 | MPI | 0.083 | 0.011 | 0.029 | 1.000
57 | Batch 4 | MPI | 1.000 | 0.005 | 0.226 | 0.622
58 | Batch 4 | MPI | 0.083 | 0.053 | 0.948 | 1.000
59 | Batch 4 | MPI | 1.000 | 1.000 | 1.000 | 1.000
60 | Batch 4 | MPI | 0.083 | 0.003 | 0.045 | 1.000
21 | Batch 1 | LCB | 0.667 | 0.000 | 1.000 | 0.739
22 | Batch 1 | LCB | 0.500 | 0.077 | 0.996 | 0.529
23 | Batch 1 | LCB | 0.833 | 0.322 | 0.998 | 0.665
24 | Batch 1 | LCB | 0.917 | 0.053 | 0.980 | 0.610
25 | Batch 1 | LCB | 0.250 | 0.038 | 0.944 | 0.692
26 | Batch 1 | LCB | 0.500 | 0.015 | 0.550 | 0.655
27 | Batch 1 | LCB | 0.500 | 0.422 | 0.996 | 1.000
28 | Batch 1 | LCB | 0.083 | 0.021 | 0.554 | 0.692
29 | Batch 1 | LCB | 0.250 | 0.020 | 0.520 | 0.656
30 | Batch 1 | LCB | 0.750 | 0.702 | 1.000 | 1.000
31 | Batch 2 | LCB | 0.000 | 0.000 | 1.000 | 1.000
32 | Batch 2 | LCB | 0.917 | 0.037 | 0.118 | 1.000
33 | Batch 2 | LCB | 1.000 | 1.000 | 1.000 | 1.000
34 | Batch 2 | LCB | 0.000 | 0.910 | 0.021 | 1.000
35 | Batch 2 | LCB | 0.833 | 0.054 | 0.985 | 0.569
36 | Batch 2 | LCB | 0.000 | 0.021 | 0.177 | 0.470
37 | Batch 2 | LCB | 0.000 | 0.969 | 0.907 | 1.000
38 | Batch 2 | LCB | 1.000 | 0.996 | 0.107 | 1.000
39 | Batch 2 | LCB | 1.000 | 0.023 | 0.831 | 0.699
40 | Batch 2 | LCB | 0.083 | 0.009 | 0.963 | 0.815
41 | Batch 3 | LCB | 0.000 | 0.000 | 1.000 | 1.000
42 | Batch 3 | LCB | 0.917 | 0.030 | 0.006 | 1.000
43 | Batch 3 | LCB | 1.000 | 1.000 | 1.000 | 1.000
44 | Batch 3 | LCB | 0.000 | 0.735 | 0.003 | 1.000
45 | Batch 3 | LCB | 1.000 | 0.032 | 0.914 | 0.607
46 | Batch 3 | LCB | 0.083 | 0.997 | 0.978 | 1.000
47 | Batch 3 | LCB | 0.083 | 0.027 | 0.021 | 1.000
48 | Batch 3 | LCB | 0.917 | 0.947 | 0.038 | 1.000
49 | Batch 3 | LCB | 0.083 | 0.006 | 0.962 | 0.666
50 | Batch 3 | LCB | 1.000 | 0.160 | 0.004 | 1.000
51 | Batch 4 | LCB | 1.000 | 0.000 | 1.000 | 0.827
52 | Batch 4 | LCB | 0.083 | 0.071 | 0.542 | 0.464
53 | Batch 4 | LCB | 1.000 | 1.000 | 0.000 | 1.000
54 | Batch 4 | LCB | 0.083 | 0.930 | 0.964 | 1.000
55 | Batch 4 | LCB | 0.833 | 0.006 | 0.060 | 1.000
56 | Batch 4 | LCB | 1.000 | 1.000 | 1.000 | 1.000
57 | Batch 4 | LCB | 0.000 | 0.869 | 0.068 | 1.000
58 | Batch 4 | LCB | 0.250 | 0.065 | 0.992 | 0.569
59 | Batch 4 | LCB | 0.000 | 0.050 | 0.016 | 1.000
60 | Batch 4 | LCB | 0.917 | 0.009 | 0.848 | 0.746
Table S-2: Microfluidics conditions for three independent Bayesian experiments of droplet structures using three different decision policies: EI, MPI, LCB. The initialization dataset is obtained via LHS. This table details all of the experiments conducted for the Bayesian experimentation and optimization of droplet structures using the microfluidic droplet deposition system. The synthesis conditions of $N=2$ control parameters: water pressure and oil pressure, all displayed as normalized values. The last column is the computer vision-computed loss, $\ell$, for each measurement. Sample | Batch | Method | Water Pressure | Oil Pressure | Loss
---|---|---|---|---|---
1 | Initialization | LHS | 0.420 | 0.040 | 1.000
2 | Initialization | LHS | 0.043 | 0.083 | 0.352
3 | Initialization | LHS | 0.255 | 0.138 | 1.000
4 | Initialization | LHS | 0.727 | 0.168 | 1.000
5 | Initialization | LHS | 0.870 | 0.218 | 1.000
6 | Initialization | LHS | 0.155 | 0.277 | 0.365
7 | Initialization | LHS | 0.141 | 0.333 | 0.319
8 | Initialization | LHS | 0.812 | 0.399 | 1.000
9 | Initialization | LHS | 0.760 | 0.442 | 1.000
10 | Initialization | LHS | 0.948 | 0.465 | 1.000
11 | Initialization | LHS | 0.303 | 0.549 | 0.320
12 | Initialization | LHS | 0.208 | 0.599 | 0.291
13 | Initialization | LHS | 0.988 | 0.618 | 1.000
14 | Initialization | LHS | 0.575 | 0.674 | 1.000
15 | Initialization | LHS | 0.540 | 0.713 | 1.000
16 | Initialization | LHS | 0.648 | 0.796 | 1.000
17 | Initialization | LHS | 0.478 | 0.824 | 1.000
18 | Initialization | LHS | 0.068 | 0.887 | 1.000
19 | Initialization | LHS | 0.683 | 0.916 | 1.000
20 | Initialization | LHS | 0.392 | 0.979 | 0.196
21 | Batch 1 | EI | 0.305 | 1.000 | 0.145
22 | Batch 1 | EI | 0.321 | 0.987 | 0.157
23 | Batch 1 | EI | 0.297 | 0.993 | 0.080
24 | Batch 1 | EI | 0.315 | 0.949 | 0.176
25 | Batch 1 | EI | 0.290 | 0.964 | 0.167
26 | Batch 1 | EI | 0.302 | 0.915 | 0.188
27 | Batch 1 | EI | 0.261 | 0.999 | 0.073
28 | Batch 1 | EI | 0.330 | 0.992 | 0.186
29 | Batch 1 | EI | 0.285 | 0.936 | 0.172
30 | Batch 1 | EI | 0.294 | 0.881 | 0.194
31 | Batch 2 | EI | 0.228 | 1.000 | 0.132
32 | Batch 2 | EI | 0.219 | 0.989 | 0.137
33 | Batch 2 | EI | 0.214 | 0.924 | 0.134
34 | Batch 2 | EI | 0.203 | 0.997 | 0.201
35 | Batch 2 | EI | 0.227 | 0.948 | 0.168
36 | Batch 2 | EI | 1.000 | 0.000 | 1.000
37 | Batch 2 | EI | 0.176 | 0.995 | 0.242
38 | Batch 2 | EI | 0.243 | 0.987 | 0.170
39 | Batch 2 | EI | 0.202 | 0.899 | 0.168
40 | Batch 2 | EI | 0.260 | 0.995 | 0.131
41 | Batch 3 | EI | 0.261 | 1.000 | 0.090
42 | Batch 3 | EI | 0.260 | 0.977 | 0.115
43 | Batch 3 | EI | 0.272 | 0.999 | 0.105
44 | Batch 3 | EI | 0.099 | 0.000 | 1.000
45 | Batch 3 | EI | 0.002 | 0.000 | 1.000
46 | Batch 3 | EI | 0.251 | 0.992 | 0.111
47 | Batch 3 | EI | 0.879 | 1.000 | 1.000
48 | Batch 3 | EI | 0.254 | 0.987 | 0.097
49 | Batch 3 | EI | 0.258 | 0.935 | 0.153
50 | Batch 3 | EI | 0.270 | 0.960 | 0.151
51 | Batch 4 | EI | 0.000 | 0.252 | 1.000
52 | Batch 4 | EI | 0.009 | 0.234 | 1.000
53 | Batch 4 | EI | 0.019 | 0.316 | 1.000
54 | Batch 4 | EI | 0.031 | 0.188 | 0.971
55 | Batch 4 | EI | 0.005 | 0.372 | 1.000
56 | Batch 4 | EI | 0.003 | 0.296 | 1.000
57 | Batch 4 | EI | 0.030 | 0.199 | 1.000
58 | Batch 4 | EI | 0.001 | 0.197 | 1.000
59 | Batch 4 | EI | 0.049 | 0.251 | 0.483
60 | Batch 4 | EI | 0.003 | 0.166 | 1.000
21 | Batch 1 | MPI | 0.382 | 0.989 | 0.180
22 | Batch 1 | MPI | 0.360 | 0.947 | 0.194
23 | Batch 1 | MPI | 0.327 | 0.886 | 0.194
24 | Batch 1 | MPI | 0.332 | 0.996 | 0.102
25 | Batch 1 | MPI | 0.316 | 0.818 | 0.216
26 | Batch 1 | MPI | 0.316 | 0.938 | 0.163
27 | Batch 1 | MPI | 0.316 | 0.744 | 0.261
28 | Batch 1 | MPI | 0.281 | 0.861 | 0.178
29 | Batch 1 | MPI | 0.371 | 0.922 | 0.203
30 | Batch 1 | MPI | 0.295 | 0.995 | 0.132
31 | Batch 2 | MPI | 0.335 | 1.000 | 0.188
32 | Batch 2 | MPI | 0.334 | 0.979 | 0.180
33 | Batch 2 | MPI | 0.291 | 0.997 | 0.079
34 | Batch 2 | MPI | 0.313 | 0.994 | 0.180
35 | Batch 2 | MPI | 0.266 | 0.982 | 0.089
36 | Batch 2 | MPI | 0.324 | 0.999 | 0.191
37 | Batch 2 | MPI | 0.325 | 0.998 | 0.125
38 | Batch 2 | MPI | 0.357 | 0.987 | 0.202
39 | Batch 2 | MPI | 0.356 | 0.993 | 0.191
40 | Batch 2 | MPI | 0.228 | 0.998 | 0.164
41 | Batch 3 | MPI | 0.268 | 1.000 | 0.095
42 | Batch 3 | MPI | 0.246 | 0.962 | 0.091
43 | Batch 3 | MPI | 0.281 | 0.999 | 0.088
44 | Batch 3 | MPI | 0.270 | 0.953 | 0.109
45 | Batch 3 | MPI | 0.279 | 0.967 | 0.118
46 | Batch 3 | MPI | 0.260 | 0.921 | 0.166
47 | Batch 3 | MPI | 0.261 | 0.989 | 0.103
48 | Batch 3 | MPI | 0.289 | 0.992 | 0.123
49 | Batch 3 | MPI | 0.260 | 0.969 | 0.110
50 | Batch 3 | MPI | 0.253 | 0.890 | 0.135
51 | Batch 4 | MPI | 0.265 | 1.000 | 0.107
52 | Batch 4 | MPI | 0.276 | 1.000 | 0.070
53 | Batch 4 | MPI | 0.276 | 0.989 | 0.047
54 | Batch 4 | MPI | 0.274 | 0.974 | 0.082
55 | Batch 4 | MPI | 0.262 | 1.000 | 0.097
56 | Batch 4 | MPI | 0.253 | 0.957 | 0.139
57 | Batch 4 | MPI | 0.000 | 0.000 | 1.000
58 | Batch 4 | MPI | 0.252 | 0.980 | 0.124
59 | Batch 4 | MPI | 0.257 | 0.986 | 0.069
60 | Batch 4 | MPI | 0.268 | 0.952 | 0.141
21 | Batch 1 | LCB | 1.000 | 0.000 | 1.000
22 | Batch 1 | LCB | 0.000 | 0.503 | 1.000
23 | Batch 1 | LCB | 0.000 | 0.280 | 1.000
24 | Batch 1 | LCB | 0.000 | 0.000 | 1.000
25 | Batch 1 | LCB | 0.254 | 1.000 | 0.097
26 | Batch 1 | LCB | 0.001 | 0.172 | 1.000
27 | Batch 1 | LCB | 1.000 | 1.000 | 1.000
28 | Batch 1 | LCB | 0.142 | 0.495 | 0.316
29 | Batch 1 | LCB | 0.006 | 0.639 | 1.000
30 | Batch 1 | LCB | 0.452 | 1.000 | 0.250
31 | Batch 2 | LCB | 0.301 | 1.000 | 0.083
32 | Batch 2 | LCB | 0.277 | 0.846 | 0.107
33 | Batch 2 | LCB | 0.226 | 0.661 | 0.276
34 | Batch 2 | LCB | 0.161 | 0.997 | 0.430
35 | Batch 2 | LCB | 0.000 | 0.125 | 1.000
36 | Batch 2 | LCB | 0.178 | 0.428 | 0.319
37 | Batch 2 | LCB | 0.373 | 0.950 | 0.207
38 | Batch 2 | LCB | 0.149 | 0.204 | 0.443
39 | Batch 2 | LCB | 0.367 | 0.761 | 0.304
40 | Batch 2 | LCB | 0.335 | 0.534 | 1.000
41 | Batch 3 | LCB | 0.298 | 1.000 | 0.150
42 | Batch 3 | LCB | 0.246 | 0.875 | 0.094
43 | Batch 3 | LCB | 0.225 | 0.724 | 0.253
44 | Batch 3 | LCB | 0.394 | 0.991 | 0.192
45 | Batch 3 | LCB | 0.180 | 0.522 | 0.303
46 | Batch 3 | LCB | 0.195 | 0.993 | 0.283
47 | Batch 3 | LCB | 0.133 | 0.294 | 0.333
48 | Batch 3 | LCB | 0.121 | 0.000 | 1.000
49 | Batch 3 | LCB | 0.358 | 0.841 | 0.171
50 | Batch 3 | LCB | 0.303 | 0.607 | 0.331
51 | Batch 4 | LCB | 0.307 | 1.000 | 0.133
52 | Batch 4 | LCB | 0.264 | 0.812 | 0.236
53 | Batch 4 | LCB | 0.192 | 0.619 | 0.285
54 | Batch 4 | LCB | 0.216 | 0.993 | 0.189
55 | Batch 4 | LCB | 0.165 | 0.422 | 0.323
56 | Batch 4 | LCB | 0.431 | 0.999 | 0.310
57 | Batch 4 | LCB | 0.321 | 0.695 | 0.306
58 | Batch 4 | LCB | 0.141 | 0.250 | 0.373
59 | Batch 4 | LCB | 0.332 | 0.895 | 0.212
60 | Batch 4 | LCB | 0.141 | 0.807 | 0.477