# Deep-learning Assisted Extraction of Fluid Velocity from Scalar Signal
Transport in a Shallow Microfluidic Channel

Xiao Zeng
School of Optoelectronic Engineering
and Instrumentation Science
Dalian University of Technology
Dalian 116024, Liaoning Province, P. R. China
<EMAIL_ADDRESS>
Chundong Xue
School of Optoelectronic Engineering
and Instrumentation Science
Dalian University of Technology
Dalian 116024, Liaoning Province, P. R. China
<EMAIL_ADDRESS>
Kejie Chen
School of Optoelectronic Engineering
and Instrumentation Science
Dalian University of Technology
Dalian 116024, Liaoning Province, P. R. China
<EMAIL_ADDRESS>
Yongjiang Li
School of Optoelectronic Engineering
and Instrumentation Science
Dalian University of Technology
Dalian 116024, Liaoning Province, P. R. China
<EMAIL_ADDRESS>
Kai-Rong Qin∗
School of Optoelectronic Engineering
and Instrumentation Science
Dalian University of Technology
Dalian 116024, Liaoning Province, P. R. China
<EMAIL_ADDRESS>Correspondence: Kairong Qin<EMAIL_ADDRESS>and Chundong
Xue<EMAIL_ADDRESS>
###### Abstract

Precise measurement of flow velocity in microfluidic channels is of importance
in microfluidic applications, such as quantitative chemical analysis, sample
preparation and drug synthesis. However, simple approaches for quickly and
precisely measuring the flow velocity in microchannels are still lacking.
Herein, we propose a deep neural networks assisted scalar image velocimetry
(DNN-SIV) for quick and precise extraction of fluid velocity in a shallow
microfluidic channel with a high aspect ratio, which is a basic geometry for
cell culture, from a dye concentration field with spatiotemporal gradients.
DNN-SIV is built on physics-informed neural networks and residual neural
networks that integrate data of scalar field and physics laws to determine the
velocity in the height direction. The underlying enforcing physics laws are
derived from the Navier–Stokes equation and the scalar transport equation.
Apart from this, dynamic concentration boundary condition is adopted to
improve the velocity measurement of laminar flow with small Reynolds Number in
microchannels. The proposed DNN-SIV is validated and analyzed by numerical
simulations. Compared to integral minimization algorithm used in conventional
SIV, DNN-SIV is robust to noise in the measured scalar field and more
efficiently allowing real-time flow visualization. Furthermore, the
fundamental significance of rational construction of concentration field in
microchannels is also underscored. The proposed DNN-SIV in this paper is
agnostic to initial and boundary conditions that can be a promising velocity
measurement approach for many potential applications in microfluidic chips.

_Keywords_ Shallow Microfluidic Channel $\cdot$ Fluid Velocity Measurement
$\cdot$ Scalar Image Velocimetry $\cdot$ Spatiotemporal Concentration Field
$\cdot$ Deep Neural Networks

##  1 Introduction

Microfluidic chip, a miniaturized device that integrates complex functions
into a single chip with various microchannels, makes traditional serial
processing and analysis in chemistry and biology be easily performed by
manipulating the fluid at the microscale. It has several compelling advantages
compared to traditional methods, such as little reagent consumption, faster
analysis and response time, multi-functional integration and massive
parallelization (Whitesides, 2006; Stone et al., 2004; Nge et al., 2013). With
the widespread application of microfluidic technology in cell culture
(Halldorsson et al., 2015; Sackmann et al., 2014; Shields IV et al., 2015),
organs-on-chips (Esch et al., 2015; Wu et al., 2020; Wilmer et al., 2016), and
many other exciting applications (Shang et al., 2017; Lam et al., 2005;
Doufène et al., 2019), the quantitative characterization of fluid in
microchannels becomes more and more important. The shallow microchannel with a
high aspect ratio (the height is much smaller than the other two dimensions)
is one of the most common structure of microchannels especially for cell
culture in microfluidic chips (Lane et al., 2012; Mehling and Tay, 2014; van
der Meer et al., 2009). In the shallow microchannel, the average flow velocity
is legitimately used as the most important parameter to characterize the flow
fields (Nauman et al., 1999; Bacabac et al., 2005; Wong et al., 2016). Thus
precise measurement of the flow velocity in the microchannel is critical for
the successful deployment of the microfluidic system (Yang et al., 2010; Ong
et al., 2008).

A variety of velocimetry techniques have developed to quantitatively
characterize the fluid velocity profile since the late 1900s (Sinton, 2004;
Williams et al., 2010; Wereley and Meinhart, 2010). The most common
velocimetry of the microfluidic flow are microscopic Particle Imaging
Velocimetry (micro-PIV) and microscopic Particle Tracking Velocimetry (micro-
PTV). The PIV method obtains the fluid velocity by measuring the displacement
of the tracer particles, which are artificially introduced into the fluid and
follow the fluid motion (Meinhart et al., 1999; Lindken et al., 2009;
Westerweel et al., 2013). In contrast, the PTV method directly measures and
analyzes the movement of individual tracer particles, thus avoiding the
Brownian motion of PIV and often achieves higher spatial accuracy (Adamczyk
and Rimai, 1988; Kreizer et al., 2010). However, such particulate-based
methods often bring about relatively large measurement errors when they are
applied in shallow microchannels with a high aspect ratio (Wang and Wang,
2009; Williams et al., 2010). The micron-sized tracers may interact with the
upper and lower walls of the microchannels that leads to the poor flow
following feature and a low measurement accuracy.

Scalar Imaging Velocimetry (SIV) is another kind of method that infers the
fluid velocity from a conserved scalar field (Su and Dahm, 1996a, b). It is
developed based on the continuous mass transfer rather than the movement of
individual particles. Thus, it avoids the interference of the flow field and
following feature error caused by the size effect. Different from the optical
flow velocimetry (Chen et al., 2015; Kucukal et al., 2021), SIV considers the
diffusion effect during the mass transfer to make velocity measurement more
accurate. The prerequisite is that the scalar field must contain more
information than the velocity field. The time scalar and the dissipation
length scalar of dynamic scalar field must be smaller than the corresponding
scalar of dynamic velocity field in the interested flow field (Gillissen et
al., 2018). Hence, the SIV method can provide an accurate estimation of the
large-scale turbulent flow with high Reynolds Number where the convection is
dominating (Gillissen et al., 2018; Heitz et al., 2010). Meanwhile, SIV has
also been applied frequently for experimentally visualizing the turbulence of
geophysical flow in meteorology, climatology, and oceanography (Kalnay, 2003;
Papadakis and Mémin, 2008).

However, it is usually hard to accurately estimate the fluid flow in
microfluidic channels using the original SIV method because the microfluidic
flow at low Reynolds Numbers $(Re<1)$ is not convection-dominated but rather
convection-diffusion-cooperated. The spatiotemporal gradient of the scalar
field is modulated by the coupling effect of the convection and diffusion.
This coupling effect becomes more complex when diffusion is more dominated.
Moreover, it has been reported (Li et al., 2013; Chen et al., 2016) that the
transport of dynamic scalar signals in a shallow microfluidic channel can be
modulated by different flow behaviors, such as the pulsatile flow. The
amplitude and frequency of concentration signal are nonlinearly modulated by
the amplitude and the frequency of the pulsatile flow (Li et al., 2018). How
the interplay between the concentration signal and flow velocity affects the
velocity reconstruction from scalar signal transport in the shallow straight
microchannels remains elusive. In fact, to the best of our knowledge, the SIV
method has rarely been successfully used for the characterization of fluid in
microfluidic chips.

The common implementation of SIV is to apply an integral minimization
algorithm of a cost function that minimizes weighted residuals of conserved
scalar transport equation combining continuity condition and conserved fluid
momentum (Heitz et al., 2010; Papadakis and Mémin, 2008). However, quite a few
studies have reported (Burman et al., 2020) that the accuracy of the velocity
reconstruction by minimizing the cost function is sensitive to noisy data of
scalar fields and has poor robustness even in flows at a high Reynolds Number.
It is likely that the effects of noise on SIV would be sharply augmented for
microscale flows at a small Reynolds Number. New algorithms need to be
proposed to address the issues of noise sensitivity and robustness. While
data-driven model has being gradually emerged in the fluid mechanics (Kou and
Zhang, 2021; Bezgin et al., 2021), physics-informed deep learning framework
(PINN) has been widely used in solving the forward and inverse problems in
fluid mechanics (Chen and Gu, 2021; Raissi et al., 2020, 2019). The deep
neural networks (DNN) have been reported to be able to break the limits of
accuracy and extensibility in modeling of fluid mechanics, showing better
performance in accuracy, stability and prediction power (Barwey et al., 2019;
Cai et al., 2021).

To address the aforementioned issues, by integrating the SIV concept and deep
learning, we propose a deep neural networks assisted scalar image velocimetry
(DNN-SIV) for quick and precise extraction of the average fluid velocity from
the spatiotemporal concentration gradients in a shallow microfluidic channel.
In order to improve velocimetry in microchannel, multiple dynamic
concentration boundary conditions are applied at the inlet of the microchannel
and the spatiotemporal profiles are generated by regulating the transport of
the dynamic concentration signals. Based on the principles of fluid mechanics
and the scalar transport, the proposed approach is validated via direct
numerical simulation studies. In addition, the effects of influential factors
including the scalar signal transport and fluid flow characteristics are
analyzed.

##  2 Method

###  2.1 Mathematical model

The typical shape of the shallow microchannel considered in the present study
is a straight rectangle with a high aspect ratio as shown in Fig 1., which is
one of the most common structures for cell culture in microfluidic chips (Lane
et al., 2012; Mehling and Tay, 2014; van der Meer et al., 2009). The channel
height $H$ is much smaller than the channel width $W$, i.e., $H/W\ll 1$. The
fluid in the microchannel is assumed to be incompressible, viscous and
Newtonian. In most microfluidic applications, the fluid flow in the
microchannels has a small Reynolds Number and a small Womersley Number. Thus,
the flow field can be described by the simplified Navier-Stokes equation as
follows (Li et al., 2013),

$\frac{\partial u}{\partial t}=-\frac{1}{\partial p}\frac{\partial p}{\partial
z}+\frac{\mu}{\rho}\frac{\partial^{2}u}{\partial y^{2}},$ (1)

where $u$ is the velocity in $z$-direction, $p$ is the pressure, $\rho$ and
$\mu$ are the density and the dynamic viscosity of the fluid, respectively.
Under the assumption of quasi-steady flow, the velocity profile
$u\left(y,t\right)$ can be derived as follows (Li et al., 2018),

$u\left(y,t\right)=\frac{3}{2}\left[1-\left(\frac{2y}{H}\right)^{2}\right]\bar{u}\left(t\right),$
(2) $\bar{u}\left(t\right)=\frac{Q\left(t\right)}{WH},$ (3)

where $\bar{u}\left(t\right)$ and $Q\left(t\right)$ denote the average
velocity along the channel height and the flow rate, respectively.

If there is some substance, such as fluorescent dyes or fluorescent protein
molecules, inside the fluid, the transport of substance is governed by the
convection-diffusion equation,

$\frac{\partial\mathbf{c}}{\partial t}+u\frac{\partial\mathbf{c}}{\partial
z}=D\nabla^{2}\mathbf{c},$ (4)

where $\mathbf{c}$ is the substance concentration and $D$ is the substance
diffusivity. Since $H/W\ll 1$, the substance can be quickly and uniformly
distributed along the channel height direction (_y_ -direction). The
concentration in _y_ -direction is thus assumed to be homogeneous. The average
concentration along the channel height, $\bar{\mathbf{c}}\left(x,z,t\right)$,
can be defined as,

$\bar{\mathbf{c}}\left(x,z,t\right)=\frac{1}{H}\int_{-H/2}^{H/2}\mathbf{c}\left(x,y,z,t\right)dy,$
(5)

which satisfies Taylor-Aris dispersion equation as follows (Li et al., 2018),

$\frac{\partial\bar{\mathbf{c}}}{\partial
t}+\bar{u}\frac{\partial\bar{\mathbf{c}}}{\partial
z}=D_{eff}\frac{\partial^{2}\bar{\mathbf{c}}}{\partial z^{2}},$ (6)
$\bar{\mathbf{c}}|_{z=0}=\bar{\mathbf{c}}_{0}\left(t\right),$ (7)
$\frac{\partial\bar{\mathbf{c}}\left(z,t\right)}{\partial t}|_{z=0}=0,$ (8)

where $D_{eff}$ is the effective diffusivity coefficient, that is usually
expressed as,

$D_{eff}=D\left[1+\frac{1}{210}\left(\frac{\bar{u}H}{D}\right)^{2}\right],$
(9)

where $\bar{\mathbf{c}}_{0}\left(t\right)$ is the concentration boundary
condition at the inlet. And the periodic concentration signal
$\bar{\mathbf{c}}_{0}\left(t\right)$ is considered in this study expressed as,

$\bar{\mathbf{c}}_{0}\left(t\right)=C_{0}\left(1+\epsilon_{c}\sin\left(\omega_{c}t\right)\right),$
(10)

where $C_{0}$ is the time-averaging concentration which is a constant,
$\epsilon_{c}$ is the amplitude of the fluctuating concentration signal, and
$\omega_{c}=2\pi f_{c}$ is the angular frequency corresponding to the
frequency $f_{c}$. If the temporal and spatial gradients of the concentration
in the microchannel, i.e., $\frac{\partial\bar{\mathbf{c}}}{\partial t}$,
$\frac{\partial\bar{\mathbf{c}}}{\partial z}$, and
$\frac{\partial^{2}\bar{\mathbf{c}}}{\partial z^{2}}$ are given, the average
velocity $\bar{u}$ can be solved based on Eq.(6).

Figure 1: Schematic and geometry of a straight shallow microchannel with a
high aspect ratio in a microfluidic chip.

###  2.2 DNN-assisted SIV method

Conventional SIV method is mainly applied in dense velocity field and
turbulent flow measurement in experimental fluid mechanics, where the flow is
convection-dominated and the Reynolds Number is high (Feng et al., 2007;
Corpetti et al., 2009). Conventional SIV extracts velocity from a scalar field
by applying an integral minimization algorithm of the cost function that
penalizes the deviation from the scalar transport equation under constraint
conditions. In a straight shallow microchannel, the Reynolds Number is small
and both the convection and diffusion play significant roles (Li et al.,
2018). Inspired by the minimization algorithm of conventional SIV, the average
velocity $\bar{u}$ can be computed by minimizing the cost function,
$\mathcal{J}\left(\hat{u}\right)$, which is constructed by an integral of the
one-dimensional Taylor-Aris dispersion equation (Eq.(6)) as,

$\mathcal{J}\left(\hat{u}\right)=\int_{\Omega}\left(\frac{\partial\bar{\mathbf{c}}}{\partial
t}+\hat{u}\frac{\partial\bar{\mathbf{c}}}{\partial
z}-D_{eff}\frac{\partial^{2}\bar{\mathbf{c}}}{\partial z^{2}}\right)dz,$ (11)

where $\Omega$ denotes the region of interest and $\hat{u}$ denotes the
height-averaging velocity extracted from the scalar concentration field. It is
worth noted that the cost function (Eq. (11)) needs no additional constraint
of continuity equation since the velocity in a straight shallow microchannel
is only time dependent.

As stated in the introduction, the accuracy of the velocity extraction by the
conventional SIV approach is sensitive to the noisy data of the scalar fields
and has poor robustness (Wallace and Vukoslavčević, 2010; Burman et al.,
2020). To address this issue, a DNN framework integrating scalar transport
equation is designed to map the scalar concentration field to the height-
averaging velocity. As depicted in Fig.2, a physics-informed residual networks
framework integrating SIV is an acyclic cascade consisting of the physics-
informed neural networks (Raissi et al., 2019) and the residual neural
networks (He et al., 2016), of which the physics-informed neural networks deal
with the physical constraints and the residual neural networks deal with the
vanishing gradients and mitigates the degradation problem. We referred to this
method as the DNN-assisted SIV method (DNN-SIV). The main idea of the proposed
DNN model is to approximate the velocity extracted from the concentration
field as,

$\hat{u}\left(t_{k}\right)=f\left[\bar{\mathbf{c}}\left(t_{k},t_{k+1}\right),D,H;\mathbf{\Theta}\right],$
(12)

where $t_{k}$ and $t_{k+1}$ are two successive moments,
$\hat{u}\left(t_{k}\right)$ is the averaging velocity predicted by the DNN at
moment $t_{k}$, $\bar{\mathbf{c}}\left(t_{k},t_{k+1}\right)$ represents the
normalized concentration in _z_ -direction at moment $t_{k}$ and $t_{k+1}$,
$D$ and $H$ are the diffusivity and height of the microchannel, respectively.
$\mathbf{\Theta}\left(\mathbf{W},\mathbf{b}\right)$ is weights and biases of
the DNN model. The DNN models is composed of a fully-connected neural network
with six hidden layers. For the fully-connected networks, the relationship
between _i_ -_th_ layer and the (_i_ +1)-_th_ layer can be expressed as,

$\sigma\left(\mathbf{W_{i}x_{i}+b_{i}}\right)\mapsto\mathbf{x_{i+1}},$ (13)

where $\mathbf{W_{i}}$ and $\mathbf{b_{i}}$ are the weights and biases,
$\mathbf{x}_{i}$ is input vector of _i_ -_th_ layer and
$\sigma\left(\cdot\right)$ denotes the Rectified Linear Unit (ReLU) activation
function.

In order to infer the averaging velocity, an additional constraint with the
underlying physics is encoded into the DNN. Specifically, we aim to minimize
the residual of the governing equation in the networks, which can be thought
of as enforcing for the feed-forward networks output. The residual of the
scalar transport equation (Eq. 6) is defined as,

$e_{1}=\bar{\mathbf{c}}_{t}+\hat{u}\bar{\mathbf{c}}_{z}-D\left(1+\frac{1}{210}\left(\frac{\hat{u}H}{D}^{2}\right)\right)\bar{\mathbf{c}}_{zz},$
(14)

where the subscripts represent the derivatives of the corresponding
quantities, $\bar{\mathbf{c}}_{t}$, $\bar{\mathbf{c}}_{z}$, and
$\bar{\mathbf{c}}_{zz}$. Specially,
$\bar{\mathbf{c}}_{t}=\frac{d\bar{\mathbf{c}}}{dt}$,
$\bar{\mathbf{c}}_{z}=\frac{\partial\bar{\mathbf{c}}}{\partial z}$, and
$\bar{\mathbf{c}}_{zz}=\frac{\partial^{2}\bar{\mathbf{c}}}{\partial z^{2}}$.
By integrating the mean square error and residual of the scalar transport
equation, the loss function, $\mathcal{L}$, for training the DNN is then
defined as:

$\mathcal{L}=\lambda_{f}\mathcal{L}_{f}+\lambda_{SIV}\mathcal{L}_{SIV},$ (15)
$\mathcal{L}_{f}=\sum\|\hat{u}\left(t_{k}\right)-\bar{u}\left(t_{k}\right)\|_{2},$
(16) $\mathcal{L}_{SIV}=\sum\|\
e_{1}\left(\hat{u},\bar{\mathbf{c}},D,H,\right)\|_{2},$ (17)

where $\lambda_{f}$ and $\lambda_{SIV}$ are penalty weighting factors,
$\|\quad\|_{2}$ is the Euclidean norm.

We train the DNN model by minimizing the loss function (Eq. 15) composed of a
data mismatch term (Eq. 16) and residual term (Eq. 17) associated with the
scalar transport equation. Instead of directly computing the averageing
velocity by minimizing the cost function (Eq. 11) during each experiment, the
DNN-SIV method firstly trains a surrogate model (Eq. 12) by minimizing the
loss functions (Eq. 15), and then use the measured the spatiotemporal
concentration to quickly predict the height-averaging velocity.

The parameters of the neural networks are initialized using the Xavier scheme
(Glorot and Bengio, 2010), and then optimized until convergence. Firstly,
forward propagation is used to obtain the prediction of the velocity
$\hat{u}$. Then the cost function is computed to acquire the Euclidean
distance between the vectors and the residual of the scalar transport
equation. Finally, the Adam algorithm (Kingma and Ba, 2014) is adopted to
compute the cost function gradients and the weights and biases of networks are
updated by the backpropagation algorithm. The total training procedure is
summarized in Algorithm 1. When the optimal parameters $\mathbf{\Theta}$ are
obtained after training, the velocity can be inferred very fast by feeding the
scalar concentration field data to the DNN model.

Figure 2: Schematic of physics-informed residual networks with fully connected
layers for extraction of height-averaging velocity from the scalar
concentration field. The input is the scalar concentration field at successive
moments and output is the height-averaging velocity. The residual of physical
constraint attached in neural networks is derived from Taylor-Aris dispersion
equation (Eq. 6).

Input: Training dataset $\left(\bar{\mathbf{c}},\bar{u}\right)$, Shape
parameters $\left(D,H\right)$,

Learning rate $\eta$, and Penalty weighting factors
$\left(\lambda_{f},\lambda_{SIV}\right)$.

Initialization: Randomly initialize $\mathbf{\Theta}$;

while  _not convergent_ do

$\mathcal{L}_{f}\leftarrow\mathbf{FORWARD}\left(\bar{\mathbf{c}},\bar{u};\mathbf{\Theta}\right)$;

$\mathcal{L}_{SIV}\leftarrow\mathbf{FORWARD}\left(\bar{\mathbf{c}},\hat{u},D,H;\mathbf{\Theta}\right)$;

$\mathcal{L}\left(\lambda_{f},\lambda_{SIV}\right)=\lambda_{f}\mathcal{L}_{f}+\lambda_{SIV}\mathcal{L}_{SIV}$;

$\mathbf{\Theta}\leftarrow\mathbf{BACKWARD}\left(\mathcal{L};\mathbf{\Theta}\right)$;

end while

Output: Return the optimal $\mathbf{\Theta}$

Algorithm 1 DNN Training Scheme.

###  2.3 Numerical simulation method

The dataset of the concentration field and the velocity profile for the
validation of the above method is obtained from forward numerical simulation.
Briefly, given the height-averaging velocity $\bar{u}$ (Eq. 3) as well as the
initial conditions and boundary conditions (Eqs. 7 – 10), the spatial and
temporal evolution of the concentration in a straight shallow channel is
solved numerically. The computer code based on the finite difference methods
is developed to acquire the numerical solution of spatial and temporal
gradients of concentration, including a central difference approximation in
spatial gradient and a forward difference approximation in time gradient.

In the forward numerical simulations, we discretize the channel length in
spatial step $\Delta z$ along the _z_ -direction. The spatial grid points are
denoted as $z_{m}(m=1\cdots M)$. Time is discretized in time step $\Delta t$
and the discretized moment is denoted as $t_{k}(k=1\cdots N)$. Dataset is
constructed through normalizing the simulation data of concentration field and
recording the corresponding velocities.

To validate the stability and robustness of the proposed method, white noise
is added in the dataset of concentration field,

$\bar{\mathbf{c}}_{noise}\left(t_{k}\right)=\bar{\mathbf{c}}\left(t_{k}\right)\left(1+\alpha
W\left(M\right)\right),$ (18)

where $\bar{\mathbf{c}}_{noise}$ denotes the concentration field added with
noise, $\bar{\mathbf{c}}$ denotes the original concentration field without
noise, $\alpha$ is a coefficient between 0 and 1 representing the intensity of
white noise and $W(M)$ is a $M$ uncorrelated random variables with zero mean
and finite variance approximating discrete-time white noise.

In order to quantitatively evaluate the accuracy of the velocity prediction by
DNN-SIV, the absolute percentage error (APE) and mean absolute percentage
error (MAPE) are used, which are defined as,

$APE=\left|\frac{T_{i}-P_{i}}{T_{i}}\right|,$ (19)
$MAPE=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{T_{i}-P_{i}}{T_{i}}\right|,$ (20)

where $N$ is the number of data points, $T_{i}$ and $P_{i}$ are the true and
prediction values of the _i_ -th data point, respectively.

All aforementioned algorithms and calculations were built in MATLAB (The Math
Works R2020b, Inc). The default values of the parameters used in baseline
simulations are listed in Table 1. Unless otherwise specified, these values
are used for all the simulations.

Table 1: Default values of the straight shallow microchannel parameters used in numerical simulations. Parameters | Values
---|---
$L\left(x-direction\right)$ | $30~{}mm$
$W\left(y-direction\right)$ | $3~{}mm$
$H\left(z-direction\right)$ | $0.15~{}mm$
$\Delta x$ | $0.5~{}mm$
$\Delta y$ | $0.5~{}mm$
$\Delta t$ | $0.001~{}s$
$D$ | $8.3\times 10^{-11}m^{2}/s$
$C_{0}$ | $1~{}mmol/m^{3}$
$\epsilon_{c}$ | $1$
$f_{c}$ |  $2$ Hz

##  3 Results

###  3.1 Velocity Extraction

#### A. Noise-free data of concentration field

Figure 3:  Comparison of different flow waveforms extracted from the noise-
free data of scalar concentration field based on conventional SIV method and
DNN-SIV. (a), (c), and (e) are distributions of normalized spatiotemporal
concentration generated by different flow waveforms and same dynamic
concentration boundary conditions as presented in Table 1. (b), (d), and (f)
are the extracted hegiht-averaging velocity profiles of sinusoidal, square,
and physiological blood-flow-like waveform, respectively. Black solid line
denotes true velocity. Red square dashed line and blue point dashed line are
the extracted velocities using conventional SIV method and DNN-SIV method
respectively.

To validate the DNN-SIV proposed in the present study, we first extract the
velocity from a noise-free concentration field in Figure 3. Three
spatiotemporal profiles of the normalized noise-free concentration field are
simulated in the microchannel and used as the datasets, as shown in Figure
3(a), (c) and (e), respectively. The corresponding pulsatile flow with three
different waveforms, i.e., sinusoidal, square and physiological blood-flow-
like waveform are referred as “Truth” in Figure 3(b), (d), and (f),
respectively. Both the integral minimization algorithm of the conventional SIV
and the DNN-SIV are used and the corresponding results are denoted as “SIV”
and “DNN-SIV”, respectively. It is clear that the extracted velocity waveforms
are very close to the true values, even for the case of physiological blood-
flow-like waveform. This demonstrates the ability of our proposed DNN-SIV in
extraction of flow velocity in the shallow microchannel.

In addition, for all the cases, no significant difference is observed between
the velocity waveforms extracted using SIV and DNN-SIV methods. It is
conceivable because the data of concentration field used here are noise-free.
However, the measurement error of concentration field is inevitable in
experiments. The performance of the two methods needs to be further evaluated
and compared for the case of noisy concentration field.

#### B. Noisy data of concentration field

Figure 4: Effect of noise on the conventional SIV method and DNN-SIV method. (a), (b), and (c) are the velocity profiles and corresponding APEs. The velocity profiles in (a), (b), and (c) are extracted from scalar field data with different levels of white noise (0.1%, 2%, and 4%) as stated in Sect. 2.3 respectively. Black solid line denotes true velocity. Red square dashed line and blue point dashed line denote conventional SIV method and DNN-SIV method respectively. The APEs of extracted velocity using convention SIV method (red solid line) and DNN-NN method (blue solid line) are presented below. Table 2: Time consumption comparison of conventional SIV method and DNN-SIV method. Data Points |  Consumption time of conventional SIV method |  Consumption time of DNN-SIV method
---|---|---
$1\times 10^{3}$ | $1.23~{}s$ | $0.14~{}s$
$5\times 10^{3}$ | $6.19~{}s$ | $0.31~{}s$
$1\times 10^{4}$ | $12.25~{}s$ | $0.51~{}s$

Figure 4 demonstrates the velocity extracted from the concentration field with
different noisy levels. The noisy concentration filed are constructed by
adding white noise to the idealized concentration filed as stated in Sect.
2.3. The results for the case of three levels of white noise, i.e., 0.1%, 1%,
and 4%, are given in Figure 4(a), (b) and (c), respectively. The corresponding
APEs for two methods is then calculated and compared in bottom of Figure 4.
For simplicity, only the results for the sinusoidal pulsating flow are
illustrated here since the sinusoidal waveform is the most basic waveform. The
other periodic waveforms can be obtained by superimposing the sinusoidal
waveform based on the Fourier transform. All the following results are
illustrated using the sinusoidal pulsating flow.

It is evident from the Fig. 4 that the APE of conventional SIV method
increases significantly as the noise grows (Fig. 4(b) and (c)). The APE of the
DNN-SIV method, in contrast, remains low at different levels of noise, showing
the robustness of the DNN-SIV. Another advantage of DNN-SIV compared to
conventional SIV method is time consumption. Although the training process of
neural networks is fairly time-consuming, once the training is completed the
efficiency of DNN-SIV method in velocity extraction has a huge improvement
over conventional SIV method (Table. 2). The fast processing speed makes it
possible to visualize the real-time flow velocity in the microchannel and
extend its application domain to those with high response time demand.

###  3.2 Effects of scalar signal transport and fluid flow characteristics

#### A. Scalar signal transport

Figure 5: Effect of concentration signal transport under different boundary
conditions of flow frequency and concentration frequency. (a), (b), and (c)
show the concentration signal at three different locations of the straight
shallow microchannel (Inlet: blue solid line; One-fifth from inlet: orange
solid line; Two-fifth from inlet: yellow solid line). The corresponding flow
frequency and concentration frequency in (a), (c), and (e) are 1Hz and 0.5Hz,
1Hz and 1Hz, and 1Hz and 1.5Hz, respectively. Those are for the case where
flow frequency is less than or equal or bigger than concentration frequency at
inlet. The corresponding true velocity profile (black solid line) and
extracted velocity profile using conventional SIV method (red square dashed
line) and DNN-SIV method (blue point dashed line) is shown in (b), (d), and
(f). The APEs of extracted velocity using convention SIV method (red solid
line) and DNN-NN method (blue solid line) are presented below. Figure 6: The
effect of scalar concentration signal transport. The MAPE of three parameters
are presented (N=3000), including normalized time averaging concentration (a),
relative amplitude fraction (b), and concentration frequency (c). The red
square dashed line denotes MAPE of conventional SIV method and the blue point
dashed line denotes the MAPE of DNN-SIV method.

It should be underscored that the construction of concentration filed is of
fundamental significance to our DNN-SIV approach. It is clearly shown in Fig 5
that different combination of flow and concentration frequency could generate
quite different scalar signals in the microchannel, as shown in Fig 5(a), (c),
and (e). The APEs of velocity extracted from different scalar signals are
significantly distinct, as shown in Fig 5(b), (d), and (f). When the
frequencies of the velocity and the concentration are equal, a flatter scalar
signal is generated in the microchannel due to the low-pass filtering effect
and nonlinear modulation (Li et al., 2018), resulting in a relatively larger
APE compared to the other two cases. This reveals that, in addition to
improving velocity extraction algorithms, the APEs of extracted velocity can
be reduced by adjusting the velocimetry parameters of SIV. In the following,
we analyze other factors including normalized time averaging concentration,
relative amplitude fraction and concentration frequency that may affect
velocity extraction.

The effect of three parameters with respect to scalar concentration signal
transport has been shown in Fig 6. The magnitude of normalized time averaging
concentration seems to have no notable effect on the MAPE of extracted
velocity as shown in Fig. 6(a). It is reasonable that the velocity extraction
using SIV method is related to the gradient of scalar field and independent of
the absolute value. It is the same reason for the influence of the relative
amplitude fraction on the MAPE as displayed in Fig. 6(b). The larger relative
amplitude fraction, the larger the gradient of the scalar field generated by
concentration signal. Thus, the MAPE decreases with increasing relative
amplitude fraction. As presented in Fig. 6(c), the MAPE first decreases and
then remains stable as the concentration frequency increases for a given flow
frequency.

#### B. Fluid flow characteristics

Figure 7: The effect of fluid flow characteristics. The MAPE of three
parameters are presented (N=3000), including time averaging flow velocity (a),
relative amplitude fraction (b), and flow frequency(c). The red square dashed
line denotes MAPE of conventional SIV method and the blue point dashed line
denotes the MAPE of DNN-SIV method.

The effects of three parameters with respect to fluid flow characteristics
have been exhibited in Fig. 7. As shown in Fig. 7(a), MAPE first decreases
rapidly and then gradually flatters as the flow rate increases. The MAPE of
the extracted velocity is bigger at low flow velocity due to the reduction of
the convective effect and the growing of diffusion effect. Besides, the
relative amplitude fraction and the MAPE of extracted velocity are
approximately linearly and positively correlated, as shown in Fig. 7(b). The
magnitude of the relative amplitude fraction represents the fluctuation range
of the flow rate, and it is obvious that the larger the fluctuation ranges,
the larger the extracted velocity error is. As for the influence of the flow
frequency, it is shown in Fig 7(c) that the MAPE rises quite fast as the flow
frequency increases. Based on Fig. 6(c) and Fig. 7(c), pulsating flow of
higher frequencies may require correspondingly higher concentration frequency
to reduce the error. These results may indicate that the existence of a
minimum concentration frequency for acquiring the smallest MAPE in SIV for a
specific flow frequency.

##  4 Discussion

The conventional SIV method for extracting height-averaging fluid velocity
from scalar signal transport in a shallow microfluidic channel using the
integral minimization algorithm works well in a noise-free dataset of scalar
field (Fig. 3), however, it is very sensitive to noisy data of scalar field
and has poor robustness ( Fig. 4). It is for the purpose of overcoming the
noise sensitivity and improving the robustness of conventional SIV method that
DNN-SIV (Fig. 2) is proposed and validated by simulation in this study. To the
best of our knowledge, such DNN-SIV method has never been proposed in the
past.

Conventional SIV method is effective on the premise that the measured scalar
field is sufficiently resolved and relatively noise free, such that the
spatial and temporal derivatives of the measured scalar field can be
accurately evaluated. However, such noise requirements are difficult to be
satisfied in the measurements of concentration field in microfluidic chips. In
the turbulence velocimetry, improved SIV approach was proposed by Gillissen
(Gillissen et al., 2018), that reconstructs not only the velocity field but
also the scalar field. And the reconstruction error of SIV is remarkably
reduced in turbulence generated by a falling soap film. Thus, we take a
completely novel approach based on the deep learning to overcome the noise
sensitivity of the conventional SIV method. The superiority of the DNN-SIV
method in this problem is illustrated in Fig. 4 compared with the conventional
SIV method. Besides, this method also demonstrates, such as advantages of
stability, efficiency and robustness (Fig. 5 – 7 and Table. 2).

The key point determining the accuracy of the velocity extraction from the
scalar signal transport in a shallow straight microchannel is the scalar field
itself without doubt. Obviously, if the concentration field in the
microchannel does not vary with space and time, it is certain that none of
velocity can be extracted from such scalar field with zero information
entropy. That is why dynamic concentration condition is particularly adopted
in the DNN-SIV. Hence, two fundamental issues about the scalar concentration
field can be summarized. What is a “good” scalar field for velocity extraction
and how to generate this scalar field in the microchannel.

For the first issue, we must refocus on the scalar transport equation which is
composed of both convection and diffusion terms (Eq. 4). The diffusion term in
scalar transport equation is dependent on the concentration gradient and
independent of the flow velocity. Thus, information for velocity extraction is
mainly enclosed in the convection term while retrieving velocity from the
concentration field. This does not mean, of course, that the diffusion term
has only negative effects. It contributes to the formation of the spatial
concentration gradients in some cases. Thus, we conclude that the velocity can
be extracted from the scalar field only if there is a spatiotemporal gradient
of scalar concentration along the direction of velocity. It is not surprising
to deduce that the more significant the spatiotemporal gradient of scalar
field is, the more accurate the extracted velocity is (Fig. 5). And absolute
value of magnitude of the scalar concentration plays an ineffective role (Fig.
7(a)).

For the second issue, the prerequisite condition is to design an appropriate
dynamic concentration signal being input from inlet of microchannel. Yet, it
needs to be emphasized that the scalar field in the microchannel is not
determined by these dynamic concentration boundary conditions alone, but a
combination of the input concentration signal and the flow characteristics.
So, we investigated and analyzed the influence of the fluid flow
characteristics and the scalar concentration signal transport in Sect 3.2. It
could be concluded from Fig. 6 and Fig. 7 that sophisticated design of scalar
concentration signal can significantly reduce the MAPE of extracted velocity.
It is instructive that the minimum error of extracted velocity may be achieved
by setting an optimal concentration boundary condition at the entrance for
specific flow characteristics in a microchannel chip.

The idea of DNN-SIV can extend to other shapes of microchannels with the high
aspect ratio in which fluid flow can be simplified to a planner flow. An
actual velocimetry system based on this idea is needed to be built for
experiments to further investigate the technical specifications of the
proposed method, such as measurement ranges, spatial and temporal resolution,
error range and so forth. We have analyzed the effect of fluid flow
characteristics and scalar concentration signal transport in Sect 3.2, but the
mathematical principles underlying it remain to be revealed. The collective
influence of multiple factors needs to be further clarified to obtain the
optimal concentration boundary conditions for a certain velocity profile
analytically. In addition, despite the physics-informed neural network
technique is adopted, the prediction performance of the trained network is
still deficient while using different parameters of flow characteristics and
input scalar concentration signal. But the issue of generalizability seems to
be the most common problem in the application of neural networks. Inspired by
the SIV theory, we believe that the proposed DNN-SIV can be further improved
to have wide generalization as conventional SIV methods while maintaining its
advantages.

##  5 Conclusions

SIV is an important flow visualization and measurement technology applied in
dense velocity field and turbulent flow. It happens to be a very suitable
technique for velocimetry of microfluidic chips with little modifications. We
firstly introduce the concept of SIV into the measurement of laminar flow with
small Reynolds Number in shallow microchannels, where diffusion is evident and
convection is not dominant. Dynamic concentration boundary conditions are
adopted at the inlet of the microchannel to improve the performance of
velocity extraction. DNN-SIV coupled with scalar transport equation is also
proposed in this paper to overcome the flaw of conventional SIV method and
proven its advantages of stability and efficiency through numerical
simulations. The velocimetry parameters of SIV have also been studied. We
believe that this classical SIV concept will be revitalized in the measurement
of fluid velocity in microfluidic chips with the application of deep learning
technology.

## Acknowledgment

The research reported here was supported by grants from National Natural
Science Foundation of China (No. 31971243 and No. 11802054), and the
Fundamental Research Foundation for the Central Universities in China (No.
DUT20YG113).

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