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††thanks: Corresponding author<EMAIL_ADDRESS>
# Machine Learning Assisted Sorting of Active Microswimmers

Abdolhalim Torrik<EMAIL_ADDRESS>Department of Physical and
Computational Chemistry, Shahid Beheshti University, Tehran 19839-9411, Iran
Mahdi Zarif<EMAIL_ADDRESS>Department of Physical and Computational
Chemistry, Shahid Beheshti University, Tehran 19839-9411, Iran

###### Abstract

Active matter systems being in a non-equilibrium state, exhibit complex
behaviors such as self-organization and giving rise to emergent phenomena.
There are many examples of active particles with biological origins, including
bacteria and spermatozoa, or with artificial origins, such as self-propelled
swimmers and Janus particles. The ability to manipulate active particles is
vital for their effective application e.g. separating motile spermatozoa from
nonmotile and dead ones, to increase fertilization chance. In this study, we
proposed a mechanism – an apparatus – to sort and demix active particles based
on their motility values (Pèclet number). Initially, using Brownian
simulations we demonstrated the feasibility of sorting self-propelled
particles. Following this, we employed machine learning methods, supplemented
with data from comprehensive simulations that we conducted for this study, to
model the complex behavior of active particles. This enabled us to sort them
based on their Pèclet number. Finally, we evaluated the performance of the
developed models and showed their effectiveness in demixing and sorting the
active particles. Our findings can find applications in various fields,
including physics, biology, and biomedical science, where the sorting and
manipulation of active particles play a pivotal role.

active matter, sorting, Machine learning, trappment

##  I INTRODUCTION

Active matter systems – living and non-living – are defined by their ability
to consume energy, generate forces, and show synchronized self-organization
[1, 2]. These systems are intrinsically nonequilibrium and show complex
behaviors and emergent phenomena [3, 4, 5, 6]. They exist in various forms and
scales, and can be biological or artificial, macroscopic or microscopic in
nature. Examples of biological active systems with macroscopic origin include:
flocks of birds [7], schools of fish [8], swarms of insects [9, 10], and herds
of mammals [11]. Microscopic examples to highlight the dynamics of biological
active matter systems include: bacteria [12, 13], sperm cells [14, 15], amoeba
cell clusters [16, 17, 18], microbial biofilms [19], marine algae [20], and
zooplanktons [21]. Self-propelled artificial swimmers (catalytic, magnetic and
chemically driven micro-motors) [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32],
Janus particles [33, 34, 35, 36], and robotic swarms [37] are examples of
artificial active matter systems. All of the preceding examples are only a
portion of the spectrum of active matter systems.

Despite advancements in understanding these systems, there’s still a
challenge: the effective sorting of active particles based on their motility
values (Pèclet number). Manipulating active particles is a vital part of their
effective application, there are many ways to achieve this, examples of such
cases are: separation and rectification of swimming bacteria [38, 39, 40],
using microswimmers for separating a mixture of colloids [41], aggregation and
segregation of microswimmers based on their motilities [42, 43], using the
confinement to manipulate microswimmers [44, 45, 46, 47, 48], separation [49]
and sorting [50, 51] of microswimmers based on their chirality, motility [52]
and separating motile spermatozoa [53, 54]. However, a mechanism that can sort
active particles based on their motility values, such as the Pèclet value, is
yet to be developed.

Machine learning algorithms learn from the data and establish connections
between the data to generate decisions/predictions for previously unobserved
data without explicit programming [55, 56, 57, 58].

In machine learning, there are three main branches: supervised, unsupervised,
and semi-supervised learning [58]. In supervised learning, algorithms are
trained to learn from input data and corresponding output data, input is
processed to map it to new data based on expected outputs. This algorithms are
used on both regression and classification tasks. Unsupervised learning
algorithms are different, learning is done without any output data for the
given input data. examples of such algorithms are clustering, PCA, etc. Semi-
supervised learning falls between supervised and unsupervised learning, in
this algorithm, a combination of small labeled data and a large amount of
unlabeled data is used to train the model [59].

Machine learning has applications across many scientific and also practical
disciplines. Within the fields of medicine and biology, machine learning
example use cases are: drug discovery [60, 61, 62, 63, 64, 65], neuroscience
[66], biomedicine [67, 68, 69, 70], cancer research [71, 72, 73, 74, 75],
medical imaging [76, 77, 78, 79, 75, 80, 81, 82, 83, 84]. Machine learning
algorithms are also being used for material design [85, 86, 87, 88, 89, 90,
91, 92, 93, 94], polymer science [95, 96, 97, 98, 99, 100, 101],
crystallography [102, 103, 104, 105], fluid dynamics [106, 107, 108, 109, 110,
111, 112, 113, 114, 115, 116], condensed matter physics [117, 118, 119].
Active mater physics also takes the advantage of machine learning [120, 121,
122], example applications include: detecting various phase transitions using
Principal Component Analysis (PCA) [123, 124], active particle control using
neural networks [125, 126], identifying anomalous diffusion using recurrent
neural networks [127], forecasting dynamics of active nematics using neural
networks [128, 129, 130], extracting the effective interaction of particles
from the system configuration [131], optimal navigation strategies for active
particles [132, 133, 134, 135, 136, 137, 138, 139, 140], characterizing
motility induced regimes [141, 142], tracking Janus particles in three
dimensions [143], and applying deep learning algorithms to plankton ecology
[144].

In their study, Galajda et al. [145], developed an array of funnel-shaped
barriers with wide and narrow openings which concentrated bacterial (E.coli)
suspensions with motility on the side with narrow openings. This was because
motile bacteria could swim into the funnels through wide openings while their
ability to do the reverse through narrow openings was severely limited.
Paoluzzi et al. Kumar et al. [146] explored the trapping phase transition of
motile polar rods in the presence of a V-shaped obstacle. They found that when
the trap angle of the obstacle was below a certain threshold, a trapping
transition would occur, while above the threshold angle, all motile particles
would escape. Ribeiro et al. [147] considered the influence of spatially
periodic potentials on the trapping and sorting of motile active particles.
Using simulations they showed that different diffusion regimes and trapping
states occur which was a result of the noise value of the active particles and
the density of the system. In their work, Ai et al. [148] investigated the
mixing and demixing of binary mixtures of polar chiral (clockwise and
anticlockwise) active particles with polar velocity alignment. They showed
that when chirality difference and the polar velocity alignment of active
particles compete with each other (neither is dominant), demixing occurs in
which particles with clockwise rotation aggregate into one cluster and
particles with anticlockwise chirality aggregate into another cluster. Thus
using chirality was a means to demix particles. Miska et al. [149] proposed a
method incorporating an acoustofluidic setup for selecting particles based on
their motility, where particles with high motility would escape from the
acoustic trap. They demonstrated their method using both simulations and also
experiments with Janus particles and human sperm and proposed that using this
method highly motile sperm could be selected for medically assisted
reproduction.

Paoluzzi et al. [150] investigated the narrow escape time of active particles
from circular domains using numerical simulations, they showed that narrow
escape time undergoes a crossover between two asymptotic regimes with control
parameters being the ratio of persistence length of the active particles and
the length scale of the circular domain, they suggested the possibility of
sorting active particles based on motility parameters. Based on this, we
propose a mechanism – an apparatus – to sort (demix) active particles based on
their motility. We use Brownian dynamics simulations and also incorporate
machine learning methods to model the active particles using the data from an
extensive number of comprehensive simulations that we conducted for this
study.

##  II METHODS

We consider a rigid circular apparatus with radius $r_{s}$ which is made up of
monomers on its circumference each having a radius of $d_{mono}$. The
apparatus is inside a planar square box, which also contains active Brownian
particles each with a radius $a$. We also pin the apparatus from its center of
mass to the center of the box. Thus it can rotate with angular velocity
$\omega$, but prevented from moving. The angular velocity value can be varied
in different simulations but has a constant value in each. We study our system
in 2D, which is convenient to reduce computational resources while being
effective in reproducing the key features of active models.

We use the over-damped Langevin equations to describe the motion of particles
as time ($t$) progresses:

$\displaystyle\dot{{\mathbf{r}}}_{i}$ $\displaystyle=$ $\displaystyle
V_{s}{\mathbf{n}}_{i}-\mu_{T}\frac{\partial
U}{\partial{{\mathbf{r}}_{i}}}+\sqrt{2D_{T}}\,{\bm{\eta}}_{i}(t),$ (1)
$\displaystyle\dot{\theta}_{i}$ $\displaystyle=$
$\displaystyle\omega+\sqrt{2D_{R}}\,\zeta_{i}(t),$ (2)

here, ${\mathbf{r}}_{i}(t)=(x_{i}(t),y_{i}(t))$ is the position of the
particle, $V_{s}$ is the self-propulsion of the particle, and
${\mathbf{n}}_{i}=(\cos\theta_{i},\sin\theta_{i})$ is direction of motion with
$\theta_{i}(t)$ as orientation angles. $\mu_{T}$ is the translational mobility
of particles, $\mathbf{f_{i}}\equiv-{\partial U}/{\partial{{\mathbf{r}}_{i}}}$
is the force acting on each particle, from interactions between the $i$th
particle and all other particles. $D_{T}=\mu_{T}k_{\mathrm{B}}T$ is the
translational diffusion coefficient of particles ($k_{\mathrm{B}}$ is the
Boltzmann constant and $T$ is the absolute temperature). $\omega$ is the
angular velocity of particles. $D_{R}$ is the rotational diffusion coefficient
of particles which satisfies the relation $D_{R}=3D_{T}/\sigma^{2}$ in low
Reynold’s regime. The symbols ${\bm{\eta}}_{i}(t)$ and ${\bm{\zeta}}_{i}(t)$
denote independent translational and rotational noises for each particle. They
are characterized by white Gaussian distributions with zero mean, such that
$\langle{\eta}_{i}^{\alpha}(t)\rangle=\langle\zeta_{i}(t)\rangle=0$.
Additionally, they exhibit two-point correlations given by
$\langle{\eta}i^{\alpha}(t){\eta}j^{\beta}(t^{\prime})\rangle={ij}\delta{\alpha\beta}\delta(t-t^{\prime})$
and $\langle\zeta_{i}(t)\zeta_{j}(t^{\prime})\rangle=\delta(t-t^{\prime})$,
where $i$ and $j$ represent active particle labels, and $\alpha$ and $\beta$
refer to the Cartesian coordinates $x$ and $y$. The steric potentials, denoted
as $V({\mathbf{r}}_{ij})$, govern the interactions among active particles and
the inclusions. These potentials are derived from modified versions of the
Weeks-Chandler-Andersen potential (WCA):

$\\!\\!\\!\\!V({\mathbf{r}_{ij}})\\!=\\!\left\\{\begin{array}[]{l
l}\\!\\!4\epsilon\\!\left[\\!\left(\frac{\sigma_{\mathrm{eff}}}{|{\mathbf{r}_{ij}}|}\right)^{12}\\!\\!-\\!2\\!\left(\frac{\sigma_{\mathrm{eff}}}{|{\mathbf{r}_{ij}}|}\right)^{6}\\!\\!+\\!1\right]&:|{\mathbf{r}_{ij}}|\\!\leq\\!\sigma_{\mathrm{eff}},\\\
\\!\\!0&:|{\mathbf{r}_{ij}}|\\!>\\!\sigma_{\mathrm{eff}},\end{array}\right.$
(3)

The Péclet number (or rescaled self-propulsion strength), $Pe_{s}$, defined as

$Pe_{s}=\frac{aV_{s}}{D_{T}}=\frac{3V_{s}}{4D_{R}a},$ (4)

where we have used $D_{R}=3D_{T}/4a^{2}$ for no-slip spheres in the low-
Reynolds-number (Stokes) regime [151].

Using Brownian dynamics methods, we discretized the simulations over
sufficiently small time steps $\Delta\tilde{t}$ to solve the equations (1) and
(2) numerically.

##  III Results

###  III.1 Demixing

Fig 1 shows an example of demixing of four particles with different $Pe_{s}$,
with system having $r_{s}$ = 10, 20, 30 and 40, with $d_{mono}$ = 1.0,
$l_{pore}$ = 2.0, and $k_{B}T$ = 1.0, with Figs 1a and 1b being the initial
and the final configurations, respectively. Demixing of particles in this
figure occurs because of the system configuration and $Pe_{s}$ values of
particles that were chosen. Determining the optimal parameters for successful
demixing necessitated a process of trial and error. Thus, with the aim of the
occurrence of demixing and the ambiguity of the relationship between system
parameters and the onset of demixing, we employ a grid search across various
parameter combinations and system configurations. By leveraging machine
learning methods we aim to gain a better understanding of the demixing
phenomenon in active matter systems.

To model the demixing of particles from the apparatus, more than 75,000
simulations were done to collect data points with various parameters, namely:
$r_{s}$, $\omega$, $d_{mono}$, $l_{pore}$, and $k_{B}T$. Data were obtained
using Brownian dynamics, simulations aimed to obtain $Pe_{s}$ values for
active particles at which they start going outside of the apparatus. The
criteria were escaping 5% of particles after 500000 steps with a timestep of
dt = 0.0001. This was the main component to have demixing for
particles/apparatus with different properties. After obtaining the data from
simulations, we proceeded with different machine learning algorithms to model
active particle motility (Pèclet), each algorithm is considered in the
following sections.

(a) Initial configuration of example system

(b) Final configuration of example system

Figure 1: Example of demixing of 4 particles with different $Pe_{s}$. System
defined have $r_{s}$ = 10, 20, 30 and 40, $d\\_mono$ = 1.0, $l_{pore}$ = 2.0,
and $k_{B}T$ = 1.0

###  III.2 Machine Learning Models

####  III.2.1 Artificial Neural Network

In this section, we use ANN to model our systems. Typical workflow for
modeling using ANN involves several steps, starting with collecting data and
preprocessing it, preparing the model, training it, and evaluating the
performance of the model. For collecting data, as stated earlier, 2500
relevant data points from a set of 75000 simulations were used. Input
parameters for the data was $r_{s}$, $\omega$, $d_{mono}$, $l_{pore}$, and
$k_{B}T$ with target values being $Pe_{s}$. Then we preprocessed our data
using the Scikit-learn package [152], this step transforms the data into a
format that is more convenient to train ANN.

Using Keras from TensorFlow, we created a dense (fully connected network),
meaning that each of the neurons in a layer is connected to every neuron in
the preceding and subsequent layers, this interconnectivity would make the ANN
capable of modeling complex and non-linear relationships between data. The
architecture of our ANN which was designed for regression is comprised of (Fig
2): an input layer, a dense layer where each unit is connected to every
feature in the input data, and a Rectified Linear Unit (ReLU) activation
function. This was followed by a Batch Normalization Layer, we included this
layer to normalize the activation of the previous layer to enhance the
training stability and convergence speed. A Dropout Layer was added to ”drop
out” (set to zero) a fraction of the neurons from the previous layer at random
while training the network. Using this we prevent the model from relying
heavily on a neuron or a feature. Thus, making the network more general and
not specialized and overly complex, in simple words prevents overfitting. This
was succeeded with another dense layer and a ReLU activation function. This
layer processes information from the input and learns to recognize complex
patterns in the data. This layer also was followed by Batch Normalization and
Dropout layers. A subsequent hidden layer with ReLU activation was included,
followed by a dropout layer. Finally, we have the output layer which contains
a single unit, appropriate for regression tasks where the goal is to predict a
continuous value.

Figure 2: ANN architecture containing the input, dense layer with 512 neurons,
a bath normalization layer, a dropout layer, dense layer with 128 neurons, a
bath normalization layer, a dropout layer, dense layer with 64 neurons, a
dropout layer, and the output layer

Following defining the ANN architecture, hyperparameters of the model were
optimized to enhance the predictive performance, optimized hyperparameters are
listed in Table 1. We trained the model utilizing a custom Adam optimizer with
the determined learning rate. The loss function we chose was MSE. We also
utilized early stopping while training the model, this technique monitors MSE
on a validation set and when the model has no improvement over a defined
number of epochs, training stops. This prevents learning noise and aids
generalization to unseen data.

After completion of training the ANN model, we assessed it on the data that we
used to train the model and also the test set. We also used another external
test set, which was not used or seen by the neural net, and we assessed it
after getting the final model to see its predictive ability. We employed
R-squared ($R^{2}$), Mean Absolute Error (MAE), Root Mean Squared Error (MSE),
and Mean Absolute Percentage Error (MAPE) as evaluation metrics, these
showcase the predictive capabilities of the model, and are defined in the
following:

$\displaystyle R^{2}$
$\displaystyle=1-\frac{{\sum\limits_{i=1}^{N}\left(y_{i}-\hat{y}_{i}\right)^{2}}}{{\sum\limits_{i=1}^{N}\left(y_{i}-\bar{y}\right)^{2}}}$
(5) $\displaystyle MAE$
$\displaystyle=\frac{{\sum\limits_{i=1}^{N}\left|y_{i}-\hat{y}_{i}\right|}}{{N}}$
(6) $\displaystyle RMSE$
$\displaystyle=\sqrt{\frac{{\sum\limits_{i=1}^{N}\left(y_{i}-\hat{y}_{i}\right)^{2}}}{{N}}}$
(7) $\displaystyle MAPE$
$\displaystyle=\frac{{\sum\limits_{i=1}^{N}\left|\frac{{y_{i}-\hat{y}_{i}}}{{y_{i}}}\right|}}{{N}}\times
100\%$ (8)

$N$ is the total number of data points that are getting evaluated, $y_{i}$ ,
$\hat{y}_{i}$, and $bar{y}$ are the real, predicted, and mean values of the
entity being evaluated, respectively. For ANN model evaluation metrics are
presented in Table 2.

####  III.2.2 Support Vector Regression

In this section, we discuss the utilization of Support Vector Regression (SVR)
to model our systems. The modeling steps for SVR are similar to those for ANN:
data collection, data preprocessing, model preparation, model training, and
model evaluation performance. Data collection and preprocessing are the same
as before. To implement the SVR model, we used the scikit-learn library,
specifically Epsilon-Support Vector Regression.

We used the Radial Basis Function (RBF) kernel in our SVR model. It is defined
by the formula: $K(x,x^{\prime})=e^{-\gamma\cdot\lVert
x-x^{\prime}\rVert^{2}}$. Here, $x$ and $x^{\prime}$ are the input vectors,
$||x-x^{\prime}||^{2}$ is the squared distance between them, and gamma is a
parameter that controls the width of the Gaussian distribution. The RBF kernel
can effectively capture complex, non-linear relationships in the data by
transforming the data using the preceding formula. Gamma is the hyperparameter
that needs to be tuned for optimal model performance.

Other hyperparameters for implementing this SVR model include C
(regularization parameter) and epsilon (insensitivity parameter).
Hyperparameter C controls how the model treats outliers. A low value of C
implies that more outliers are allowed, while a high value of C implies that
fewer outliers are tolerated. Hyperparameter epsilon is given by
$L(y,f(x),\epsilon)=\max(0,|y-f(x)|-\epsilon)$, where $y$ represents the true
target value, $f(x)$ is the predicted target value and $\epsilon$ denotes the
width of the insensitive zone, controlling the tolerated error. Optimum
hyperparameters and evaluation metrics for our SVR model are listed in Tables
1, 2, respectively.

####  III.2.3 Kernel Ridge Regression

Similar to the SVR, Kernel Ridge Regression (KRR) transforms the input data
into a higher-dimensional space using a nonlinear kernel function. This makes
it possible for the algorithm to learn complex relationships between the input
data and the outputs. In this section, we will introduce and use KRR to model
our systems.

The objective function in KRR is defined by the equation:

$\displaystyle\hat{\bm{\alpha}}=\arg\underset{\bm{\alpha}\in\mathbb{R}^{N}}{\min}\
\|\mathbf{y}-\mathbf{K}\bm{\alpha}\|_{2}^{2}+\lambda\bm{\alpha}^{T}\mathbf{K}\bm{\alpha}.$
(9)

KRR is designed to fit the training data while controlling overfitting through
regularization. The objective is to find the vector $\hat{\bm{\alpha}}$ that
minimizes the squared Euclidean norm between $\mathbf{y}$ and
$\mathbf{K}\bm{\alpha}$ plus a regularization term
($\|\mathbf{y}-\mathbf{K}\bm{\alpha}\|_{2}^{2}+\lambda\bm{\alpha}^{T}\mathbf{K}\bm{\alpha}$).
Here, $\bm{\alpha}$ is the coefficients (weights) assigned to each training
sample in the kernelized feature space which is getting optimized.
$\mathbf{y}$ is the real (target) value. $\mathbf{K}$ is the kernel matrix
where each element is the result of applying a kernel function to the pair of
input samples and $\mathbf{K}\bm{\alpha}$ is the transformed input data. The
squared Euclidean norm between $\mathbf{y}$ and $\mathbf{K}\bm{\alpha}$
measures the deviation between real and predicted values. The regularization
term is $\lambda\bm{\alpha}^{T}\mathbf{K}\bm{\alpha}$, where $\lambda$ is a
constant parameter that specifies the strength of the regularization to
prevent overfitting.

We implemented the KRR using the scikit-learn library and performed a grid
search to obtain the optimal values of the hyperparameters. The
hyperparameters included in the optimization were regularization strength,
kernel type (Linear, polynomial, RBF, and sigmoid), gamma, which denotes the
kernel coefficient (the width of the kernel) that is specific to each kernel
being used, and the polynomial degree for the polynomial kernel (ignored by
other kernels). Best performing hyperparameters obtained for the KRR model and
the evaluation metrics for the KRR model are listed in Tables 1, 2,
respectively.

####  III.2.4 Gaussian Process Regression

Gaussian Process Regression (GPR) is a probabilistic modeling technique used
to capture non-linear relationships in data to perform regression tasks.
Unlike conventional methods, GPR treats the target variable as a distribution
over functions, rather than yielding single-point estimates. This allows for
making predictions with associated uncertainty.

To perform a regression task using GPR, initially, we establish a prior
belief, characterized by a mean function $m(x)$ and a covariance function
(kernel) $(k(x,x^{\prime})$, before any data is observed. This represents the
initial understanding of the function space. Combining the prior belief with
the observed data results in a posterior distribution (Bayes’ rule) which is
Gaussian. The posterior is a refined understanding of the function space. This
procedure is iterative, and with each new data point, the model’s accuracy
increases.

During training, we compute the posterior mean $\mu$ and covariance $\Sigma$
using the following equations:

$\displaystyle\mu(X_{*},X,\mathbf{y})=K(X_{*},X)[K(X,X)+\sigma_{n}^{2}I]^{-1}\mathbf{y}$
(10) $\displaystyle\Sigma(X_{*},X)=K(X_{*},X_{*})$ (11)
$\displaystyle-K(X_{*},X)[K(X,X)+\sigma_{n}^{2}I]^{-1}K(X,X_{*})$

$K$ is the kernel matrix with $K_{ij}=k(x_{i},x_{j})$. This captures the
pairwise similarities between input points. $X_{*}$ represents new input
points, $\mathbf{y}$ is the vector of observed values, $\sigma_{n}^{2}$ is the
variance of the noise, and $I$ is the identity matrix. When making predictions
for new inputs, the posterior mean $\mu$ is the predicted value, while
covariance $\Sigma$ indicates the associated uncertainty.

We used the scikit-learn library to implement GPR for modeling our systems.
Specifically, we used the method therein called GaussianProcessRegressor with
a custom kernel function which is a combination of the product of a constant
kernel (C) by an RBF kernel plus a white kernel to account for noise in data
plus a ”DotProduct” kernel with power four to act as a four-degree polynomial.
Both the RBF kernel and DotProduct kernel (with power 4) account for the non-
linear relationship between the input data and the targets. We used default
hyperparameters for our GPR, listed in Table 1, the reason being that all of
these hyperparameters are optimized during the training by the implementation
of GPR in the scikit-learn library. Evaluation metrics for our GPR model are
also listed in Table 2.

Table 1: ML models hyperparameters Model |  Hyperparameter |  Details
---|---|---
ANN |  optimizer |  Adam
|  max_iter |  10000
|  early stopping patience |  2000
|  learning_rate |  0.001
|  batch_size |  512
|  hidden neurons |  512, 128, 64
|  activations |  relu
|  loss |  mse
SVR |  kernel |  rbf
|  C |  100000
|  epsilon |  0.9
|  gamma |  0.05
KRR |  kernel |  polynomial
|  degree |  4
|  $\lambda$ |  0.5
|  gamma |  0.2
GPR |  kernel |  Constant*rbf+ WhiteKernel+ $\text{DotProduct}^{4}$
|  alpha |  0.1
|  n_restarts_optimizer |  10

####  III.2.5 Comparing the ML methods

Here, we compare the overall performance of all models using evaluation
metrics listed in Table 2. The $R^{2}$ values for all models are high,
approaching unity, across all three datasets, suggesting a strong correlation
between the variables and the target values. We also note that a high $R^{2}$
value for the training set would indicate overfitting, considering the table,
$R^{2}$ values for test and external test sets are high and near the values of
$R^{2}$ for their respective training set, this would mitigate the overfitting
problem, thus, validating the generalizability of our models. The MAPE, MAE,
and RMSE evaluation metrics too, further showcase consistent values across all
three datasets, with minor variations, this too denotes the generalizability
and consistency in the predictive power of our models.

Table 2: Evaluation Metrics for Different Models Model | Eval. | R2 | MAPE | MAE | RMSE
---|---|---|---|---|---
NN | Train | 0.9997 | 1.9539 | 1.1007 | 1.5084
| Test | 0.9997 | 2.1825 | 1.2077 | 1.5331
| Ext. | 0.9997 | 1.9725 | 1.2286 | 1.6334
SVR | Train | 0.9996 | 1.4968 | 1.1768 | 1.8964
| Test | 0.9995 | 1.6141 | 1.3136 | 2.0397
| Ext. | 0.9993 | 1.7029 | 1.4192 | 2.4055
KRR | Train | 0.9994 | 2.1512 | 1.5838 | 2.4108
| Test | 0.9992 | 2.0918 | 1.6233 | 2.4612
| Ext. | 0.9991 | 2.2392 | 1.7889 | 2.7267
GPR | Train | 0.9995 | 2.7235 | 1.5265 | 2.2082
| Test | 0.9994 | 2.6323 | 1.5563 | 2.2458
| Ext. | 0.9993 | 2.7110 | 1.6136 | 2.4577

(a) NN

(b) SVR

(c) KRR

(d) GPR

Figure 3: Plots of prediction vs observed value for our models: a)NN, b)SVR,
c)KRR, and d)GPR.

We also included Fig. 3, which illustrates the predicted values plotted
against the observed values for each ML method we used, it includes the data
for all three datasets – training, testing, and external validation – and also
a guideline to show the perfect predictions (benchmark). Each plot shows how
closely predictions from each of the ML models match up to the actual target
values across the datasets.

Overall, based on the evaluation metrics and also plots of predicted vs
observed values, the predictive power of the ML methods from the highest
performing to the lowest one – with a minimal variation – is: ANN, SVR, GPR,
and SVR respectively.

####  III.2.6 Model Interpretation

All the models we used, as well as many ML methods, present challenges to
rationalizing and interpreting the results of predictions, this is due to
their intrinsic black-box nature. To overcome this shortcoming we will use the
SHapley Additive exPlanations (SHAP) method, utilizing SHAP we can measure the
contribution of each feature on the model’s predictions [153].

Figure 4: Feature importance using SHAP analysis. The x-axis represents the
SHAP values and the y-axis represents the feature names. Feature values change
from cyan to blue as they increase from low to high. The most important
feature is placed at the top. Here, a positive (negative) SHAP value for a
feature means that the prediction increases (decreases) with an increase
(decreases) in the given feature value. Figure 5: Last snapshots of the
simulations, radial probability distributions, and also number of particles
going outside of their respective confinements. a, c, e for system with
$\omega$=1.0, $k_{B}T$=1.0, $l_{pore}$=2.0, $d_{mono}$=1.0, and $Pe_{s}$:
27.28, 66.11, 105.65, 145.47. b, d, f for systems with $\omega$=3.0,
$k_{B}T$=1.0, $l_{pore}$=4.0, $d_{mono}$=2.0, and $Pe_{s}$: 70.57, 162.37,
250.83, 338.16. Colors indicate particle types with specific Péclet values
that are predicted with ANN. In the legend for e and f, the first number
indicates the number of particles that escaped their confinement, and the
second value is their ratio (same color).

In Figure 4, $x$ coordinates show the SHAP values and the y coordinate depicts
the feature names, there is also color coding for the values of each feature
from low to high when going from cyan to blue. This plot also shows the
importance of features, with the most important one placed at the top. We also
note that a positive (negative) SHAP value for one feature means the
prediction, namely the Pèclet value, increases (decreases) with the given
feature value. Feature importance from the figure is:
$r_{s}>\omega>l_{pore}>d_{mono}>k_{B}T$. Further, from the color coding in the
figure, for radius ($r_{s}$), increasing the feature values increases Pècelt
values. This is in line with the apparatus design and configuration of our
systems, higher radius ($r_{s}$) values would mean a higher linear velocity
for the monomers of the apparatus while all other parameters including angular
velocity ($\omega$) are kept constant, this translates into higher Pèclet
values needed for active particles to escape outside of the apparatus. This is
the same for the angular velocity of the apparatus, an increase in this
feature would lead to an increase in the velocity needed for particles to pass
through the pore of the apparatus, as it is shown from color coding for
$\omega$. Higher monomer diameter $d_{mono}$ also shows increased SHAP values,
considering that when all parameters are kept constant, a higher $d_{mono}$
would mean smaller pore length and bigger contact area for the swimmers to
pass through the pore. For pore length $l_{pore}$ this means that a higher
value of the feature would decrease the Pèclet needed for active particles to
go outside the apparatus, as evident from the figure. Finally, as it can be
seen from the figure, increasing $k_{B}T$ decreases the SHAP values, it is
intuitive that increasing $k_{B}T$ values would mean a higher activity of the
swimmers, and the Pèclet values needed to escape get smaller.

####  III.2.7 Demixing using ML

Here, we showcase how our ML models define our systems capable of demixing. We
consider our best-performing model, namely ANN. Figures 5a and 5b show the
last snapshots of simulations of two separate systems in which we modeled the
system using our ANN model. This figure also includes the radial probability
distributions of the particles for their respective systems, Figures 5c and
5d. Further, we show the number of particles that escape their respective
confinements and the ratio of escaped particles to the total particles of the
respective type in Figures 5e and 5f. Figure 5a is the last snapshot of the
system with $\omega$=1.0, $k_{B}T$=1.0, $l_{pore}$=2.0, $d_{mono}$=1.0.
$Pe_{s}$ values predicted for this system using our GPR model are $Pe_{s}$:
27.28, 66.11, 105.65, 145.47. We used these parameters and $Pe_{s}$ values to
simulate the system, and from the figure, we see the onset of demixing. Figure
5c shows the Radial Probability Distribution for this system, from this, we
see that most of the particles remain in between their respective
confinements. From Figure 5e, 95% of particles of type A remain below the
radius 10 and only 5% of them go outside. For other types of particles, namely
B, C, and D, less than 5% of each goes above their respective confinement
radii, namely 20, 30, and 40 respectively.

Other system with $\omega$=3.0, $k_{B}T$=1.0, $l_{pore}$=4.0, $d_{mono}$=2.0
and predicted values of $Pe_{s}$: 70.57, 162.37, 250.83 and 338.16 showcase a
similar pattern (see Figures 5b,5d,5f), showing effectiveness of our ANN to
predict Péclet values for demixing.

##  IV Discussion

The trapping phenomena of active particles, as observed in various
experimental and numerical studies, have been instrumental in advancing our
understanding of how geometrically designed boundaries can be leveraged for
effective substance purification and microorganism control. These studies have
demonstrated the potential of active [154, 147, 146] and chiral active [49,
50, 148, 155, 156, 157] particles to navigate through geometrically designed
boundaries, offering promising applications in the purification of substances
by selectively eliminating unwanted microorganisms. A notable aspect of these
studies is the exploration of narrow escape times of active particles from a
circular domain, which was achieved using numerical simulation [150]. In
addition, trapping particles using an array of funnel-shaped barriers with
wide and narrow openings was developed to concentrate bacteria (E. coli)
suspensions with motility on the side with narrow openings [145]. Another
study investigated the impact of spatially periodic potentials on trapping and
sorting motile active particles [147]. Furthermore, a study proposed a method
that incorporates an acoustofluidic setup for selecting particles based on
their motility, where particles with high motility would escape from the
acoustic trap [149]. This method was demonstrated using both simulations and
experiments with Janus particles and human sperm, proposing that this method
could be used to select highly motile sperm for medically assisted
reproduction. All of the previous research focused on entrapment and sorting
active particles, however, one key challenge that remained unexplored was
sorting microorganisms with various propulsions with a high selectivity.

In the present study, we have developed a novel mechanism to sort active
particles based on their motility values, specifically the Pèclet number,
utilizing a combination of Brownian dynamics simulations and machine learning
methods. Our proposed mechanism is based on a circular spinning apparatus with
pores allowing active particles with certain Pèclet numbers to pass through
it. Initially, we devised a system with a configuration that would demix and
separate active particles with four different motilities. To better understand
the system configurations required to separate motile particles, considering
their complex and non-equilibrium dynamics, the utilization of machine
learning methods was necessary.

Using Brownian dynamics we conducted an extensive number of simulations to
generate a dataset to be used by the machine learning methods to capture the
complex dynamics of active particles. After gathering the data, four machine
learning methods, namely: artificial neural network, support vector
regression, kernel ridge regression, and Gaussian process regression were
developed, with the best performing one being the artificial neural network.
Further, we showcased the onset of demixing and sorting based on the Pèclet
numbers predicted by the neural network, simulation snapshots, plots of radial
probability distribution, and also plots of the escape number of particles
from the apparatus were included as visual and numerical supplements to our
findings. We also employed the SHAP method to consider the importance of
system configurations and to interpret the model’s predictions, this revealed
that the radius ($r_{s}$) and angular velocity ($\omega$) of the apparatus,
the monomer diameter ($d_{mono}$), and the pore length ($l_{pore}$)
significantly influenced the Pèclet numbers required for active particles to
escape the apparatus, while $k_{B}T$ values, indicating the noise part of the
motion, resulted in lower required Pèclet numbers and had the lowest impact.

Although our system was confined to two dimensions, such a mechanism could
easily be implemented in 3D experimental setups. However, further research is
needed to explore the effectiveness of this mechanism with different motility
parameters and in real-world systems. Further, with significant advancements
in microfluidic device fabrication techniques [158, 159], and the introduction
of 3D printing technologies, rapid and single-step production of intricate
microfluidic devices is not only more accessible but has also expanded the
design possibilities [160, 161], [162]. Such devices show great potential as
advanced systems for manipulating and immobilizing small organisms. Therefore,
we believe that our proposed mechanism and apparatus can be designed and
fabricated with an autonomous mechanism, opening up possibilities for
applications in various fields such as physics, biology, and drug delivery.

###### Acknowledgements.

This work was supported by the Iran National Science Foundation (INSF) via
Grant No. 4003065. Computational resources were provided by the Center for
High-Performance Computing (SARMAD) at Shahid Beheshti University, Tehran. We
acknowledge useful discussions with R. K. Bowles.

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