# Physics Informed Neural Networks for Free Shear Flows

Siddharth Raghu
Department of Applied Mechanics
Indian Institute of Technology Delhi
&Rajdip Nayek
Department of Applied Mechanics
Indian Institute of Technology Delhi
<EMAIL_ADDRESS>
&Vamsi K Chalamalla
Department of Applied Mechanics
Indian Institute of Technology Delhi
<EMAIL_ADDRESS>

###### Abstract

The transformative impact of machine learning, particularly Deep Learning
(DL), on scientific and engineering domains is evident. In the context of
computational fluid dynamics (CFD), Physics-Informed Neural Networks (PINNs)
represent a significant innovation, enabling data-driven fluid simulations
while incorporating physics-based laws described by partial differential
equations (PDEs). While PINNs have demonstrated efficacy in various fluid flow
scenarios, a noticeable gap exists in their application to simulate jet
flows—an essential category in engineering. Jets, crucial for downburst
outflow, ventilation, and heat transfer, lack comprehensive exploration
through PINNs in existing literature.

This study addresses this gap by focusing on the application of PINNs to
simulate steady jet flows, specifically 2D planar turbulent jet flow
scenarios. The novelty lies not only in adapting PINNs for simulating jet
flows but also in overcoming challenges such as poor convergence during
training, attributed to imbalances in loss terms. We propose a novel PINN
architecture for Reynolds-Averaged Navier–Stokes (RANS) simulation of steady
turbulent jet flows, without the need for turbulence models and simulation
data. Additionally, an extended dynamic weighting strategy is introduced to
enhance the balance of loss term contributions in the PINN, resulting in
improved convergence and more accurate predictions.

_K_ eywords PINN $\cdot$ Turbulent jet $\cdot$ Extended dynamic weights
$\cdot$ Free shear flows $\cdot$ Non-dimensionalization

##  1 Introduction

The recent surge in machine learning (ML) has significantly influenced various
scientific and engineering fields [1]. Deep Learning (DL), as a pivotal
component of this ML wave, has found applications across diverse fields,
including astronomy [2], climate modeling [3], solid mechanics [4], and
chemistry [5]. In the context of scientific applications related to
computational fluid dynamics (CFD), two predominant approaches existed:
physics-driven methods relying on traditional numerical solvers for solving
physical laws expressed through coupled partial differential equations [6, 7,
8], and data-driven methods, particularly deep learning, capable of
constructing cheap surrogate models of flow phenomena when trained with ample
data.

However, generating training data for deep neural networks typically involves
time-intensive simulations of fluid dynamics using numerical solvers. With the
rise of modern deep neural networks, the ML community became interested in the
potential of conducting data-driven fluid simulations without extensive
training data, while still incorporating physics-based laws described by PDEs.
Physics-informed neural networks (PINNs) have emerged as a class of physics-
informed-deep-learning approaches that address these requirements. At their
core, PINNs feature a deep neural network that approximates PDE solutions,
effectively transforming the problem of solution approximation into one
focused on minimizing a loss function. This composite loss function includes
terms addressing initial and boundary conditions, as well as the PDE residual
at specific collocation points within the domain. Post-training, the network
can generate solutions for a given input point within the integration domain
of a differential equation. What sets PINNs apart from generic neural networks
is the incorporation of a physics-based residual error in the loss function.
This inclusion represents a noteworthy innovation, enhancing the network’s
ability to capture and integrate the underlying physics, eliminating the need
for training data derived from previous simulations or experiments.

The exploration of PINNs has given rise to several innovative deep
architectures, offering promising applications in the simulation of fluid
dynamics. Noteworthy studies by Raissi et al. [1] demonstrated the efficacy of
PINNs in addressing laminar and turbulent flows across various scenarios,
including cylinder flow, 2D channel flow with a spherical obstacle, and 3D
intracranial aneurysm. Sun et al. [9] contributed to PINN development by
introducing hard boundary conditions and experimenting with activation
functions and adaptive learning rates, resulting in PINNs demonstrating
excellent agreement with CFD simulations, particularly in predicting
aneurysmal flow. The integration of convolutional neural networks into PINNs,
as demonstrated by Zhu et al. [10], showcased accurate predictions for 2D
Darcy flow, surpassing the performance of conventional fully-connected PINNs.
Jin et al. [11] highlighted the versatility of PINNs in handling diverse flow
scenarios, spanning Kovasznay, cylinder wake, Beltrami, and turbulent channel
flows. In the realm of additive manufacturing, Zhu et al. [12] applied PINNs
with hard boundary-constraint enforcement to predict temperature and dynamics
in melt pool fluid dynamics, illustrating their capability to address complex,
dynamic systems with moderate datasets. Exploiting the potential of CNN-based
PINNs, Wang et al. [13] introduced TF-Net for accurate 2D multiscale turbulent
flow predictions, emphasizing PINNs’ ability to capture multiscale phenomena
in fluid dynamics. Pioch et al. [14] effectively employed PINNs to reconstruct
recirculating flow over a backward step, a well-known problem in fluid
dynamics. Lucor et al. [15] used PINNs for the forward solution of 3D
Rayleigh-Bénard convection within a cavity, achieving reasonably accurate
predictions for the flow. However, simulating turbulent flows using
instantaneous Navier-Stokes equations poses computational challenges.
Addressing this, Eivazi et al. [16] proposed an alternative approach by
employing Reynolds-Averaged Navier-Stokes (RANS) equations without turbulence
models in PINNs to predict turbulent flows. Their methodology involved
predicting not only velocities and pressure but also Reynolds stresses,
proving effective across various turbulent flow fields, from boundary layer
flows to airflow over an airfoil, achieving excellent predictions. Accurate
forward solutions with PINNs have been noted to be more difficult to obtain
compared to inverse PINNs where data also acts as a regularizer.

Despite these advancements, there is a critical gap in exploring PINNs for
simulating jet flows — a significant category of free-shear flows with
paramount engineering significance. Jets are pivotal in scenarios such as
downburst outflow impact on structural components, ventilation systems, and
heat transfer processes. Many studies have investigated turbulent jets through
experimental and numerical approaches [17, 18, 19, 20, 21, 22, 23]. Despite
their importance, the literature lacks a comprehensive utilization of PINNs
for simulating jet flows.

In many engineering applications, especially those involving steady flows, the
Reynolds-averaged Navier–Stokes (RANS) equations offer a suitable framework
for fluid simulation. Consequently, our study is geared towards addressing
this critical gap by honing in on the potential of PINNs in simulating steady
jet flows. Specifically, our focus lies on 2D planar jet flow scenarios,
chosen for their simplified yet representative nature. This strategic choice
enables an effective exploration of PINNs’ capabilities in capturing the
underlying physics of jet flows.

It is well known that vanilla PINNs, as introduced by Raissi et al. [1], face
limitations detailed in [24]. A significant challenge arises from the poor
convergence of PINNs during training, stemming from imbalances in the loss
terms. In fluid dynamics, governed by the nonlinear Navier-Stokes (NS)
equations — which is a system of multiple coupled PDEs based on mass and
momentum conservation principles —, the loss function for training PINNs
includes multiple residual terms, each corresponding to one PDE, along with
contributions from boundary and initial conditions. Imbalances in loss terms
occur when the magnitude of one loss term significantly outweighs others,
leading to premature convergence in the composite loss function. Although
these individual losses are typically combined through a weighted summation
with static weights – kept fixed throughout training –, tuning these weights
proves challenging. Wang et al. [25] observed substantial differences in the
gradient values of each constraint during backpropagation, which contributes
to the imbalance. To tackle this issue, they proposed a method of dynamically
adjusting the weights between different components in the loss to improve
training convergence.

In the pursuit of simulating steady jet flows using PINNs, this paper
introduces a novel PINN architecture for forward RANS simulation of a steady
turbulent jet flow without reliance on a turbulence model and simulation data.
Additionally, an extended dynamic weighting strategy is proposed to better
balance the contribution of each loss term in the PINN. Inspired by the
dynamic weighting scheme of [25], this strategy uses trainable weights for
each PDE, updated based on the gradient information of the corresponding loss
term. The proposed architecture, coupled with the extended dynamic weighting
strategy, enhances loss convergence and yields more accurate predictions
compared to PINN models previously proposed by Eivazi et al. [16] and Pioch et
al. [14].

##  2 Free Shear Flow: Planar Jet Flow

###  2.1 Simulation Setup

Figure 1: A schematic of the computational domain for the present study. The
top and side boundaries are considered to be outlet boundaries. The bottom
boundary is considered to be a wall except at the center where the jet fluid
enters the computational domain.

In this study, we conduct two-dimensional simulations of steady laminar and
turbulent planar jets using different architectures of PINN. The spatial
configuration of the problem is shown in Fig 1, where the domain extends from
-1 m to 1 m in the radial direction (x-axis) and from 0 to 5 m in the axial
direction (z-axis). A continuous stream of the jet is introduced at the bottom
center into a quiescent environment of fluid. The jet width of 0.05 meters is
maintained constant while the inlet velocity of the jet is varied to obtain a
range of Reynolds numbers spanning from laminar to fully turbulent regime. The
simulation domain for the jet was chosen to align with the domain used by
Agarawal et. al. [21]. Due to the utilization of a uniform mesh in this study,
the jet width was increased to ensure an adequate distribution of collocation
points along the jet entrance. Consequently, the axial extent of the
simulation domain was also enlarged to capture the self-similar region of the
jet. The domain is discretized into a uniform mesh with 1000 points in the
axial direction and 400 points in the radial direction, and this mesh remains
consistent across all Reynolds numbers considered for the jet simulations. The
selection of the mesh was based on the initial examination of the mesh
dependence in PINN results for the jet. Due to the absence of a precise
correlation between mesh refinement and PINN accuracy, the decision was made
to adopt a mesh similar to that used in CFD simulations. The working fluid
chosen for these simulations is water, characterized by a density ($\rho$) of
$1000$ $kg/m^{3}$ and a dynamic viscosity ($\mu$) of $0.001Pa.s$.

###  2.2 Theoretical Background

The planar laminar and turbulent jet flow considered in this study are
governed by two-dimensional incompressible Navier-Stokes equations and
Reynolds-Averaged Navier-Stokes equations respectively. In PINN simulations
for the laminar jet, the flow variables are scaled to non-dimensional values
as follows:

$u^{*}=\frac{u}{w_{i}},\;w^{*}=\frac{w}{w_{i}},\;p^{*}=\frac{p}{\rho
w_{i}^{2}},\;x^{*}=\frac{x}{D},\;z^{*}=\frac{z}{D}$ (1)

Here, $w_{i}$ is inlet velocity and $D$ is inlet width of the jet. Here,
$u^{*},w^{*},p^{*}$ are non-dimensional quantities, and $u,w,p$ are
dimensional quantities. For simplicity, the asterisks are dropped for non-
dimensional quantities for the rest of the manuscript. Following are the non-
dimensional Navier-Stokes equations in two-dimensional space:

$\displaystyle\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}\;$
$\displaystyle=\;0,$ (2) $\displaystyle u\frac{\partial u}{\partial
x}+w\frac{\partial u}{\partial z}$ $\displaystyle=-\frac{\partial p}{\partial
x}+\frac{1}{\textit{Re}}\left(\frac{\partial^{2}u}{\partial
x^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\right),$ (3) $\displaystyle
u\frac{\partial w}{\partial x}+w\frac{\partial w}{\partial z}$
$\displaystyle=-\frac{\partial p}{\partial
z}+\frac{1}{\textit{Re}}\left(\frac{\partial^{2}w}{\partial
x^{2}}+\frac{\partial^{2}w}{\partial z^{2}}\right).$ (4)

The Reynolds number of the jet is defined using the inlet velocity and the jet
width given by $Re=\rho w_{i}D/\mu$. For the turbulent jet simulations, we
consider Reynolds Averaged Navier Stokes equations (RANS) as we are primarily
interested in predicting the mean flow statistics of the jet using PINN. RANS
is a widely used tool for studying turbulent flows at a much smaller
computational cost when compared to more elaborate large-eddy simulations and
direct numerical simulations. However, modeling the unclosed Reynolds stress
terms in RANS equations remains a significant challenge to date. In this
study, we utilize PINN to find the solution of turbulent jet flow, using RANS
equations, without any underlying turbulence models to model the Reynolds
Stress terms. RANS variables are non-dimensionalized as shown below:

$\overline{u^{*}}=\frac{\overline{u}}{{w}_{i}},\;\overline{w^{*}}=\frac{\overline{w}}{{w}_{i}},\;\overline{p^{*}}=\frac{\overline{p}}{\rho{{w}_{i}}^{2}},\;x^{*}=\frac{x}{D},\;z^{*}=\frac{z}{D},\;\overline{{u^{\prime}u^{\prime}}^{*}}=\frac{\overline{u^{\prime}u^{\prime}}}{{w_{i}}^{2}},\;\overline{{u^{\prime}w^{\prime}}^{*}}=\frac{\overline{u^{\prime}w^{\prime}}}{{w_{i}}^{2}},\;\overline{{w^{\prime}w^{\prime}}^{*}}=\frac{\overline{w^{\prime}w^{\prime}}}{{w_{i}}^{2}}.$
(5)

Here, the overbar for each variable represents the time-averaged variable. The
corresponding non-dimensional RANS equations after dropping the asterisk are
shown below:

$\displaystyle\frac{\partial\overline{u}}{\partial
x}+\frac{\partial\overline{w}}{\partial z}\;$ $\displaystyle=\;0,$ (6)
$\displaystyle\overline{u}\frac{\partial\overline{u}}{\partial
x}+\overline{w}\frac{\partial\overline{u}}{\partial z}$
$\displaystyle=-\frac{\partial\overline{p}}{\partial
x}+\frac{1}{\textit{Re}}\left(\frac{\partial^{2}\overline{u}}{\partial
x^{2}}+\frac{\partial^{2}\overline{u}}{\partial
z^{2}}\right)-\frac{\partial\overline{u^{\prime}u^{\prime}}}{\partial
x}-\frac{\partial\overline{u^{\prime}w^{\prime}}}{\partial z},$ (7)
$\displaystyle\overline{u}\frac{\partial\overline{w}}{\partial
x}+\overline{w}\frac{\partial\overline{w}}{\partial z}$
$\displaystyle=-\frac{\partial\overline{p}}{\partial
z}+\frac{1}{\textit{Re}}\left(\frac{\partial^{2}\overline{w}}{\partial
x^{2}}+\frac{\partial^{2}\overline{w}}{\partial
z^{2}}\right)-\frac{\partial\overline{u^{\prime}w^{\prime}}}{\partial
x}-\frac{\partial\overline{w^{\prime}w^{\prime}}}{\partial z}.$ (8)

The Reynolds stress terms are usually modelled using a turbulence model with
the help of Boussinesq hypothesis as shown in Equation 9

$\overline{u_{i}^{\prime}u_{j}^{\prime}}=-\nu_{t}\left(\frac{\partial\overline{u_{i}}}{\partial
x_{j}}+\frac{\partial\overline{u_{j}}}{\partial
x_{i}}\right)-\frac{2}{3}k\delta_{ij}$ (9)

Where $k$ is the turbulent kinetic energy, which can be described as below for
the 2D incompressible flow, and $\delta_{ij}$ is the Kronecker Delta.

$k=\frac{1}{2}\overline{u_{i}^{\prime}u_{i}^{\prime}}$ (10)

Substituting the expression for $k$ in the Equation 9 and solving for the
Reynolds Stress terms, the Boussinesq hypothesis reduces to Equation 11.

$\overline{u_{i}^{\prime}u_{j}^{\prime}}=-\nu_{t}\left(\frac{\partial\overline{u_{i}}}{\partial
x_{j}}+\frac{\partial\overline{u_{j}}}{\partial x_{i}}\right)$ (11)

Extensive research on planar jets has been conducted in the past, as evident
by studies such as those by Gutmark et. al. Heskestad et. al. and Bradbury et.
al. [26, 27, 28]. These investigations aimed to ascertain flow statistics, and
collectively, they established that self-similarity in a planar jet is
attained after an axial distance of $30\times D$. Thus, for a laminar jet, it
is now well established that the centerline velocity and the half-width of the
jet as a function of $z$, as described in Equation 12 and 13, in the self-
similar region.

$\displaystyle w_{0}(z)$ $\displaystyle\propto z^{-\frac{1}{3}}$ (12)
$\displaystyle l_{\frac{1}{2}}(z)$ $\displaystyle\propto z^{\frac{2}{3}}$ (13)

Where, $w_{0}$ is the center-line velocity of the laminar jet,
$l_{\frac{1}{2}}$ is the half-width of the jet computed from axial velocity of
the laminar jet. The corresponding self-similar profiles for a turbulent jet
are given in the Equation 14 and 15.

$\displaystyle\overline{w}_{0}(z)$ $\displaystyle\propto z^{-\frac{1}{2}}$
(14) $\displaystyle\overline{l}_{\frac{1}{2}}(z)$ $\displaystyle\propto z$
(15)

Where, $\overline{w}_{0}$ is the mean center-line velocity of the turbulent
jet, $\overline{l}_{\frac{1}{2}}$ is the half-width of the jet computed from
mean axial velocity of the turbulent jet.

The above mentioned flow statistics will be used in this study to evaluate the
accuracy of flow predictions by the PINNs for both laminar and turbulent jets.

###  2.3 CFD Simulations

To compare the results obtained from PINN, CFD simulations using two-
dimensional RANS equations are performed for different values of Reynolds
numbers. To validate the accuracy of benchmark CFD simulations, the results
were validated against the established theory of jets [29, 30].

The CFD simulations were set up using Ansys FLUENT in a 2D computational
domain, with domain and boundary conditions consistent with PINN
configuration. For the laminar jet, a laminar flow solver was adopted, while
for the turbulent jet, the standard $k-\epsilon$ turbulence model was employed
to model Reynolds stresses. The parameters of the $k-\epsilon$ model were
selected based on previous studies [17, 18, 19, 20, 21]. The simulations were
performed until the residuals of all the equations reached below $10^{-6}$.The
boundary conditions for laminar and turbulent jets are outlined in Tables 1
and 2 respectively.

Table 1: Boundary conditions for laminar jet simulations. Variables | Inlet | Outlet | Wall
---|---|---|---
u | Zero | Zero-gradient | Zero
w | $w_{i}$ | Zero-gradient | Zero
p | Zero | Zero-gradient | Zero-gradient
Table 2: Boundary conditions for turbulent jet simulations. Variables | Inlet | Outlet | Wall
---|---|---|---
$\overline{u}$ | Zero | Zero-gradient | Zero
$\overline{w}$ | $w_{i}$ | Zero-gradient | Zero
$\overline{p}$ | Zero | Zero-gradient | Zero-gradient
$\overline{u^{\prime}u^{\prime}}$ | Zero | Zero-gradient | Zero
$\overline{u^{\prime}w^{\prime}}$ | Zero | Zero-gradient | Zero
$\overline{w^{\prime}w^{\prime}}$ | Zero | Zero-gradient | Zero

##  3 Methodology: PINN

###  3.1 PINN Architectures

For PINN solutions to the laminar jet case, the neural network architecture
comprises $100\times 10$ hidden layers, taking two spatial inputs ($x,z$) and
producing three outputs ($u,w,p$), as illustrated in Fig 2.

Figure 2: A schematic of PINN for solving Navier-Stokes equations for steady
laminar jet.

For turbulent jets, four different architectures have been analysed:

  * A1

: In this case, $l_{m}$ is an additional output of the neural network,
alongside the three outputs ($u,w,p$). The architecture, shown in Fig 3,
includes two spatial inputs ($x,z$) and four flow variables
($\overline{u},\overline{w},\overline{p},l_{m}$) as outputs, with $100\times
10$ hidden layers. The governing equations include only the dominant Reynolds
stress gradient term ($\overline{u^{\prime}w^{\prime}}$), as described in the
theory of jets, modeled using the Prandtl’s mixing length model (Equations 16
and 17).

$\displaystyle\overline{u^{\prime}w^{\prime}}=-\nu_{t}\frac{\partial\overline{w}}{\partial
x}$ (16)
$\displaystyle\nu_{t}=l_{m}^{2}\left|\frac{\partial\overline{w}}{\partial
x}\right|$ (17)

Figure 3: A schematic of PINN for solving RANS equations, with only
$\overline{u^{\prime}w^{\prime}}$ Reynolds stress term which is modeled using
Prandtl’s mixing length model, with mixing length, $l_{m}$, as an output of
the neural network, for the steady turbulent jet.

  * A2

: In this case, the PINN architecture is presented in Fig 4, with two spatial
inputs ($x,z$) and six flow variables
($\overline{u},\overline{w},\overline{p},\overline{u^{\prime}u^{\prime}},\overline{u^{\prime}w^{\prime}},\overline{w^{\prime}w^{\prime}}$)
as outputs and $100\times 10$ hidden layers. The governing equations are based
on RANS equations (Equations 6, 7, and 8), with Reynolds stress terms as
additional outputs of the neural network. This architecture was introduced by
Eivazi et. al. [16].

Figure 4: A schematic of PINN for solving RANS equations, with the Reynolds
stresses being the output of the neural network, for the steady turbulent jet.

  * A3

: In this case, the Boussinesq hypothesis is utilized which relates the
Reynolds stresses to mean velocity gradients with the turbulent viscosity as
the proportionality constant. The PINN architecture, shown in Fig 5, includes
two spatial inputs ($x,z$) and four flow variables
($\overline{u},\overline{w},\overline{p},\nu_{t}$) as outputs, with $100\times
10$ hidden layers. This architecture was utilized by Pioch et. al. [14]. The
governing equations are based on RANS equations (Equations 6, 7, 8) and the
Boussinesq hypothesis (Equation 11).

Figure 5: A schematic of PINN for solving RANS equations, with Reynolds
stresses being modeled using the Boussinesq hypothesis where the turbulent
viscosity is also considered as part of the PINN outputs, for the steady
turbulent jet.

  * A4

: In this case, the Boussinesq hypothesis is utilized as in A3. The PINN
architecture, shown in Fig 6, includes two spatial inputs ($x,z$) and three
flow variables ($\overline{u},\overline{w},\overline{p}$) as outputs, with
$100\times 10$ hidden layers. In contrast to A3, it incorporates a second
neural network with four velocity gradients as inputs and turbulent viscosity
($\nu_{t}$) as output, with $50\times 5$ hidden layers. The governing
equations are based on RANS equations (Equations 6, 7, 8) and the Boussinesq
hypothesis (Equation 11).

Figure 6: A schematic of PINN for solving RANS equations, with Reynolds
stresses being modeled using the Boussinesq hypothesis where the turbulent
viscosity is made to be a function of velocity gradients using a second neural
network, for the steady turbulent jet.

###  3.2 Composite Loss Function and Training of PINN

The loss function in this study is the Mean Squared Error (MSE) loss for
training the neural networks. The total loss function ($L$) is a combination
of the PDE loss ($L_{PDE}$) and boundary loss ($L_{b}$) with their respective
weights ($\alpha$), as shown in Equation 18. The gradients in the PDE losses
are computed using the automatic differentiation functionality of PyTorch.

$\displaystyle L$ $\displaystyle=L_{PDE}+\alpha L_{b}$ (18) $\displaystyle
L_{PDE}$ $\displaystyle=L_{c}+L_{x}+L_{z}$ (19)

where, $L_{PDE}$ is the residual loss of Navier-Stokes/RANS equations, as
defined in Equation 19.The total PDE loss is the sum of losses from continuity
($L_{c}$), x-momentum ($L_{x}$), and z-momentum ($L_{z}$) equations defined by
Equations 2, 3 and 4 respectively. $L_{PDE}$ is a combination of residual loss
of different equations in Navier-Stokes/RANS equation as shown in

Typically, a gradient descent algorithm is employed to train neural network
parameters using the following update rule:

${\theta}^{k+1}={\theta}^{k}-\eta{\nabla}_{\theta}L_{PDE}-\eta\alpha{\nabla}_{\theta}L_{b}.$
(20)

Here, $k$ denotes the current iteration step, ${\theta}$ represents the neural
network parameters (weights and biases), $\eta$ is the learning rate,
${\nabla}_{\theta}L_{PDE}$ is the gradient of PDE loss with respect to neural
network parameters, and ${\nabla}_{\theta}L_{b}$ is the gradient of boundary
loss with respect to the neural network parameters.

The parameter $\alpha$ represents the boundary loss weight and is typically
kept constant (or static) in standard PINN configuration. It serves as a
hyper-parameter that requires careful tuning for good results. However,
maintaining a static weight through the training iterations poses a challenge
of gradient pathologies, leading to inaccurate results in the case of forward
PINN. To overcome this issue, Wang et al. [25, 31] proposed a dynamic weight
strategy, expressed as follows:

$\displaystyle\hat{\alpha}^{k+1}$
$\displaystyle=\frac{\overline{|{\nabla}_{\theta}L_{PDE}|}}{\overline{|{\nabla}_{\theta}L_{b}|}}$
(21) $\displaystyle\alpha^{k+1}$
$\displaystyle=(1-\lambda)\alpha^{k}+\lambda\hat{\alpha}^{k+1}$ (22)

Here, $k$ is the current iteration step, ${\theta}$ represents the neural
network weights, $\lambda$ is a hyperparameter defining the contribution from
the previous weights, $\overline{|{\nabla}_{\theta}L_{PDE}|}$ is the mean of
$|{\nabla}_{\theta}L_{PDE}|$, and $\overline{|{\nabla}_{\theta}L_{b}|}$ is the
mean of $|{\nabla}_{\theta}L_{b}|$.

Given that the PDE loss $L_{PDE}$ inherently comprises multiple components, it
is intuitive to allocate weights to each individual element of the PDE loss
and dynamically optimize them. Thus, in this study, we propose an extended
version of the dynamic weights algorithm where we introduce weights for each
component of the PDE loss in the composite PDE loss:

$\displaystyle L$ $\displaystyle=\alpha L_{b}+L_{c}+\gamma L_{x}+\zeta L_{z}$
(23) $\displaystyle\hat{\alpha}^{k+1}$
$\displaystyle=\frac{\overline{|{\nabla}_{\theta}L_{c}|}}{\overline{|{\nabla}_{\theta}L_{b}|}},\>\>\>\;\hat{\gamma}^{k+1}=\frac{\overline{|{\nabla}_{\theta}L_{c}|}}{\overline{|{\nabla}_{\theta}L_{x}|}},\>\>\>\;\hat{\zeta}^{k+1}=\frac{\overline{|{\nabla}_{\theta}L_{c}|}}{\overline{|{\nabla}_{\theta}L_{z}|}}$
(24) $\displaystyle\alpha^{k+1}$
$\displaystyle=(1-\lambda)\alpha^{k}+\lambda\hat{\alpha}^{k+1}$ (25)
$\displaystyle\gamma^{k+1}$
$\displaystyle=(1-\lambda)\gamma^{k}+\lambda\hat{\gamma}^{k+1}$ (26)
$\displaystyle\zeta^{k+1}$
$\displaystyle=(1-\lambda)\zeta^{k}+\lambda\hat{\zeta}^{k+1}$ (27)

The loss weights are initialized at $\alpha=100$, $\gamma=1$, and $\zeta=1$.
Furthermore, $\lambda$ is set to 0.1 for all the cases. A $\mathtt{tanh}$
nonlinear activation is applied at each hidden layer. The entire PINN code is
implemented using PyTorch libraries. For training the PINN, the ADAM optimizer
with a learning rate of $5\times 10^{-4}$ is utilized for 10000 epochs.
Additionally, the Xavier initialization technique is employed to initialize
neural network parameters at the start of the training process.

To compare PINN predictions with 2D RANS simulations, two different error
metrics are employed: Root Mean Square Error (RMSE) and $l_{2}$ relative error
($\epsilon_{v}$), which represents a Euclidean norm error. The definitions of
these metrics are provided in Equations 29 and 30.

$RMSE=\frac{1}{N}{\parallel V-\hat{V}\parallel}_{2}$ (29)
$\epsilon_{v}\;(\%)=\frac{{\parallel V-\hat{V}\parallel}_{2}}{{\parallel
V\parallel}_{2}}\times 100$ (30)

where $V$ is the target or ground truth value of the variable and $\hat{V}$ is
the prediction of the variable using PINN.

##  4 Results

###  4.1 Effect of Non-Dimensionalization

Previous studies on PINNs applied to fluid flow problems used both dimensional
and non-dimensional forms of the Navier-Stokes equations. For example,
Chengping et. al. [32] used the dimensional Navier-Stokes equations to
simulate the flow past the cylinder using PINN with an accuracy of 1.8%
$l_{2}$ relative error. On the other hand, Jin et. al. [31] used non-
dimensional Navier-Stokes equations to simulate Kovasznay flow, 2D cylinder
wake, and 3D Beltrami flow using PINN with an accuracy of 0.08% $l_{2}$
relative error. Thus, it is necessary to compare the ability of PINN to
approximate the non-dimensional Navier-Stokes and standard Navier-Stokes
equations. The dimensional form of Navier-Stokes equations are given in
Equations 31,32, and 33. The non-dimensional form of the Navier-Stokes
equation is given in Equations 2,3, and 4. To compare the effect of non-
dimensionalization, we considered the architecture given in Fig 2 with
$100\times 10$ hidden layers with extended dynamic weights for the laminar
jet. The inlet velocity in this case is $w_{i}=0.004$ m/s which corresponds to
a Reynolds number of $Re=200$. The PINN was run considering both dimensional
and non-dimensional equations for 10,000 epochs with $\lambda=0.1$,
$\alpha=100$, $\gamma=1$, and $\zeta=1$ as the initial weights. The results of
the two simulations are shown in Fig 7.

$\displaystyle\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}\;$
$\displaystyle=\;0,$ (31) $\displaystyle u\frac{\partial u}{\partial
x}+w\frac{\partial u}{\partial z}$
$\displaystyle=-\frac{1}{\rho}\frac{\partial p}{\partial
x}+\nu\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial
z^{2}}\right),$ (32) $\displaystyle u\frac{\partial w}{\partial
x}+w\frac{\partial w}{\partial z}$
$\displaystyle=-\frac{1}{\rho}\frac{\partial p}{\partial
z}+\nu\left(\frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial
z^{2}}\right).$ (33)

Figure 7: Plots of (a) Dimensional PINN $w$ prediction (b) Dimensional 2D CFD
$w$ prediction, and (c) Non-Dimensional PINN $w$ prediction (d) Non-
Dimensional 2D CFD $w$ prediction for laminar jet flow at $Re=200$.

Observing Fig 7 suggests that the non-dimensional variant of the PINN tends to
outperform its dimensional counterpart. Notably, the dimensional form appears
to face challenges in capturing the jet phenomenon. It can be inferred that
the non-dimensionalization acts analogous to normalization used in machine
learning to enhance the training procedure and therefore improves the
performance of the PINN.

###  4.2 Effect of loss weights

The significance of effectively tuning the weights for each loss component in
training PINNs becomes apparent when dealing with diverse flow fields.
Traditional PINNs assume larger weights for boundary and initial condition
losses and keep these weights constant during the training process. However,
the static weights in PINNs might not yield satisfactory results. To address
this issue, the method of dynamically tuning the loss weights was introduced
by [25], which optimizes the loss weights during each epoch using the loss
gradients.

The dynamic weights method was used only to optimize the boundary and initial
condition loss weights. However, the PDE loss also has various components such
as x-momentum, z-momentum, and continuity losses. Hence, it is pertinent to
introduce dynamic weights to individual components of PDE losses as well.
Thus, we formulated a novel extended version of dynamic weights for individual
PDEs while keeping the continuity equation loss as a base loss for Navier-
Stokes/RANS equations.

To assess the impact of the dynamic weights method, the non-dimensional form
of the Navier Stokes equations was used with PINN for a laminar jet at
$Re=200$. The simulation was run for 10,000 epochs with the architecture of
$100\times 10$. In the static method, the loss weights were kept the same for
all epochs: $\alpha=100$, $\gamma=1$, and $\zeta=1$. For the dynamic and
extended dynamic weights, these values of weights were taken as the initial
values which were then optimized with the help of loss gradients with
$\lambda=0.1$. The results of different formulations of loss weights for
predicting $w$ are illustrated in Fig 8.

Figure 8: Plots of (a) static-weights-PINN $w$ prediction, (b) dynamic-
weights-PINN $w$ prediction, and (c) extended-dynamic-weights-PINN $w$
prediction (d) 2D CFD $w$ prediction for laminar jet flow at $Re=200$

From Fig 8 and Table 3, it can be seen that the extended dynamic weights are
performing better than the static and dynamic weights counterparts. This can
be attributed to the fact that it is necessary to distribute the loss
gradients of individual PDEs as well, apart from boundary and initial
condition losses.

Figure 9: Comparison between the center-line $w$ velocity decay for different
loss weights formulation with the 2D CFD prediction for laminar jet flow at
$Re=200$ beyond $z=30\times D$ Figure 10: Comparison between the jet half-
width for different loss weights formulation with the 2D CFD prediction for
laminar jet flow at $Re=200$ beyond $z=30\times D$

To further support the claim that the extended dynamic weights perform better
than their counterparts, the center-line velocity decay is shown in Fig 9, and
the half-width of the jet is shown in Fig 10. It can be seen that the PINN
results using extended dynamic weights closely agree with the CFD results.

Table 3: Comparison of errors of different flow variables between different loss weights formulations for laminar jet flow at $Re=200$ beyond $z=30\times D$ Variables | Static Weights | Dynamic Weights | Extended Dynamic Weights
---|---|---|---
RMSE | $\epsilon_{v}(\%)$ | RMSE | $\epsilon_{v}(\%)$ | RMSE | $\epsilon_{v}(\%)$
$w_{0}$ | 0.51 | 82 | 0.599 | 96.3 | 0.041 | 6.6
$l_{1/2}$ | 1.381 | 80.6 | 1.309 | 224.3 | 0.2 | 11.6

###  4.3 Effect of Architectures on Turbulent Jet

So far we have discussed the laminar 2D planar jet and the challenges in
predicting flow variables using PINN. In the subsequent discussions, we
consider turbulent 2D planar jets. Reynolds-Averaged Navier Stokes (RANS)
equations are considered in this study to predict mean flow variables using
PINN architecture. However, the RANS equations have their own set of
challenges such as the Reynolds stress closure problem. The Reynolds stresses
in RANS equations are unknown and require additional modeling. Several
turbulence models are currently in use to model the Reynolds stresses, and the
degree of complexity across different turbulence models varies depending on
the problem being considered.

In this study, to model the Reynolds stresses, different PINN configurations
were considered as described above. Four different architectures, A1, A2, A3
and A4, are considered and results from these configurations are compared with
the CFD results to assess their accuracy.

From the results of the laminar jet, it was evident that the extended dynamic
weights formulation for loss weights was more accurate when compared to static
and dynamic loss weights formulation. Hence, extended dynamic weights
formulation for loss weights is used for all cases of turbulent jet. To
simulate the turbulent jet, PINN was run for 10,000 epochs with $\lambda=0.1$
and loss weights of $\alpha=100$, $\gamma=1$ and $\zeta=1$ as initial loss
weights for A1 and A2 configurations. For A4 architecture, PINN was run for
7000 epochs with $\lambda=0.01$ due to high oscillations observed during
convergence towards the end.

In the first architecture A1, a simple one-equation turbulence model is
considered, namely the mixing length model, to predict the dominant component
of the Reynolds stress $\overline{u^{\prime}w^{\prime}}$ as a function of
mixing length and velocity gradients. Along with mean velocity components and
mean pressure, mixing length is also one of the outputs of PINN as shown in
Fig 3. The mean axial velocity predicted by A1 is shown in Fig 11 (a).
Further, the respective errors of different flow variables are given in Table
4.

Figure 11: Comparison between the (a) A1 PINN $\overline{w}$ prediction, and (b) A2 PINN $\overline{w}$ prediction from Eivazi [16] with the (c) 2D CFD $\overline{w}$ prediction for turbulent jet flow at $Re=5000$ Table 4: Comparison of errors of different flow variables between A1, and A2 of PINN for the turbulent jet at $Re=5000$ beyond $z=30\times D$ Variables | A1 | A2
---|---|---
RMSE | $\epsilon_{v}(\%)$ | RMSE | $\epsilon_{v}(\%)$
$\overline{w}_{0}$ | 0.175 | 52.7 | 0.226 | 68.3
$\overline{l}_{1/2}$ | 4.198 | 58.6 | 7.163 | 100

Different turbulence models have their shortcomings, such as the inability of
the $k-\epsilon$ model to predict the slip wall conditions, and may provide
different levels of accuracy in results depending on the type of flow
considered. To circumvent this problem in turbulence modeling, Eivazi et. al.
[16] proposed to predict the Reynolds stresses directly from PINN as shown in
Fig 4. We considered this method in A2, where all the 3 components of Reynolds
stresses are outputs of the PINN. The mean axial velocity obtained from this
PINN configuration is shown in Fig LABEL:fig:turb-w (b). The predictions of
mean flow variables seem to be poor when compared to results from A1. This
phenomenon may be attributed to the non-closure issue, characterized by three
equations and six unknowns that need to be resolved simultaneously.

To further improve the results from A2, Pioch et. al. [14] proposed to
leverage the Boussinesq hypothesis, given Equation 11, and predict the
turbulent viscosity $\nu_{t}$ as a part of PINN outputs in the A3
configuration, as shown in Fig 5. The mean axial velocity obtained from this
PINN configuration is shown in Fig 12 (a). The predictions of mean flow
variables seem to be better when compared to results from A1 and A2.
Nevertheless, despite these modifications, the non-closure issue persists in
the A3 configuration, indicating the necessity for further refinements to
achieve accurate results.

Figure 12: Comparison between the (a) A3 PINN $\overline{w}$ prediction from Pioch [14], and (b) Proposed A4 PINN $\overline{w}$ prediction with the (c) 2D CFD $\overline{w}$ prediction for turbulent jet flow at $Re=5000$ Table 5: Comparison of errors of different flow variables between A3, and A4 of PINN for the turbulent jet at $Re=5000$ beyond $z=30\times D$ Variables | A3 | A4
---|---|---
RMSE | $\epsilon_{v}(\%)$ | RMSE | $\epsilon_{v}(\%)$
$\overline{w}_{0}$ | 0.102 | 31.1 | 0.008 | 2.3
$\overline{l}_{1/2}$ | 1.951 | 27.2 | 1.921 | 26.8

Finally, in the fourth configuration A4, we propose a novel technique where
two separate neural networks are considered. In the first neural network, mean
velocities and pressure are the outputs. Further, the auto-differentiation
technique is used to compute the velocity gradients which act as inputs to the
second neural network. Turbulent viscosity $\nu_{t}$ is the output of the
second neural network and the Boussinesq hypothesis is used to finally compute
the Reynolds stresses. The novel PINN architecture proposed in A4 is shown in
Fig 6. In A4, the second neural network serves as a means to address the
turbulent viscosity closure, thereby enhancing the predictive capabilities of
the PINN. The mean axial velocity obtained from this configuration is shown in
Fig LABEL:fig:turb-w (b) and the respective errors are given in Table 5.

From Fig 11, and 12 and Table 4, and 5, it can be seen that the proposed A3
architecture works better compared to configurations A1 and A2. Qualitatively,
the contours of mean axial velocity are very similar to the CFD result. Also,
centerline velocity decay and half-width of the jet match closely with the CFD
result in the self-similar region $z/D>30$ as shown in figures 13 and 14,
respectively. From the error metrics shown in Table 4, and 5, it is evident
that A4 has the least error and closest agreement with the CFD result.

Figure 13: Comparison between the center-line axial velocity for different
cases of PINN with the 2D RANS prediction for turbulent jet flow at $Re=5000$
beyond $z=30\times D$ Figure 14: Comparison between the jet half-width for
different cases of PINN with the 2D RANS prediction for turbulent jet flow at
$Re=5000$ beyond $z=30\times D$

##  5 Discussion and Summary

This work is focused on predicting the mean statistics of a turbulent jet
using the PINN with various configurations and neural network architectures.
In the first part, the laminar jet problem was considered using various
configurations. To begin with, a comparison was made between the predictions
obtained from PINN using dimensional and non-dimensionalized Navier-Stokes
equations as loss functions. We found that PINN with non-dimensional equations
significantly improved the flow prediction. The effect of loss weights is also
considered by dynamically varying the weights of boundary as well as PDE
losses. Further, we introduced a new technique called extended dynamic weights
where weights for each component of the PDE loss are also varied dynamically.
We found a significant improvement in the results using the proposed extended
dynamics weights technique. In the second part of the study, the turbulent jet
was considered using the non-dimensional equations along with the extended
dynamic weights technique. Further, we proposed a novel two-neural network
architecture for predicting the turbulent viscosity to address the closure
problem without assuming any underlying turbulence models.

The major conclusions drawn from this study are as follows:

  * •

Non-dimensionalizing the inputs, outputs, and PDEs greatly helps the PINN to
accurately predict the flow field since it acts as normalization in machine
learning that is used to boost the training performance of the model.

  * •

The weights of different losses play a significant role in predicting the
flow. In this study, to overcome the problem of gradient pathology faced by
PDEs, we proposed an extended version of dynamic weights for the PDEs which
improved the prediction of mean flow statistics significantly.

  * •

For turbulent jets, we proposed a novel architecture where two neural networks
are used. The first one predicts mean flow velocity components and mean
pressure which act as inputs to the second neural network. Turbulent viscosity
$\nu_{t}$ is the output of the second neural network and finally, the
Boussinesq hypothesis is used to compute the Reynolds stress components.

With the proposed novel techniques of extended dynamic weights and two neural
networks in conjunction with the Boussinesq hypothesis, we were able to
predict the flow statistics of laminar and turbulent jets accurately. However,
there is scope for further improvement. The errors in the proposed
architecture A4 primarily stem from the implementation of boundary conditions
and pressure field predictions which cause the jet to be asymmetric about the
center-line. Future studies can focus on the implementation of pressure-
velocity coupling in the PINN architecture to further improve the predictions.
Also, the generalizability of the proposed PINN architecture is still to be
explored, to whether the proposed architecture will work for different sets of
boundary conditions and other fluid flows will be investigated in future
studies.

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