# Identification of Settling Velocity with Physics Informed Neural Networks
For Sediment Laden Flows

Mickaël Delcey<EMAIL_ADDRESS>Yoann Cheny  Jean-Baptiste
Keck  Adrien Gans  Sébastien Kiesgen de Richter

###### Abstract

Physics-Informed Neural Networks (PINNs) have shown great potential in the
context of fluid dynamics simulations, particularly in reconstructing flow
fields and identifying key parameters. In this study, we explore the
application of PINNs to recover the dimensionless settling velocity for
sedimentation flow. The flow involves sediment-laden fresh water overlying
salt water, which is described by Navier-Stokes equations coupled with
sediment concentration and salinity transport equations. Two cases are
investigated: one where the training data contains the salinity and sediment
concentration fields, and another where it contains the velocity field. For
both cases, we investigate several flow regimes and show that the model is
capable of inferring the unknown parameter and reconstructing the hydrodynamic
field of the flow. The quality of the model inference is assessed by comparing
it with numerical simulations from a high-fidelity semi-Lagrangian solver. We
demonstrate the model’s robustness to noise by training it with data corrupted
by noise of varying magnitudes, highlighting the potential of PINNs for real-
world applications.

###### keywords:

Physics-Informed Neural Networks, Deep Learning, Sediment Laden Flows, Fluid
Dynamics

††journal: arxiv

[lemta]organization=LEMTA, CNRS, Université de Lorraine, addressline=2, Avenue
de la Forêt de Haye, city=Vandœuvre-lès-Nancy, B.P. 160, postcode=54504,
country=France [iuf]organization=Institut Universitaire de France (IUF),
city=Paris, country=France

[JK]organization=Laboratoire Jean Kuntzmann,city= Saint-Martin-d’Hères ,
country=France

## Introduction

The settling of granular particles in a fluid is a widespread natural
phenomenon with significant industrial implications. For instance, in mining,
it aids in mineral extraction by segregating particles according to density,
thereby improving efficiency. Pharmaceutical and food processing industries
employ sedimentation in centrifugation processes to effectively separate
components and refine liquids for quality assurance. In civil engineering,
sedimentation principles in soil mechanics help comprehend groundwater flow
dynamics, crucial for construction projects. Additionally, sedimentation plays
a vital role in environmental engineering, notably in wastewater treatment,
contributing to cleaner ecosystems and sustainable practices. Rivers serve as
major conduits for transporting sediment from land to coastal oceans [1],
carrying pollutants such as microplastics [2]. Understanding the fundamental
transport properties of these sediments is critical for addressing
environmental concerns. Particularly, in situations where a lighter fluid
laden with particles (freshwater) meets a denser clear fluid (saltwater), such
as in estuarine environments, instabilities like Rayleigh-Taylor instability
and double diffusivity emerge [3], leading to the formation of sedimentation
fingers. The resulting downward sediment flux can be orders of magnitude above
the one that would normally be obtained with only gravitational settling [4,
5]. The critical factor influencing these instabilities has been identified as
the ratio between the settling velocity of particles and the diffusive
spreading velocity of the initial stable stratification layer [6]. While the
settling velocity of particles can be estimated based on their physical
properties, the diffusivity of salinity ($\kappa_{s}$) relies entirely on the
Schmidt number ($S_{c}$). The upward spreading velocity of the layer is
influenced not only by $S_{c}$ but also by other dimensionless parameters of
the system, such as the stability ratio ($R_{s}$) and the diffusivity ratio
($\tau$) between salinity and sediments.

The settling velocity is a key parameter which is commonly required in
sediment transport models. However, determining this parameter accurately is
still a major challenge [2]. Furthermore, it has been shown that in estuaries,
many factors such as sediment concentration, salinity and turbulent mixing can
drastically affect the effective settling velocity [7].

Physics informed neural networks (PINNs) [8], although recent, has been
established as a powerful tool to solve forward and inverse problems of
partial differential equations (PDE) [9]. In a PINN model, a neural network is
used to approximate the solution of a PDE system. To this end, the residuals
of the governing equations computed by automatic differentiation and embedded
in the loss function of the network as a regularizing term. As a consequence,
by minimizing the loss, the network is constrained to satisfy the physical
conservation laws. PINNs are an ideal solution for tackling inverse problems
since it can handle complexe ill-posed problems where training data is
incomplete and noisy, while also working well in complex geometries. PINNs has
been widely used to tackle such inverses problems in computational science and
engineering (e.g. fluid mechanics [10], heat transfert [10], power systems
[11]). Among the inverse problems, the identification of unknown parameters
involves inferring not only the components of the solution vector $\bm{u}$ and
their derivatives, but also constant parameters in the PDEs (e.g. coefficient
such as viscosity and diffusivity), for example. In [12], add two extra
weights to PINN model to infer unknow constant in the boundary condition for
stephan problem. In [13], the permeability and viscoelastic modulus are
identified on the Cahn-Hilliard and Navier-Stokes equations, highlighting the
possibility to infer phisiological parameters with PINNs.

In this work, we employ physics informed neural networks models to infer the
dimenssionnless settling velocity in the context of particle-laden flows,
governed by Navier-Stokes coupled with sediment concentration and salinity
transport equations. This is to our knowledge the first attempt to identify
settling velocity using the paradigm of PINNs. We investigate several flow
regimes with varying settling velocity and Schmidt number, with different sets
of training data as well as the influence of noise on the results accuracy. In
the section 1, we describe the governing equations of the flow and the
construction of references datasets for for several flow regimes involving
various settling velocity $V_{p}$ and Schmidt number $Sc$. In the following
section, we explain the principles of physics informed neural networks for
flow reconstruction and its extention to parameter identification for double
diffusive sedimentation. The section three introduce the results of the
inference for each flow regimes. The final section focuses on one specific
flow regime to dive further in the study of results.

##  1 Problem setup

###  1.1 Governing equations

We consider the same problem set-up as discussed in [14] and [6]. In this
configuration, a layer of particle-laden fresh water is placed above clear,
saline water. The horizontal directions are denoted $(x,y)$ in 3D and $x$ in
2D, while the vertical coordinate is represented by z. Initially, the
interface between the two layers is situated at z = 0. The problem set-up is
illustrated Fig (1).

Figure 1: Two-Dimensionnal problem configuration. $\bm{\Gamma}$ refers to the
walls of the domain.

The overall density $\rho$ is assumed to have a linear dependence on both
salinity S and sediment concentration C, characterized by density expansion
coefficients $\alpha$ and $\gamma$, respectively.

$\displaystyle\rho=\rho_{0}(1+\alpha S+\gamma C).$ (1)

The sediment consists of small monodisperse particles with negligible inertia,
which are carried by the combination of the fluid velocity $\bm{u}$ and the
dimensionless settling velocity $V_{p}$.

Under the Boussinesq approximation and within the framework of the velocity-
vorticity formulation, the dimebsionless system is governed by the following
dimensionless governing equations :

$\displaystyle\dfrac{\partial\bm{\omega}}{\partial
t}+(\bm{u}\cdot\nabla)\bm{\omega}=\Delta\bm{\omega}+(\bm{\omega}\cdot\nabla)\bm{u}-\nabla\times(R_{s}+C)\bm{e_{z}},$
(2a) $\displaystyle\dfrac{\partial S}{\partial
t}+(\bm{u}\cdot\nabla)S=\dfrac{1}{S_{c}}\Delta S,$ (2b)
$\displaystyle\dfrac{\partial C}{\partial
t}+(\bm{u}\cdot\nabla)C-V_{p}\dfrac{\partial C}{\partial z}=\dfrac{1}{\tau
S_{c}}\Delta C,$ (2c) $\displaystyle\omega=\nabla\times\bm{u},$ (2d)
$\displaystyle\nabla\cdot\bm{u}=0,$ (2e)

where $\bm{w}=(\omega_{x},\omega_{y},\omega_{z})$ denotes the vorticity field.
In 2D, the vorticity field 2D the vorticity field is reduced to a scalar
field. The dimensionless parameters $R_{s},~{}S_{c},~{}V_{p}$ and $\tau$ are
introduced in table (1).

Stability ratio | Diffusivity ratio | Schmidt number | Settling velocity
---|---|---|---
$R_{s}$ = $\dfrac{\alpha S_{max}}{\gamma C_{max}}$ |  $\tau$ = $\dfrac{\kappa_{s}}{\kappa_{c}}$ | $S_{c}=\dfrac{\nu_{0}}{\kappa_{c}}$ | $V_{p}$
Table 1: Dimensionless parameters. Here $\alpha S_{\textrm{max}},~{}\gamma
C_{\textrm{max}}$ refer to the maximum added density due to salinity and
particles, $\kappa_{s}$, $\kappa_{c}$ are the diffusivity coefficients of
salinity and sediments, and $\nu_{0}$ is the kinematic viscosity of fresh
water.

###  1.2 Numerical Setup

The boundary conditions in the horizontal directions, denoted by $\Gamma_{l}$
and $\Gamma_{r}$ in fig. (1) are periodic. Top and bottom walls denoted
$\Gamma_{t}$ and $\Gamma_{b}$ respectively are slip walls with no penetration.
Furthermore, we impose no-flux boundary condition on those walls for the
salinity field. For the sediment concentration, no-fluw boundary condition
with respect to the settling velocity on $\Gamma_{t}$ and no-flux boundary
condition on $\Gamma_{b}$.

$\displaystyle\dfrac{\partial\bm{u}_{||}}{\partial
z}=0~{}\textrm{and}~{}u_{z}=0~{}\textrm{on}~{}\Gamma_{t}\cup\Gamma_{b},$ (3a)
$\displaystyle V_{p}C-\dfrac{1}{\tau S_{c}}\dfrac{\partial C}{\partial
z}=0~{}\textrm{on}~{}\Gamma_{t}~{}\textrm{and}~{}\dfrac{\partial C}{\partial
z}=0~{}\textrm{on}~{}\Gamma_{b},$ (3b) $\displaystyle\dfrac{\partial
S}{\partial z}=0~{}\textrm{on}~{}\Gamma_{t}\cup\Gamma_{b}.$ (3c)

Here, $\bm{u}_{||}$ represents the horizontal components $(u_{x},u_{y})$ in 3D
and $u_{x}$ in 2D.

The initial condition field is given by a smoothed step profile. The initial
salinity field is obtained by computing : $S=1-C$.

$\displaystyle\bm{u}_{0}(x,z)=\bm{0}~{}\textrm{and}~{}\omega_{0}(x,z)=0,$ (4a)
$\displaystyle
C_{0}(x,z)=\dfrac{1}{2}\left[1+erf\left(\dfrac{z-\delta(x)}{l_{0}}\right)\right],$
(4b) $\displaystyle S_{0}(x,z)=1-C_{0}(x,z).$ (4c)

Here $\delta$ is a random perturbation uniformly distributed. The parameter
$l_{0}$ is set to 1.5 for this simulation. As depicted in Fig. (2) the initial
sediment concentration and salinity denoted $C_{0}$ and $S_{0}$, are shown
with respect to the vertical coordinate $z$.

Figure 2: Initial vertical sediment profile for $l_{0}=1.5$, $\delta=0.$ .

We solve the 2D version of equations (2) with boundary conditions (3) and
initial condition (4) using the highly accurate semi-lagrangian method
described in [14]. The numerical solution is obtained on the dimensionless
computational domain $\Omega=[0,128]\times[-128,64]$ for $t\in[0,100]$. The
evolution of sediment concentration and salinity fields is illustrated Fig.
(3).

Several numerical simulations are performed using fixed parameters $Rs=2$ and
$\tau=25$, while varying the Schmidt number and settling velocities as
detailed in Table (2).

$S_{c}$ | 0.7 | 3.5 | 7.0 | 7.0 | 7.0 | 7.0 | 14.0 | 28.0
---|---|---|---|---|---|---|---|---
$V_{p}$ | 0.04 | 0.04 | 0.01 | 0.04 | 0.08 | 0.16 | 0.04 | 0.04
Table 2: Flow configuration  Figure 3: Snapshot of the sediment concentration
(first line) and salinity (second line) at dimensionless times
$t=0,~{}50,~{}75,~{}100~{}$. Parameters : $V_{p}=0.04$, $S_{c}=0.7$,
$R_{s}=2$, $\tau=25$.

###  1.3 Syntethic datasets

For each case, we construct a reference dataset within the same spatio-
temporal framework, which is utilized for training PINNs. We select a spatial
subdomain $\Omega_{1}\subset\Omega$, and the subset $\Omega_{1}\times[0,T]$ is
discretized into a uniform Cartesian grid consisting of $N_{x}\times
N_{z}\times N_{t}=342\times 128\times 125$ points. We denote the set
containing all the spatio-temporal coordinates of the discretized grid as
$\mathcal{C}={\mathbf{x}_{i},t_{i}}$, and
$\mathcal{D}={S_{i},C_{i},\mathbf{u}_{i},\omega_{i}}$ represents the set
containing the corresponding solution fields for given $S_{c}$ and $V_{p}$
parameters.

As we plan to conduct multiple PINNs trainings across all flow regimes
outlined in Table 2, the dataset $\mathcal{D}$ is intentionally kept smaller
than the numerical simulation dataset to reduce computational costs.
Furthermore, we have trimmed the spatial domain to exclude non-essential
values from the training process.

Figure 4:

##  2 Physics Informed Neural Networks

###  2.1 Inverse problem

In this section, we delve into the principles of Physics-Informed Neural
Networks (PINNs) within the context of sedimentation flow. Our objective with
the PINN model is to deduce missing information about the flow dynamics by
leveraging the governing equations (2) along with available observations of
the solution field, specifically $(S,C,\mathbf{u},\mathbf{\omega})$.

To this end, an Artificial Neural Network (ANN) denoted $\mathcal{N}$, is used
to infer the solution of (2) while approximating a set of observed data.
The network is a function that take as inputs $(\bm{x},t)\in\Omega\times[0,T]$
and predicts :

$\displaystyle\mathcal{N}(\bm{x},t;\theta)=(S_{\mathcal{N}},C_{\mathcal{N}},\bm{u}_{\mathcal{N}},\bm{\omega}_{\mathcal{N}})(\bm{x},t;\theta),$
(5)

the parameter $\theta$ contains the trainable weigths and biases of the
network $\mathcal{N}$.
The loss function of the ANN contains two terms : The first is the
contribution of the data, denoted $\mathcal{L}_{\textrm{data}}$ which
penalizes the mismatch between the measured data and the corresponding network
predictions. The second, denoted $\mathcal{L}_{\bm{f}}$ is associated to the
conservations laws.

We define :

$\displaystyle\displaystyle\dfrac{\partial\bm{\omega}_{\mathcal{N}}}{\partial
t}+(\bm{u}_{\mathcal{N}}\nabla)\bm{\omega}_{\mathcal{N}}-(\bm{\omega}_{\mathcal{N}}\cdot\nabla)\bm{u}_{\mathcal{N}}=\Delta\bm{\omega}_{\mathcal{N}}-\nabla\times(R_{s}S_{\mathcal{N}}+C_{\mathcal{N}})\bm{e_{z}}$
$\displaystyle\dfrac{\partial S_{\mathcal{N}}}{\partial
t}+(\bm{u}\cdot\nabla)S=\frac{1}{S_{c}}\Delta S$ $\displaystyle\dfrac{\partial
C_{\mathcal{N}}}{\partial
t}+(\bm{u}_{\mathcal{N}}\cdot\nabla)C_{\mathcal{N}}-V_{p}\frac{\partial
C_{\mathcal{N}}}{\partial z}=\frac{1}{\tau S_{c}}\Delta C_{\mathcal{N}}$
$\displaystyle\displaystyle\bm{\omega}_{\mathcal{N}}=\nabla\times\bm{u}_{\mathcal{N}}$
$\displaystyle\nabla\cdot\bm{u}_{\mathcal{N}}$

The function $\bm{f}_{\mathcal{N}}$ is the residuals of equations (2) given by
an approximation of equations (2) by $\mathcal{N}$, the derivatives involved
in $\bm{f}_{\mathcal{N}}$ are computed by Automatic Differentiation [15],
which allows pointwises derivatives without tuncation errors. The loss
function is expressed as :

$\displaystyle\mathcal{L}(\theta)=\lambda\mathcal{L}_{\textrm{data}}(\theta)+\mathcal{L}_{\bm{f}}(\theta),$
(6)

where

$\displaystyle\mathcal{L}_{f}(\theta)=\dfrac{1}{N_{\bm{f}}}\displaystyle\sum_{i=1}^{N_{\bm{f}}}\big{|}\big{|}\bm{f}(\bm{x}^{f}_{i},t^{f}_{i};\theta)\big{|}\big{|}_{2}^{2}.$
(7)

The parameter $\lambda$ is a constant set manually to balance the two error
terms.

The training phase consists in finding the parameters of $\theta^{*}$ which
minimizes the loss $\mathcal{L}$.

$\displaystyle\theta^{*}=\underset{\theta}{argmin~{}\mathcal{L}(\theta)}.$ (8)

In practice, the minimization process consists in iterating the gradient
descent algorithm :

$\displaystyle\theta_{n+1}=\theta_{n}-\eta\nabla_{\theta}\mathcal{L},$ (9)

where $\eta$ is the learning rate.
In the field of machine learning, the stochastic gradient descent algorithm
[16] is commonly chosen to mitigate computational costs. This method involves
updating the model’s parameters iteratively, using batches of data. Each batch
provides an estimation of the overall gradient based on a subset of the
dataset. This contributes to faster convergence and more efficient use of
computational resources compared to the traditional gradient descent approach,
where the entire dataset is used for each update. The term used to describe
the process of completing one pass through the entire dataset is commonly
referred to as an _epoch_.

###  2.2 Parameter Identification

Parameter identification is a special case of inverse problems for PINNs,
where missing data are not only components of the hydrodynamic field, but
could also be constant parameters in (2) such as $R_{s},~{}B_{p},~{}S_{c}$ or
$\tau$. In our case, the objective is to deduce the dimensionless settling
velocity $V_{p}$ from partial data on the flow fields.

Thus, we assume that $V_{p}$ is now an unknown variable. In the PINN
framework, we introduce a trainable parameter into the network, denoted
$\tilde{V_{p}}$ so that the function $\bm{f}_{\mathcal{N}}$ is now expressed
as :

$\displaystyle\displaystyle\dfrac{\partial\bm{\omega}_{\mathcal{N}}}{\partial
t}+(\bm{u}_{\mathcal{N}}\nabla)\bm{\omega}_{\mathcal{N}}-(\bm{\omega}_{\mathcal{N}}\cdot\nabla)\bm{u}_{\mathcal{N}}=\Delta\bm{\omega}_{\mathcal{N}}-\nabla\times(R_{s}S_{\mathcal{N}}+C_{\mathcal{N}})\bm{e_{z}}$
$\displaystyle\dfrac{\partial S_{\mathcal{N}}}{\partial
t}+(\bm{u}\cdot\nabla)S=\frac{1}{S_{c}}\Delta S$ $\displaystyle\dfrac{\partial
C_{\mathcal{N}}}{\partial t}+\bm{u}_{\mathcal{N}}\cdot\nabla
C_{\mathcal{N}}-\tilde{V_{p}}\frac{\partial C_{\mathcal{N}}}{\partial
z}=\frac{1}{\tau S_{c}}\Delta C_{\mathcal{N}}$
$\displaystyle\displaystyle\bm{\omega}_{\mathcal{N}}=\nabla\times\bm{u}_{\mathcal{N}}$
$\displaystyle\nabla\cdot\bm{u}_{\mathcal{N}}$

Here, $\Theta=(\theta,\tilde{V_{p}})$ contains the previous weights of the
network $\mathcal{N}$ but also a new trainable weigth $\tilde{V_{p}}$ used to
approximate the real value of $V_{p}$ during training.

As a consequence, this parameter is involved in the second loss term
$\mathcal{L}_{f}$ and the global loss function of the network stay the same as
(6).

We can see an illustration of the network $\mathcal{N}$ in figure (5).

Figure 5: Physics-informed neural network structure : a fully connected neural
network take as input $(\bm{x},t)\in\mathbb{R}^{3}$ and predicts
$\bm{\mathcal{N}}(\bm{x},t)=(S_{\mathcal{N}},C_{\mathcal{N}},u_{\mathcal{N}},w_{\mathcal{N}},\omega_{\mathcal{N}})$.
The residuals of the governing equations $e_{1,6}$ are computed by automatic
differentiation and $e_{0}$ denotes the mismatch between the observational
data $\bm{V}$ and the predictions $\bm{V}_{\mathcal{N}}$, which are combined
in the loss function $\mathcal{L}$. Finally, the weights and biases of the
network are updated iteratively with gradient descent.

##  3 Identification of settling velocity accross flow regimes

In this section, we use a physics-informed neural network model to deduce the
settling velocity across various flow regimes outlined in the table (2) using
incomplete data. The training process involves two distinct configurations of
the training datasets.

  * 1.

In the first configuration, we assume that the observation data available are
salinity and sediment concentration fields. In practical situations, these
fields might correspond to measurements obtained through the PILAT technique,
as discussed in [17]. This case is refered as PILAT case.

  * 2.

In the second, we use the velocity field as observationnal data. In real-world
scenarios, these fields may correspond to measurements obtained using the so
called PIV111Particle Image Velocimetry technique [18]. This case is refered
as PIV case.

Therefore, for each training configuration, the loss function is formulated as
in (6) where the term $\mathcal{L}_{\textrm{data}}$ depends on the training
data. For the PILAT case, $\mathcal{L}_{\textrm{data}}$ addresses the
discrepancy between the observed salinity and sediment concentration and the
corresponding fields predicted by the model :

$\displaystyle\mathcal{L}_{\textrm{data}}$
$\displaystyle=\mathcal{L}_{S}+\mathcal{L}_{C}=\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}S_{\mathcal{N}}(\bm{x}_{i},t_{i})-S_{i}\big{|}\big{|}^{2}+\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}C_{\mathcal{N}}(\bm{x}_{i},t_{i})-C_{i}\big{|}\big{|}^{2},$
(10)

while for the PIV case, $\mathcal{L}_{\textrm{data}}$ is the mismatch between
observed and predicted velocity :

$\displaystyle\mathcal{L}_{\textrm{data}}$
$\displaystyle=\mathcal{L}_{u}+\mathcal{L}_{w}=\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}u_{\mathcal{N}}(\bm{x}_{i},t_{i})-u_{i}\big{|}\big{|}^{2}+\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}w_{\mathcal{N}}(\bm{x}_{i},t_{i})-w_{i}\big{|}\big{|}^{2}.$
(11)

Here $\forall i\in\llbracket
1,N\rrbracket,~{}u_{i},v_{i},S_{i},C_{i}\in\mathcal{D}$, we use the same set
of points $(\bm{x}_{i},t_{i})\in\mathcal{C}$ to compute the residuals
$\mathcal{L}_{f}$ and $\mathcal{L}_{\textrm{data}}$ for each cases.
Furthermore, the parameter $\lambda$ is set to $1000$.

For each training, we use a fully connected ANN with 9 hidden layers of 195
neurons each.
The activation function _swish_ [19] is selected for all layers except the
last one where no activation function is imposed. The loss function is
optimized during training using Adam method [20] with a batch size of 1024 for
70 epochs. The learning rate of the gradient descend decreases from $5\times
10^{-4}$ to $1\times 10^{-5}$. Finally, we create an additionnal trainable
parameter $\tilde{V_{p}}$ which refers to the inferred settling velocity.

The results of the training are inverstigated using the relative $L_{2}$ error
defined by :

$\epsilon_{V}=\frac{100}{\sup_{\mathbf{x}\in\mathcal{C}}\left|V_{\mathbf{x}}\right|}\sqrt{\frac{\sum_{\mathbf{x}\in\mathbf{X}}\left|V_{\mathcal{N}}(\mathbf{x})-V_{\mathbf{x}}\right|^{2}}{\left|\mathbf{X}\right|}}$
(12)

where $V\in(s,c,u,w,\omega)$ and $\mathbf{X}$ defines the domain where the
error is computed. We choose $\sup_{\mathbf{x}\in\Omega_{obs}\times
T_{obs}}\left|V_{\mathbf{x}}\right|$ as reference value to avoid the division
near to zero issue reported in [21].

$S_{c}$ | $V_{p}$ | $\tilde{V_{p}}$ | $\epsilon_{s}$ | $\epsilon_{c}$ | $\epsilon_{u}$ | $\epsilon_{w}$ | $\epsilon_{\omega}$
---|---|---|---|---|---|---|---
3.5 | 0.04 | 0.04 | 0.08% | 0.30% | 0.91% | 0.92% | 1.10%
7 | 0.01 | 0.0098 | 0.06% | 0.21% | 0.79% | 0.57% | 1.64%
7 | 0.04 | 0.039 | 0.07% | 0.23% | 0.98% | 0.60% | 1.25%
7 | 0.08 | 0.079 | 0.14% | 0.58% | 0.98% | 0.97% | 1.48%
7 | 0.16 | 0.158 | 0.36% | 1.53% | 1.12% | 0.81% | 1.48%
14 | 0.04 | 0.039 | 0.08% | 0.26% | 0.89% | 0.50% | 1.34%
28 | 0.04 | 0.039 | 0.08% | 0.25% | 0.90% | 0.42% | 1.10%
Table 3: Results for the differents flow regimes. $\tilde{V_{p}}$ is the
predicted dimensionless settling velocity and $\epsilon_{\cdot}$ refers to the
$L_{2}-$ relative error (12) between reference and predicted fields for PILAT
case. Error $\epsilon$ is computed on all the spatio-temporal domain
$\Omega_{1}\times[0,T]$.

The outcomes presented in table (3) demonstrate that the model consistently
converges to the accurate settling velocity across various scenarios, while
also precisely inferring flow characteristics. Given that salinity and
sediment concentration fields were included in the training data, their
associated errors were anticipated to be minimal, as observed (below 2%).
Remarkably, errors related to velocity and vorticity fields also remained
under 2%, underscoring the model’s great accuracy in reconstructing these
fields.

Notably, alterations in the Schmidt number, while maintaining a constant
settling velocity, did not significantly impact the outcomes. Conversely, an
increase in the settling velocity $V_{p}$ led to augmented reconstruction
errors across all fields. This trend may be attributed to the accelerated
sedimentation at higher $V_{p}$ values, which intensifies flow patterns and
instabilities, thereby explaining the observed increase in network
reconstruction error for elevated settling velocities.

To rigorously evaluate the robustness of our model under varying conditions of
data quality within the parameter identification framework, we introduced
controlled levels of noise into the training datasets across all examined flow
regimes. This approach was designed to simulate real-world data inaccuracies
and test the resilience of our model’s predictive capabilities.

We augmented the training data with Gaussian noise characterized by a zero
mean ($\mu=0$) and a standard deviation ($\sigma$) proportionate to the
maximum value of the training field. Specifically, we varied $\sigma$ to
represent 0%, 15%, 25%, and 35% of the field’s maximum value, denoted as
$\mathcal{N}(0,\sigma^{2})$. This stratification allowed us to systematically
analyze the impact of noise on the model’s performance across a spectrum of
data quality scenarios. Thus, for PILAT case,

$\displaystyle\mathcal{L}_{\textrm{data}}=\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}S_{\mathcal{N}}(\bm{x}_{i},t_{i})-\tilde{S}_{i}\big{|}\big{|}^{2}+\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}C_{\mathcal{N}}(\bm{x}_{i},t_{i})-\tilde{C}_{i}\big{|}\big{|}^{2},$
(13)

where $\tilde{S}=S+\xi_{s},~{}\xi_{s}\sim\mathcal{N}(0,\sigma^{2})$ and
$\tilde{C}=C+\xi_{c},~{}\xi_{c}\sim\mathcal{N}(0,\sigma^{2})$ where
$S,C\in\mathcal{D}$ and $\sigma$ is the standard deviation of the gaussian
law.

Figure 6: Visualisation of sediment concentration field data at $t=100$
corrupted by gaussian noise $\xi_{C}\sim\mathcal{N}(0,\sigma^{2})$, where
$\sigma=0\%,~{}15\%,~{}25\%,~{}35\%$

For PIV case,

$\displaystyle\mathcal{L}_{\textrm{data}}=\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}u_{\mathcal{N}}(\bm{x}_{i},t_{i})-\tilde{u}_{i}\big{|}\big{|}^{2}+\dfrac{1}{N}\sum_{i=1}^{N}\big{|}\big{|}w_{\mathcal{N}}(\bm{x}_{i},t_{i})-\tilde{w}_{i}\big{|}\big{|}^{2},$
(14)

where $\tilde{u}=u+\xi_{u},~{}\xi_{u}\sim\mathcal{N}(0,\sigma^{2})$ and
$\tilde{w}=w+\xi_{w},~{}\xi_{w}\sim\mathcal{N}(0,\sigma^{2})$ where
$u,w\in\mathcal{D}$.
A visualisation of the noisy sediment concentration field can be seen figure
(6).

The results are presented in Table (4) for PILAT and in (5) for PIV. Theses
results showcase the robustness of PINNs when tackling the identification
problem with noisy data. While for both cases it is generally observed that
lower noise levels yield more accurate inferences, even under high levels of
noise, the outcomes consistently align with the reference value of $V_{p}$.

However, as outlined in table (5), despite achieving close results to the
PILAT cases in the absence of noise, the performance of the PIV case undergoes
a significant drop when noise is introduced. This observation becomes apparent
when examining the flow regime $Sc=0.7$, $V_{p}=0.16$. In this instance, the
inferred value of Vp for the PILAT case is 0.1534, even with 35% noise, while
it drops to 0.14 for the PIV cases.

The decrease in performance may be attributed to the position of $V_{p}$
within the equations (2). Indeed, Equation (2c) is the only one that involves
$V_{p}$, and in this equation, the sediment concentration field holds a
significant role. Notably, the quantity $\frac{\partial C}{\partial z}$ is
directly multiplied by Vp. Consequently, it might be more challenging for the
network to infer the temporal and spatial derivatives of $C$ from noisy
velocity data than for sediment concentration field itself.

$S_{c}$ | $V_{p}$ | $\tilde{V_{p}}$
---|---|---
| $\sigma=0\%$ | $\sigma=15\%$ | $\sigma=25\%$ | $\sigma=35\%$
0.7 | 0.04 | 0.0401 | 0.0397 | 0.0399 | 0.0388
3.5 | 0.04 | 0.0402 | 0.04078 | 0.04011 | 0.0384
7.0 | 0.01 | 0.00994 | 0.00957 | 0.0082 | 0.0077
7.0 | 0.04 | 0.0401 | 0.0395 | 0.0388 | 0.0385
7.0 | 0.08 | 0.0803 | 0.0799 | 0.0792 | 0.0759
7.0 | 0.16 | 0.1604 | 0.1573 | 0.1568 | 0.1534
14.0 | 0.04 | 0.0400 | 0.0393 | 0.0384 | 0.0389
28.0 | 0.04 | 0.0397 | 0.0388 | 0.0389 | 0.0367
Table 4: Comparison between reference $V_{p}$ and $\tilde{V_{p}}$ predicted for all flow regimes of (2). Salinity et sediment concentrations $S$ and $C$ (PILAT cases) from reference dataset $\mathcal{D}$ are used as training data and corrupted by gaussian noise of varying magnitudes $\mathcal{N}(0,\sigma)$ with $\sigma=15\%,~{}25\%$ and $35\%$. $S_{c}$ | $V_{p}$ | $\tilde{V_{p}}$
---|---|---
| $\sigma=0\%$ | $\sigma=15\%$ | $\sigma=25\%$ | $\sigma=35\%$
0.7 | 0.04 | 0.0395 | 0.0402 | 0.0375 | 0.0362
3.5 | 0.04 | 0.0397 | 0.0354 | 0.0351 | 0.0345
7.0 | 0.01 | 0.0097 | 0.0082 | 0.0089 | 0.0075
7.0 | 0.04 | 0.0390 | 0.0329 | 0.0295 | 0.0289
7.0 | 0.08 | 0.0789 | 0.0698 | 0.0682 | 0.0675
7.0 | 0.16 | 0.1593 | 0.133 | 0.132 | 0.140
14.0 | 0.04 | 0.0394 | 0.0338 | 0.0317 | 0.0295
28.0 | 0.04 | 0.0392 | 0.0313 | 0.0258 | 0.0312
Table 5: Comparison between reference $V_{p}$ and $\tilde{V_{p}}$ predicted
for all flow regimes of (2). Velocity components $u$ and $w$ (PIV case) are
used as training data and corrupted by gaussian noise of varying magnitudes
$\mathcal{N}(0,\sigma)$ with $\sigma=15\%,~{}25\%$ and $35\%$.

Additionally, in Figure (7), we can observe the progression of the
$L_{2}$-relative error for each studied flow regime in response to noise in
the PILAT case. This error is specifically computed for velocity and vorticity
fields that are not part of the training dataset. We can see that in the
majority of cases, higher levels of noise lead to deteriorating results, which
is consistent with the conclusion regarding the influence of noise on $V_{p}$
identification.

Figure 7: $L_{2}$-relative error (12) predicted and reference velocity and
vorticity. The error is computed on the discrete set of spatio-temporal
coordinates $\mathcal{C}$ corresponding to the training domain
$\Omega_{obs}\times[0,T_{obs}]$ of figure (4). Each flow regimes of (2) are
investigated.

A more precise study of the identification process and reconstruction results
will be undertaken in the next section.

##  4 Case study : $V_{p}=0.04$, $Sc=0.7$.

In this section, we delve further into the analysis of the identification
problem, we focus on the scenario where $Sc$ equals 0.7 and $V_{p}$ is set to
0.04, which serves as a reference for the canonical scenario discussed in [6].
We keep the same network parameters as in section (3) and the loss function
(10) which is computed on the cartesian grid $\mathcal{C}$.

Figure 8: Comparison of horizontal profiles. Snapshots of salinity, sediment
concentration velocity and vorticity at $t=100$ for the reference solution
(first row) and the PINN prediction (seconde row). Horizontal profiles at
$z=-28.0$ (red blue line on the snapshots) are compared between CFD reference
solution (red dashed line) and PINN inference (blue dashed line).

In Fig. (8), we compare the reference solution and the PINN inference by
examining horizontal profiles at two distinct locations. In both cases, the
predictions closely match the solution within the main domain. However, it is
worth pointing out that at the boundaries of the $x$-domain, subtle
discrepancies between the prediction and the reference become apparent for
velocity and vorticity fields. This difference can likely be attributed to the
absence of explicitly defined boundary conditions for the PINN model.

Figure 9: Comparison of horizontal profiles. Snapshot of salinity, sediment
concentration velocity and vorticity at $t=100$ for the reference solution
(first row) and the PINN prediction (seconde row). Horizontal profiles at
$z=28.0$ (red blue line on the snapshots) are compared between CFD reference
solution (red dashed line) and PINN inference (blue dashed line). salinity and
sediment concentration are normalized using (15).

For the PIV case, the salinity and sediment concentration field predicted cant
be compared directly with the corresponding reference field because they are
defined to within one constant. Indeed, only the derivatives in time and space
of these field are involved in the equations (2). Thus modify the predicted
and reference $S$ and $C$ fields as follows :

$\displaystyle\hat{V}(\bm{x},t)=\dfrac{V(\bm{x},t)-\inf_{(\bm{x},t)\in
X}V}{\sup_{(\bm{x},t)\in X}V-\inf_{(\bm{x},t)\in X}V}$ (15)

By normalizing the data, it becomes feasible to assess the comparison between
these fields, this comparison can be seen Fig. (9). While the velocity and the
vorticity fields match perfectly with the corresponding references fields, it
is hard to recognize the correspondance between CFD and PINN salinity field.
The profiles show a big discrepancy for this field and, although the
difference is less marked, the sediment concentration field also shows many
differences in the horizontal profiles. These results can partially explain
the difference in quality between PILAT and PIV case for the identification of
$V_{p}$.

Figure 10: Comparison of horizontal profiles. Snapshots of salinity, sediment
concentration velocity and vorticity at $t=100$ for the reference solution
(first row). Horizontal profiles at $z=-43$ (blue line on the snapshots) are
compared between CFD reference solution (red line) and PINN inference (blue
dashed line) for varying level of gaussian noise where $\sigma=15\%,~{}25\%$
and $35\%$ from second to last line.

For PILAT case, the effect on noise can be seen Fig. (10), in this figure, we
observe the evolution of the same profile as more noise is introduced. The
model exhibits a high degree of robustness, however, noticeable discrepancies
emerge, particularly around $x$ = 128 for the $w$-component of velocity. Even
in the absence of noise, the model struggles to reach the maximum values.
Moreover, as the noise level increases, the disparities between the predicted
values and the ground truth become more pronounced.

In the following, we study how the quality of $V_{p}$ identification evolves
with networks of varying architectures. In Fig. (11), we can see the evolution
of predicted $\tilde{V_{p}}$ during training for different number of neurons
per layers. We can also qualitatively compare the reconstruction of the
different models. In Fig. (11(a)) a one hidden layer deep network is used
while in Fig. (11(b)), a nine hidden layers deep architecture is investigated.
As we can see, all models seems to converge to the reference value
$V_{p}=0.04$. In the single-layer configuration, the more units, the longer
the convergence time. Moreover, for each case the one layer architecture
performs poor results on the reconstruction of $C$. In the other hand, For 9
layers, the convergence is slightly faster than for 1 layer, but is the same
regardless of the number of neurons. The reconstruction is more accurate than
for 1 layers but only 9 layers with 256 units can perform a good
reconstruction while inferrinf the right value of $V_{p}$.

While each model successfully captures the $V_{p}$ parameter, the ability to
achieve accurate reconstruction holds significant importance. In practical
experiments, the capacity to reconstruct effectively serves as a valuable
indicator for validating the identification process. Conversely, we would
hesitate to place trust in the identification of a model that fails to
adequately reconstruct the training fields.

(a) One hidden layer architecture.

(b) Nine hidden layers architecture.

Figure 11: Analyzing diverse network structures : architectures with gradually
increasing neurons (32, 64, 128, 256 units). For each case, we monitore the
evolution of $\tilde{V_{p}}$ during training and assessing sediment
concentration predictions at t=100 against the reference solution.

Fig. (12) presents a snapshot at $t=100$ depicting the residuals $e_{1}$ and
$e_{2}$ in comparison to the partial derivative of $C_{\mathcal{N}}$ with
respect to $z$. In addition, horizontal profiles at two different locations.
The residual $e_{2}$ corresponds to the vorticity equation and exhibits
minimal values. Conversely, the residual $e_{2}$, which is related to sediment
concentration transport equation involving $V_{p}$, displays values ten times
higher. Notably, these residuals appear to share a similar shape with
$\frac{\partial C_{\mathcal{N}}}{\partial z}$. Moreover, in the horizontal
profile, we can see that pic values for $\frac{\partial
C_{\mathcal{N}}}{\partial z}$ corresponds to pic values for this residuals. As
$\tilde{V_{p}}$ infered is directly linked to $\frac{\partial
C_{\mathcal{N}}}{\partial z}$. This means that the shape of this residual
might be caused by the error of prediction of $\tilde{V_{p}}$. In this cases,
this is a key result, because the shape of the residuals can be an indirect
way to monitor the quality of $V_{p}$ identification.

Figure 12: Visualisation of $\dfrac{\partial C}{\partial z}$, and residuals
$e_{2}$ and $e_{1}$ at $t=100$ (First column). Corresponding horizontal
profiles at $z=-28$ and $z=28$ (Second column).

## Conclusion

This paper demonstrates the effectiveness of Physics-Informed Neural Networks
(PINNs) in inferring dimensionless settling velocity in sedimentation flows
using synthetic data. Beyond addressing a challenging reconstruction problem
involving the coupling of Navier-Stokes equations with the evolution of
sediment concentration and salinity, the study successfully identify settling
velocity across various flow regimes that illustrate distinct key features of
sedimentation processes. The robustness and applicability of this framework
are validated through two sets of training data corrupted with varying levels
of noise, one containing salinity and sediment concentration fields and the
other containing velocity fields. These results showcase the versatility and
reliability of PINNs, particularly in experimental contexts.

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URL https://doi.org/10.1063/5.0136886


††thanks<EMAIL_ADDRESS>Present
address: Institute for Solid State Physics, University of Tokyo, Kashiwa
277-8581, Japan††thanks<EMAIL_ADDRESS>Present
address: Physics Division, Sophia University, Chiyoda-ku, Tokyo 102-8554,
Japan††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
#  Effective Tight-Binding Model of Compensated Ferrimagnetic Weyl Semimetal
with Spontaneous Orbital Magnetization

Tomonari Meguro1 Akihiro Ozawa2 Koji Kobayashi1 Kentaro Nomura1 1Department of
Physics, Kyushu University, Fukuoka 819-0395, Japan  2Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan

###### Abstract

The effective tight-binding model with compensated ferrimagnetic inverse-
Heusler lattice Ti2MnAl, candidate material of magnetic Weyl semimetal, is
proposed. The energy spectrum near the Fermi level, the configurations of the
Weyl points, and the anomalous Hall conductivity are calculated. We found that
the orbital magnetization is finite, while the total spin magnetization
vanishes, at the energy of the Weyl points. The magnetic moments at each site
are correlated with the orbital magnetization, and can be controlled by the
external magnetic field.

##  I Introduction

Topological semimetals are new classes of materials characterized by the
topologically-nontrivial gapless nodes. One of the representative systems is
the Weyl semimetal (WSM), a gapless semiconductor with non-degenerate point
nodes called Weyl points [1, 2, 3]. In momentum space, these nodes behave as a
source or sink of a fictitious magnetic field (Berry curvature), which is
distinguished by the sign degrees of freedom, so-called chirality [4, 5]. The
Weyl points with the opposite chirality must appear in pairs [6, 7].
Originating from the Weyl points, distinctive magnetoelectric effects, such as
the chiral magnetic effect [8, 9, 10, 11], arise. Generally, to realize the
WSM, one has to break either the inversion or time-reversal symmetry of Dirac
semimetal [12, 13, 14], which has degenerate gapless linear dispersions. As
early theoretical works, the WSM phases in an antiferromagnetic pyrochlore
structure [2] and a topological insulator multilayer [3] were proposed.
Especially, in Ref. 3, Burkov et al. discussed the anomalous Hall effect (AHE)
in the WSMs with broken time-reversal symmetry, so-called magnetic Weyl
semimetals (MWSMs). It was shown that the anomalous Hall conductivity (AHC) is
proportional to the distance between the pair of Weyl points with the opposite
chirality.

After these theoretical predictions, exploring for the WSM phases in specific
materials has been demonstrated. In the early stage of the experimental
studies, non-magnetic WSMs with the broken inversion symmetry, such as TaAs,
were examined [15, 16]. On the other hand, a great deal of effort has been
devoted to realize the MWSMs. After then, both theoretical and experimental
studies have succeeded in discovering the MWSM phases in some systems, such as
layered-kagome [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40] and Heusler systems [41, 42, 43, 44, 45,
46, 47]. The layered-kagome systems attract much attention from viewpoints of
anomalous transport phenomena [18, 19, 20, 21, 22, 23, 24, 25, 26, 31, 32, 38]
and various magnetic orderings [17, 34, 36, 39, 40]. Antiferromagnetic Mn3Sn
shows the AHE even without net magnetization [18, 19, 20, 21, 22, 23, 24].
Ferromagnetic Co3Sn2S2 shows the giant AHE and small longitudinal
conductivity, resulting in the large anomalous Hall angle reaching 20$\%$ [26,
27, 28, 29, 30, 33, 35, 37, 39]. As other promising candidates, recent studies
reported ferromagnetic Heusler systems with relatively high Curie temperature
$T_{\rm C}$ compared to those of the layered-kagome systems. For example,
Co2MnGa [41, 42, 43] with $T_{\rm C}\approx$ 694 K [44] and Co2MnAl [45, 46]
with $T_{\rm C}\approx$ 724 K [47] are also studied. In addition to the
layered-kagome and Heusler systems, other systems such as EuCd2As2 [48, 49,
50] and $Ln$Al$Pn$ [51, 52] ($Ln=$ lanthanides, $Pn=$ Ge, Si) are also
reported as candidate materials of MWSMs. A wide variety of candidates for
MWSMs has been explored and has influenced on both the field of topological
materials and magnetism.

The magnetic Weyl semimetal phase has been reported in compensated
ferrimagnetic inverse Heusler alloy Ti2MnAl [53] by first-principles
calculations. In Ti2MnAl, magnetic moments at two Ti sites are anti-parallel
to that at the Mn site, showing zero net magnetization. The transition
temperature is determined to be 650 K [54], which is comparable to those of
other Heusler Weyl systems, such as Co2MnGa. Besides, Ti2MnAl exhibits a large
AHE despite its vanishing total spin magnetization, similar to Mn3Sn. However,
in contrast to Mn3Sn, the density of states at the Weyl points in Ti2MnAl is
negligibly small, resulting in the manifestation of a large anomalous Hall
angle compared to other MWSMs. These properties indicate that the unique
electronic and magnetic structures of Ti2MnAl provide distinctive
magnetoelectric response and spintronic functionalities compared to
conventional materials.

In order to study the magnetoelectric responses specific to Ti2MnAl, a
quantitative analysis is necessary. First-principles calculations are widely
used to precisely calculate the electronic structure, considering all the
electron orbitals. However, with the first-principles calculations, it is
generally difficult to study the magnetoelectric responses related to the
complicated spin texture, such as the magnetic domain wall. This is because
the matrix of the Hamiltonian becomes huge due to the lack of translational
symmetry. Therefore, a simple tight-binding model describing the Weyl points
of Ti2MnAl is beneficial to calculate these magnetoelectric responses.

In this paper, we construct an effective tight-binding model of Ti2MnAl using
a few orbitals. We consider the single orbitals of two Ti and Mn, spin-orbit
coupling, and exchange interaction between compensated ferrimagnetic ordering
and itinerant electron spin. Using our model, we study the electronic
structure, AHE, spin and orbital magnetizations, and magnetic anisotropy. We
show that our model describes the configuration of the Weyl points that is
consistent with the results obtained by the first-principles calculations.
Also, we discuss the control of compensated ferrimagnetic ordering by orbital
magnetization.

##  II Model

In this section, we introduce a simple tight-binding model of the compensated
ferrimagnetic Weyl semimetal Ti2MnAl. The crystal structure of Ti2MnAl is
shown in Fig. 1(a). Each sublattice (Ti1, Ti2, Mn, and Al) forms a face-
centered-cubic (FCC) lattice, and thus the primitive unit cell is the FCC type
[Fig. 1(b)]. By focusing on pairs of the sublattices, Ti1-Al (orange and gray)
and Ti2-Mn (red and blue) form the rocksalt structure. The rest of the
combinations, e.g., Ti1-Ti2 or Ti1-Mn, form diamond or zincblende structures.

Figure 1:  (a) Conventional unit cell of Ti2MnAl. Ti and Mn are responsible
for the ferrimagnetic ordering. (b) Primitive unit cell of Ti2MnAl. In our
model, only Ti1 (A), Ti2 (B), and Mn (C) sublattices are taken into account.
(c) Inter-sublattice nearest-neighbor hoppings. (d) Intra-sublattice nearest-
neighbor hoppings. Every sublattice has the same hopping vectors.

Our model consists of a single orbital from each of Ti1 (A), Ti2 (B), and Mn
(C) atoms. We neglect Al orbitals, which are not responsible for the
magnetism, for simplicity. We explain our model Hamiltonian $H$ by dividing
into three components, $H=H_{\textrm{t}}+H_{\textrm{exc}}+H_{\textrm{SOC}}$,
where $H_{\textrm{t}}$ represents the hopping, $H_{\textrm{exc}}$ the exchange
coupling, and $H_{\textrm{SOC}}$ the spin-orbit coupling (SOC).

The hopping component $H_{\textrm{t}}$ reads

$\displaystyle H_{\textrm{t}}=$
$\displaystyle-\\!\sum_{\braket{ij}s}\\!\left(t_{\textrm{AB}}a^{\dagger}_{is}b_{js}+t_{\textrm{BC}}b^{\dagger}_{is}c_{js}+t_{\textrm{CA}}c^{\dagger}_{is}a_{js}+{\rm
h.c.}\\!\right)$
$\displaystyle-\\!\sum_{\braket{ij}s}\\!\left(t_{\textrm{AA}}a^{\dagger}_{is}a_{js}+t_{\textrm{BB}}b^{\dagger}_{is}b_{js}+t_{\textrm{CC}}c^{\dagger}_{is}c_{js}\right)$
$\displaystyle+\\!\sum_{is}\left(\epsilon_{\textrm{A}}a^{\dagger}_{is}a_{is}+\epsilon_{\textrm{B}}b^{\dagger}_{is}b_{is}+\epsilon_{\textrm{C}}c^{\dagger}_{is}c_{is}\right),$
(1)

where $a_{is}$, $b_{is}$, and $c_{is}$ are the annihilation operators for
electrons at Ti1 (A), Ti2 (B), and Mn (C) sites, respectively. The first line
corresponds to the inter-sublattice nearest-neighbor hopping [Fig. 1(c)]. The
second line corresponds to the intra-sublattice nearest-neighbor hopping [Fig.
1(d)]. The third line is the on-site energy.

The exchange component is

$\displaystyle H_{\rm exc}$
$\displaystyle=-\sum_{i}{\bm{m}}\cdot\left(J_{\textrm{A}}{\bm{s}}_{{\rm
A},i}+J_{\textrm{B}}{\bm{s}}_{{\rm B},i}-J_{\textrm{C}}{\bm{s}}_{{\rm
C},i}\right),$ (2)

where ${\bm{s}}_{{\rm
A},i}=a^{\dagger}_{is}\left({\bm{\sigma}}\right)_{ss^{\prime}}a_{is^{\prime}}$
is the itinerant spin operator of A site, and the same applies to
${\bm{s}}_{\textrm{B}}$ and ${\bm{s}}_{\textrm{C}}$. ${\bm{m}}$ is the unit
vector that is parallel to the magnetic moment of Ti (A and B) and
antiparallel to that of Mn (C). $J_{\alpha}\ (\alpha=\mathrm{A,B,C})$ are the
coupling strength.

The SOC $H_{\textrm{SOC}}$ originates from the broken inversion symmetry of
the crystal structure. The dominant symmetry breaking comes from the imbalance
between TI1 and Al sublattices. We assume the amplitudes of the SOC terms for
the Ti2-Ti2 and Mn-Mn hoppings are the same, for simplicity. Since the atomic
number of Ti and Mn are close to each other, compared with Al, we neglect the
SOC for the Ti1-Ti1 hopping. The SOC term can be described in a Fu-Kane-Mele-
like form [55],

$\displaystyle
H_{\textrm{SOC}}=-i\frac{8\lambda_{\textrm{SOC}}}{\sqrt{2}a^{2}}$
$\displaystyle\sum_{\braket{ij}}\left[b^{\dagger}_{is}(\bm{d}^{{\rm
B}ij}_{1}\times\bm{d}^{{\rm
B}ij}_{2})\cdot(\bm{\sigma})_{ss^{\prime}}b_{js^{\prime}}\right.$
$\displaystyle\left.+~{}c^{\dagger}_{is}(\bm{d}^{{\rm
C}ij}_{1}\times\bm{d}^{{\rm
C}ij}_{2})\cdot(\bm{\sigma})_{ss^{\prime}}c_{js^{\prime}}\right].$ (3)

Here $\lambda_{\textrm{SOC}}$ is the strength of the SOC. $\bm{d}^{\alpha
ij}_{1,2}$ are the two nearest-neighbor hopping vectors from the site $i$ to
$j$ of the sublattice $\alpha$. Note that $\bm{d}^{{\rm
B}ij}_{1,2}={\bm{n}}_{l_{ij}}$ and $\bm{d}^{{\rm
C}ij}_{1,2}=-{\bm{n}}_{l_{ij}}\ (l_{ij}=1,2,3,4)$ with
${\bm{n}}_{1}=\frac{a}{4}(1,-1,-1)$, ${\bm{n}}_{2}=\frac{a}{4}(-1,1,-1)$,
${\bm{n}}_{3}=\frac{a}{4}(-1,-1,1)$, and ${\bm{n}}_{4}=\frac{a}{4}(1,1,1)$.

We set the lattice constant $a=1$ for simplicity. The hopping parameters are
set to $t_{\textrm{AB}}=1.1t_{0}$, $t_{\textrm{BC}}=0.4t_{0}$,
$t_{\textrm{CA}}=1.2t_{0}$, $t_{\textrm{AA}}=0.05t_{0}$,
$t_{\textrm{BB}}=0.85t_{0}$, and $t_{\textrm{CC}}=-0.05t_{0}$. On-site
energies
$\epsilon_{\textrm{A}}=\epsilon_{\textrm{B}}=\epsilon_{\textrm{C}}=-2.15t_{0}$.
The strengths of the exchange coupling
$J_{\textrm{A}}=J_{\textrm{B}}=0.7t_{0}$, $J_{\textrm{C}}=1.7t_{0}$. The
strength of the SOC $\lambda_{\textrm{SOC}}=-0.2t_{0}$. We set the energy unit
$t_{0}=0.33$ eV. The parameters are set so that the energy bands, density of
states, and AHCs become consistent with first-principles calculations as
discussed later.

##  III Electronic structure

Figure 2:  (a) Band structure along the high symmetry lines. (b) The density
of states for (red) up- and (blue) down-spin states and (c) anomalous Hall
conductivities $\sigma_{xy}$, $\sigma_{yx}$, $\sigma_{zx}$ as a function of
energy. (d) The Brillouin zone and high symmetry lines of the system. (e)
Configuration of the Weyl points with and without spin-orbit coupling and (f)
the Berry curvature with spin-orbit coupling on the $k_{x}$-$k_{z}$ plane.

Now, we study the electronic structure of this model. The momentum
representation of the Hamiltonian is explicitly expressed as,

$\displaystyle\mathcal{H}(\bm{k})$
$\displaystyle=\begin{pmatrix}-t_{\textrm{AA}}f({\bm{k}})-\\!J_{\textrm{A}}{\bm{m}}\cdot{\bm{\sigma}}&-t_{\textrm{AB}}g({\bm{k}})&-t_{\textrm{CA}}g^{*}({\bm{k}})\\\
-t_{\textrm{AB}}g^{*}({\bm{k}})&-t_{\textrm{BB}}f({\bm{k}})+\lambda_{\textrm{SOC}}{\bm{R}}({\bm{k}})\cdot{\bm{\sigma}}-\\!J_{\textrm{B}}{\bm{m}}\cdot{\bm{\sigma}}&-t_{\textrm{BC}}\sum_{i=x,y,z}\cos\frac{k_{i}}{2}\\\
-t_{\textrm{CA}}g({\bm{k}})&-t_{\textrm{BC}}\sum_{i=x,y,z}\cos\frac{k_{i}}{2}&-t_{\textrm{CC}}f({\bm{k}})-\lambda_{\textrm{SOC}}{\bm{R}}({\bm{k}})\cdot{\bm{\sigma}}+\\!J_{\textrm{C}}{\bm{m}}\cdot{\bm{\sigma}}\end{pmatrix}\\!.$
(4)

Here, $f({\bm{k}})$ and $g({\bm{k}})$ correspond to the FCC and diamond
hoppings, respectively, and are defined as,

$f({\bm{k}})\\!=\\!4\\!\left(\\!\cos\\!\frac{k_{x}}{2}\\!\cos\\!\frac{k_{y}}{2}\\!+\\!\cos\\!\frac{k_{y}}{2}\\!\cos\\!\frac{k_{z}}{2}\\!+\\!\cos\\!\frac{k_{z}}{2}\\!\cos\\!\frac{k_{x}}{2}\\!\right)\\!,$
(5)
$g({\bm{k}})=\exp\left(-i\sum^{4}_{\mu=1}{\bm{n}}_{\mu}\cdot{\bm{k}}\right).$
(6)

${\bm{R}}({\bm{k}})$ corresponds to the FCC hopping with SOC and is defined
as,

$\displaystyle{\bm{R}}({\bm{k}})$
$\displaystyle=\begin{pmatrix}\sin\frac{k_{x}}{2}\left(\cos\frac{k_{y}}{2}-\cos\frac{k_{z}}{2}\right)\\\
\sin\frac{k_{y}}{2}\left(\cos\frac{k_{z}}{2}-\cos\frac{k_{x}}{2}\right)\\\
\sin\frac{k_{z}}{2}\left(\cos\frac{k_{x}}{2}-\cos\frac{k_{y}}{2}\right)\end{pmatrix}.$
(7)

We use the magnetic moment pointing in the out-of-plane direction as
${\bm{m}}=(0,0,1)$. By solving the eigenvalue equation
$\mathcal{H}(\bm{k})|n,\bm{k}\rangle=E_{n\bm{k}}|n,\bm{k}\rangle$, the
eigenvalues $E_{n\bm{k}}$ and eigenstates $|u_{n\bm{k}}\rangle$ are obtained.
Here, $n=1,2,...,6$ is the band index labeled from the bottom. Figures 2(a)
and 2(b) show the band structure along the high-symmetry lines and the density
of states (DOS) as a function of energy, respectively. The high-symmetry lines
are shown in Fig. 2(d). We assume that the Fermi level $E_{\rm F}$ is the
energy which is satisfied $4/6$ filling, and is being set as $E/t_{0}=0$. At
the energy $E_{\rm F}$ , we have the Weyl points as discussed later. In Fig.
2(a), red and blue lines indicate up and down spin band, respectively, in the
absence of SOC. Green lines indicate those in the presence of SOC. Here, we
focus on the spin up bands near $E/t_{0}=0.0$. As shown in Fig. 2(b), the spin
up bands show the local minimum of the DOS near the Fermi level
($E/t_{0}=0.0$). This minimum is consistent with the result obtained by the
first-principles calculations and may be significant to the large anomalous
Hall angle [53]. As shown in the insets of Fig. 2(b), the energy spectrum of
these majority spin bands show the gapless linear dispersions, corresponding
to the Weyl points, as discussed later. On the other hand, down-spin state
(blue) is gapped at the Fermi level ($E/t_{0}=0.0$).

Next we study the Weyl points structure in Brillouin zone. We show that our
model describes the Weyl points structure similarly located to the result
obtained by the first-principles calculations. From our model Hamiltonian
$\mathcal{H}(\bm{k})$ 24 gapless nodes (Weyl points) are obtained between
$n=4$ band and $n=5$ band. The $k_{x}$-$k_{z}$ plane has the Weyl points as
shown in Fig. 2(e). Each Weyl points with chirality $+$ ($-$) is labeled by
$W^{+}_{\alpha}$ ($W^{-}_{\alpha}$). $\alpha$ distinguish the pairs of the
Weyl points. To characterize these nodes by chirality, we compute the Berry
curvature $\bm{b}_{n\bm{k}}=\nabla_{\bm{k}}\times\bm{a}_{n\bm{k}}$ of the
$n=4$ band. Here, $\bm{a}_{n\bm{k}}$ is the Berry connection, defined as
$\bm{a}_{n\bm{k}}=-i\langle n,\bm{k}|\nabla_{\bm{k}}|n,\bm{k}\rangle$. Figure
2(f) shows the Berry curvature distribution on $k_{x}$-$k_{z}$ plane. The
strength of the shade of the arrows is the amplitude of $\bm{b}(\bm{k})$. The
red and blue circles indicate the sources ($+$) and sinks ($-$) of the Berry
curvature. The configuration of the Weyl points is consistent with those
obtained by the first-principles calculations [53].

##  IV Anomalous Hall effect

Let us study the AHE in this section. Using the Kubo formula [56], the
anomalous Hall conductivities (AHCs) $\sigma_{ij}\ (i\neq j)$ can be expressed
as follows,

$\begin{split}\sigma_{ij}&=-\frac{ie^{2}}{\hbar}\int_{\rm
BZ}\frac{d^{3}\bm{k}}{(2\pi)^{3}}\sum_{m\neq
n}\frac{f(E_{n\bm{k}})-f(E_{m\bm{k}})}{E_{n\bm{k}}-E_{m\bm{k}}+i\eta}\\\ &\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\frac{\langle n,\bm{k}|\hbar
v_{i}|m,\bm{k}\rangle\langle m,\bm{k}|\hbar
v_{j}|n,\bm{k}\rangle}{E_{n\bm{k}}-E_{m\bm{k}}}.\end{split}$ (8)

Here, $f(E)=1/(e^{\beta(E-E_{\rm F})}+1)$ is the Fermi-Dirac distribution
function and $\hbar\bm{v}=\frac{\partial\mathcal{H}(\bm{k})}{\partial\bm{k}}$
is the velocity operator. Figure 2(c) shows $\sigma_{xy}$, $\sigma_{yz}$, and
$\sigma_{zx}$ as a function of energy. Near the energy of the Weyl points, the
$\sigma_{xy}$ is maximized , while $\sigma_{yz}$ and $\sigma_{zx}$ vanish. The
value of $\sigma_{xy}$ at the peak is $\sigma_{xy}\approx
0.80~{}[e^{2}/ah]=496~{}{\rm S/cm}$, which reasonably agrees with the result
obtained by first-principles calculations, $550~{}{\rm S/cm}$ [53].
Then we study the relation between the AHCs and the configuration of the Weyl
points. As discussed in the introduction, when the Fermi level is located at
the Weyl points, the AHCs can be calculated with the distances between the
Weyl points with opposite chirality
$\Delta\bm{Q}^{\alpha}=({\bm{K}}^{+}_{\alpha}-{\bm{K}}^{-}_{\alpha})$ [3],

$\begin{split}\sigma^{\rm
Weyl}_{ij}&=\frac{e^{2}}{(2\pi)^{2}\hbar}\sum_{\alpha=1}^{12}\sum_{k}\epsilon_{ijk}\Delta
Q_{k}^{\alpha}.\end{split}$ (9)

Here ${\bm{K}}^{+(-)}_{\alpha}$ is the position of the Weyl points with
chirality $+(-)$. This $\sigma^{\rm Weyl}_{xy}$ is being $\approx 0.75\
[{e^{2}/ah}]$, which is in good agreement with the maximized value obtained by
the Kubo formula, shown in Fig. 2(c). Therefore, the AHE in our model mainly
originates from the Weyl points.

Next, we discuss the role of SOC for the Weyl points and the AHC. In the
absence of SOC, the Weyl points are distributed symmetrically, as represented
by empty circles in Fig. 2(e). This can be anticipated by the crystal symmetry
of the system [53]. In this case, the sum $\sum_{\alpha}\Delta\bm{Q}^{\alpha}$
cancels, and thus AHC $\sigma^{\rm Weyl}_{xy}$ vanishes. On the other hand, in
the presence of SOC, the positions of the Weyl points are shifted as
represented by filled circles in Fig. 2(e). For the pairs
($W^{+}_{1}$,$W^{-}_{1}$) and ($W^{+}_{2}$,$W^{-}_{2}$), the distance between
the Weyl points $\Delta Q^{\alpha=1,2}_{z}$ becomes longer. Whereas, for the
pairs of ($W^{+}_{3}$,$W^{-}_{3}$) and ($W^{+}_{4}$,$W^{-}_{4}$), $\Delta
Q^{\alpha=3,4}_{z}$ becomes shorter. Therefore, the cancellation of the sum
$\sum_{\alpha}\Delta\bm{Q}^{\alpha}$ is broken, giving rise to the finite AHC
$\sigma^{\rm Weyl}_{xy}$. We note that the Weyl points in other planes are
shifted in a similar manner to those in the $k_{x}$-$k_{z}$ plane.

Figure 3:  (a) Spin magnetization $M^{\rm spin}_{z}$ and (b) each component of
orbital magnetization ${\bm{M}}^{\rm orbit}$ as a function of energy for the
magnetic moment $\bm{m}$ parallel to the $z$ axis.

##  V Spin and Orbital magnetizations

In this section, we study spin and orbital magnetizations in our model. The
spin magnetization is calculated by the following equation,

$\begin{split}M^{\rm spin}_{z}&=\mu_{\rm B}\int^{E_{\rm
F}}_{E_{0}}[D_{\uparrow}(\epsilon)-D_{\downarrow}(\epsilon)]d\epsilon.\end{split}$
(10)

Here, $\mu_{\rm B}$ is the Bohr magneton, and $E_{0}$ is the lower band edge.
$D_{\uparrow}(\epsilon)$ [$D_{\downarrow}(\epsilon)$] is the DOS for the up-
spin (down-spin). The spin magnetization as a function of energy is shown in
Fig. 3(a). Recall that the magnetic moments of Ti and Mn are parallel to the
$z$ axis. $M^{\rm spin}_{z}$ vanishes at $E=E_{\rm F}$, indicating the
compensated ferrimagnetic phase. Owing to this characteristic DOS, finite spin
magnetization may be obtained by tuning $E_{\rm F}$, using a gate voltage, for
instance. When $E_{\rm F}$ is increased (decreased), finite and positive
(negative) spin magnetization might be generated. This indicates that, in
Ti2MnAl, one can induce and switch the spin magnetization electrically.
Recently, the electrical control of ferrimagnetic systems has been studied
from viewpoint of a functional magnetic memory, for example in Ref. 57.

Next we study the orbital magnetization [58, 59, 60],

$\begin{split}M^{\rm orbit}_{k}&=\frac{e}{2\hbar}\int_{\rm
BZ}\frac{d^{3}\bm{k}}{(2\pi)^{3}}\sum_{i,j}\epsilon_{ijk}\\\ &{\rm
Im}\sum_{m\neq n}f(E_{n\bm{k}})\frac{\langle n,\bm{k}|\hbar
v_{i}|m,\bm{k}\rangle\langle m,\bm{k}|\hbar
v_{j}|n,\bm{k}\rangle}{(E_{m\bm{k}}-E_{n\bm{k}})^{2}}\\\ &\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\times}(E_{n\bm{k}}+E_{m\bm{k}}-2E_{\rm
F}).\end{split}$ (11)

Figure 3(b) shows each component of the orbital magnetization ${\bm{M}}^{\rm
orbit}$ as a function of energy. We find that $M^{\rm orbit}_{z}$ can be
finite, while $M^{\rm orbit}_{x}$ and $M^{\rm orbit}_{y}$ vanish. Although
orbital magnetization is usually much smaller than spin magnetization, $M^{\rm
orbit}_{z}$ is finite at $E_{\textrm{F}}$, where $M^{\rm spin}_{z}$ is fully
compensated. As discussed later, this small but finite orbital magnetization
might be used to switch the directions of magnetic moments on Ti and Mn sites.

##  VI Magnetic Isotropy

Figure 4:  The magnetization-angle dependence of the total energy of electron,
Eq. (12), when one rotates the magnetizations of Mn and Ti uniformly (solid
red line) or the magnetization of Mn solely (dotted blue line).

In this section, we study the magnetic anisotropy and angular dependences of
the AHCs and orbital magnetization. We start to study magnetic anisotropy.
Figure 4 shows the total energy of electrons, computed as

$\displaystyle E_{\textrm{e}}$
$\displaystyle=\frac{1}{N_{\bm{k}}}\sum_{n=1}^{6}\sum_{{\bm{k}}}E_{n{\bm{k}}}f(E_{n{\bm{k}}}),$
(12)

where $N_{\bm{k}}$ is the number of $\bm{k}$-mesh. Here, we consider the two
tilted configurations in which the magnetic moments of (a) Mn only or (b) both
Mn and Ti are tilted in $x$-$z$ plane. Their tilting angles are denoted by the
angles (a) $\theta_{\rm Mn}$ and (b) $\theta_{\rm Mn}=\theta_{\rm Ti}$. The
schematic figures for these situations are shown in the insets of Fig. 4. In
case (a), as the blue dotted line indicates, the electron energy is maximized
at $\theta_{\rm Mn}=180^{\circ}$, indicating the stability of the compensated
ferrimagnetic ordering. On the other hand, in case (b), the electron energy is
independent of $\theta_{\rm Mn}=\theta_{\rm Ti}$ as the red solid line
indicates. Therefore, in our model, the compensated ferrimagnetic ordering
does not show the easy-axis magnetic anisotropy. With these results, the
ferrimagnetic interaction between Ti and Mn can be estimated as
$0.1t_{0}\approx 33~{}{\rm meV}$, when $t_{0}$ = 0.33 eV is assumed.

Figure 5:  The magnetization-angle dependence of (a) orbital magnetization and
(b) AHCs at $E=E_{\rm F}$ when one changes the direction of magnetic moment
uniformly in the $x$-$z$ plane.

Then, we study a correlation between magnetic moments and orbital
magnetization. In the previous section, we showed that, at the energy of the
Weyl points, our model shows compensated spin magnetization and finite orbital
magnetization. In the following, we tilt the magnetic moments on both Ti and
Mn sites in the $x$-$z$ plane, as shown in the inset of Fig. 5(a). Figure 5(a)
shows each component of the orbital magnetization as a function of $\theta$.
We find that the orbital magnetization follows the direction of the magnetic
moments. This feature can be used for the control of the magnetic moments, as
similarly discussed in Refs. 24, 40. Under an external magnetic field,
magnetic moments on each site and orbital magnetization couple with the field
via Zeeman interaction. However, as discussed in the previous paragraph, the
ferrimagnetic coupling between the magnetic moments on Ti and Mn is stronger
than the typical strength of the Zeeman interaction ($\approx 0.15~{}{\rm
meV}$ at 1 T). Thus, the Zeeman interaction of the magnetic moments on both
sites cancels each other. This cancellation implies that only the orbital
magnetization couples with the external magnetic field. In addition, owing to
SOC, the directions of the magnetic moments are locked with the orbital
magnetization, as shown in Fig. 5(a). Consequently, the direction of the
magnetic moments can be controlled by an external magnetic field, even without
net magnetization. The changes in the direction of the magnetic moments may be
probed by the AHE. Figure 5(b) shows the $\theta$ dependence of the AHCs. We
find the relation $\sigma_{ij}\propto-\sum_{k}\epsilon_{ijk}{M}^{\rm
orbit}_{k}$. This relation indicates that the directions of the magnetic
moments are experimentally determined by measuring the AHCs.

##  VII Conclusion

In this paper, we constructed an effective model of compensated ferrimagnetic
Weyl semimetal Ti2MnAl. The Weyl points configuration in our model reasonably
agrees with those obtained by the first-principles calculations. The finite
AHC in the presence of SOC can be understood by the shifting of the Weyl
points. At the energy of the Weyl points, the orbital magnetization is finite
while the total spin magnetization vanishes. The magnetic moments at each site
are correlated with the orbital magnetization, and can be controlled by the
external magnetic field.

###### Acknowledgements.

The authors would like to appreciate Y. Araki and A. Tsukazaki for valuable
discussions. This work was supported by JST CREST, Grant No. JPMJCR18T2 and by
JSPS KAKENHI, Grant Nos. JP20H01830 and JP22K03446. A. O. was supported by GP-
Spin at Tohoku University.

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