# A General Framework for Uncertainty Quantification via Neural SDE-RNN

Shweta Dahale, , Sai Munikoti, , and
Balasubramaniam Natarajan,
S. Dahale and B. Natarajan are with the Electrical and Computer Engineering,
Kansas State University, Manhattan, KS-66506, USA (e-mail<EMAIL_ADDRESS>bala@ksu.edu). S.Munikoti is with the Data Science and Machine Intelligence
group, Pacific Northwest National Laboratory, USA, (e-mail:
sai.munikoti@pnnl.gov). This material is based upon work supported by the
Department of Energy, Office of Energy Efficiency and Renewable Energy (EERE),
Solar Energy Technologies Office, under Award Number DE-EE0008767.

###### Abstract

Uncertainty quantification is a critical yet unsolved challenge for deep
learning, especially for the time series imputation with irregularly sampled
measurements. To tackle this problem, we propose a novel framework based on
the principles of recurrent neural networks and neural stochastic differential
equations for reconciling irregularly sampled measurements. We impute
measurements at any arbitrary timescale and quantify the uncertainty in the
imputations in a principled manner. Specifically, we derive analytical
expressions for quantifying and propagating the epistemic and aleatoric
uncertainty across time instants. Our experiments on the IEEE 37 bus test
distribution system reveal that our framework can outperform state-of-the-art
uncertainty quantification approaches for time-series data imputations.

###### Index Terms:

Neural ordinary differential equations, uncertainty propagation, linearization
approximation, imputation

##  I Introduction

Irregularly sampled multivariate time-series data are found in several domains
spanning from bio-medical systems to the smart grid. This is primarily the
outcome of two factors. The first source is the inherent system design where
data generation happens at different rate. For instance, in a smart grid,
different sensors sense and transmit measurements at different time scales,
i.e., advanced metering infrastructure (AMI) samples at 15-min interval, data
acquisition (SCADA) sensors at 1-min interval, etc.[1]. The second factor
corresponds to system malfunctions that causes frequent unavailability of data
samples. For example, communication impairments in the smart grid result in
loss of data, further aggravating this issue [2]. It is important to reconcile
such multi-scale intermittent measurements at a common timescale in order to
enhance the situational awareness of the grid. Typically, this kind of
irregularly sampled data is first imputed (i.e., estimate missing values) to a
uniform scale so that an effective analysis can be carried out with the
processed data. To this end, several methods are being proposed in the
literature such as linear interpolation [3], kNN [4] and multitask Gaussian
process [1, 5]. Recently, neural ODES (NeuODES) have shown to be a very
effective framework for handling irregular sampled data [6, 7, 8, 9]. This is
primarily due to the fact that NeuODES can estimate values in a continuous
space (using dynamics) unlike discrete space (proportional to the number of
layers) in a vanilla neural network.

However, most of the data imputation frameworks solely offer a point estimate
of the missing value, ignoring their uncertainties. It is crucial to provide
confidence scores with prediction so that an informed decision can be made
while deployed in the real-world applications. Uncertainty quantification (UQ)
of the imputed measurements can be helpful in various ways, including (i)
variance-informed state estimation where variance (confidence interval) of the
imputed measurement’s can be used to modulate the inputs of the estimator [2].
For instance, high variance input is disregarded or corrected before being fed
to the estimator. (ii) Effective data sampling where UQ helps in selecting
appropriate measurements from high or low-fidelity sensors.

Uncertainty in deep neural network-based models such as NeuODES arises due to
two components. The first factor is with respect to the model parameters and
it arises due to the model either being blind to some portion of the data
distribution which has not been seen while it’s training or due to its over-
parameterization. There are various ways to handle this and can be quantified
in a computationally efficient manner. On the other hand, the second class of
uncertainty is related to the input data, also known as aleatoric uncertainty.
It is inherent in nature and cannot be alleviated by any means of model
engineering [10]. Modeling and quantifying these uncertainties are critical
with regard to time series imputation since the sensors are often associated
with noise, and one cannot cover the entire data distribution while training
the models.

There exist a few works in the literature that quantify uncertainty in
NeuODES-based frameworks, [11, 12, 13, 14]. Authors in [14] proposed a neural
Jump ODE framework for modeling the conditional expectation of a stochastic
process in an irregularly observed time series. However, most of these works
only retrieve the epistemic part and ignore the aleatoric part. It is
important to incorporate the aleatoric part, especially for critical complex
systems where sensor-generated data is pivotal in various operational
decision-making. Therefore, we propose a novel framework to quantify the
uncertainties in a comprehensive manner. In this regard, we first formulate an
SDE-RNN framework that combines the stochastic differential equation (SDE)
form of NeuODES and recurrent neural network (RNN) for imputation.
Specifically, SDE is chosen since it can propagate both mean (prediction) and
variances (uncertainty) along the NeuODES framework in an efficient manner.
Furthermore, RNN helps us to capture the inputs at observed time instances,
and effectively carry forward the information to future states via its memory
cell. Altogether, their combination offers an powerful and efficient approach
to quantify and propagate both uncertainties from the input to final
predictions.

The main contributions of this paper are summarized below:

  * •

We formulate a novel SDE-RNN which combines the principle of stochastic
differential equations and neural networks to model the irregularly sampled
time series.

  * •

The proposed SDE-RNN approach allows us to quantify both aleatoric and
epistemic uncertainty in the imputed measurements. Analytical results
capturing these sources of uncertainty have been derived. We derive the
theoretical expressions of uncertainty propagation for Gated Recurrent Unit
(GRU) model.

  * •

Simulation results on the power distribution test system (IEEE 37 network)
demonstrate the effectiveness of the proposed approach compared to the classic
baseline (RNN + MC dropout).

The paper is organized as follows. Section II presents a background of NeuODES
with a literature review. Section III formulates our fundamental SDE-RNN
framework followed by the detailed process of uncertainty quantification in
Section IV. Experiments are discussed in Section V with conclusions and future
work in Section VI.

##  II Background and Related work

This section provides an overview of the UQ literature for unevenly sampled
time series imputation models. Additionally, the mathematical background of
neural ODES, which serves as a fundamental module in our proposed approach, is
provided.

###  II-A Related work

Recurrent neural networks (RNN) form the first choice for modeling high
dimensional, regularly sampled time series data. The success of RNNs is due to
its memory and cell state modules that can capture long range dependencies. A
natural extension of RNNs to unevenly sampled time series data is to divide
the timeline into equally-sized intervals [15], [16], and impute or aggregate
observations using averages. This pre-processing stage maligns the
information, particularly about the timing of measurements, as they constitute
a piece of important information about the data. Another approach uses a
simple exponential decay between observations, and updates the hidden states
accordingly [16]. But the states of RNN-decay model approach zero if the time
gaps between observations are high. However, Neural ODE, which is a novel
framework combining deep learning principles and differential equations has
been found to be suitable for modeling irregularly sampled time series [6].
Neural ODEs can systematically accommodate continuous time series data using a
technique that captures the continuous nature of hidden states. Recently, [17]
combines neural ODEs with RNN, leading to a method referred to as ODE-RNN. The
hidden states in these ODE-RNN models obey an ODE between consecutive
observations and are only updated at observed time instances. The ODE-RNN
models are suitable for sparse observations and thus very effective for
imputation and one-step prediction. Authors in [11] propose a Latent SDE with
an adjoint method for continuous time series modeling. Neural SDEs driven by
stochastic processes with jumps is proposed in [12] to learn continuous and
discrete dynamic behavior. Generative models built from SDEs whose drift and
diffusion coefficients are samples from a Gaussian process are introduced in
[13]. A neural Jump ODE framework for modeling the conditional expectation of
a stochastic process in an irregularly observed time series is proposed in
[14]. However, all neural ODE driven methodologies only offer a point estimate
and fail to quantify the uncertainty associated with the estimates.

UQ approaches in general neural networks are mainly categorized into two
classes: Bayesian and Ensemble methods. Bayesian approaches, such as Bayesian
neural nets proposed in [18] quantify the uncertainty by imposing probability
distributions over model parameters. Though, these approaches provide a
principled way of uncertainty quantification, the exact inference of the
parameter posteriors is intractable. Also, specifying the priors for the
parameters of deep neural networks becomes challenging when size of the
network increases. To deal with these challenges, approximation methods like
variational inference [19], Laplace approximation [20], Assumed density
filtering [21] and stochastic gradient MCMC [22] are used. Authors in [10]
proposed to use Monte-Carlo Dropout (MC-dropout) during inference to quantify
the uncertainty in neural networks. However, MC dropout is computationally
expensive for larger network and only capture epistemic uncertainty. Non-
Bayesian approaches have also been used for UQ but they also demand a large
computational effort [23]. A different route has been taken in [24] where SDEs
are used to quantify both aleatoric and epistemic uncertainty by training the
SDE model based on out-of-distribution (OOD) training data. However, this
approach fails to evaluate uncertainty in a principled manner.

###  II-B Background: Neural ODE

Consider time-series sensor data $\\{x_{i},t_{i}\\}_{i=0}^{N}$, where the
measurements $\\{x_{i}\\}_{i=0}^{N}\in\mathbb{R}^{d}$ are obtained at times
$\\{t_{i}\\}_{i=0}^{N}$. The goal of timeseries imputation is to reconcile the
unevenly sampled measurements at at the finest time resolution. Neural ODE
exploits the continuous-time dynamics of variables from input state to final
predictions, unlike a standard deep neural network which only performs a
limited number of transformations depending on the number of layers chosen in
the architecture. The state transitions in RNNs are generalized to continuous
time dynamics specified by a neural network. We denote this model as ODE-RNN.
The hidden state in ODE-RNN is a solution to an ODE initial-value problem,
given as:

$\frac{d\mathbf{h}_{t}}{dt}=f(\mathbf{h}_{t},\theta)$ (1)

where, $f$ is a neural network parameterized by $\theta$ that defines the ODE
dynamics. $\mathbf{h}_{t}$ is the hidden state of the Neural ODE. Thus,
starting from an initial point $\mathbf{h}_{0}$, the transformed state at any
time $t_{i}$ is given by integrating an ODE forward in time, given as,

$\displaystyle\mathbf{h}_{i}=\mathbf{h}_{0}+\int_{t_{0}}^{t_{i}}\frac{d\mathbf{h}_{t}}{dt}dt$
(2) $\displaystyle\mathbf{h}_{i}=ODESolve(\mathbf{h}_{0},(t_{0},t_{i}),f)$

(2) can be solved numerically using any ODE Solver (e.g., Euler’s method). In
order to train the parameters of the ODE function $f$, an adjoint sensitivity
approach is proposed in [6]. This approach computes the derivatives of the
loss function with respect to the model parameters $\theta$ by solving a
second augmented ODE backward in time. Some of the advantages of using Neural
ODE solvers over other conventional approaches are: (1) Memory efficiency: The
adjoint sensitivity approach allows us to train the model with constant memory
cost independent of the layers in the ODE function $f$; (2) Adaptive
computation: Adaptive number of updation steps by ODE solver is more effective
than fixed number of steps in vanilla DNN. (3) Effective formulation: Finally,
the continuous dynamics of the hidden states can naturally incorporate data
which arrives at arbitrary times.

Based on the foundations of NeuODEs, ODE-RNN was developed. ODE-RNN is
autoregressive in nature unlike the generative form of NeuODES [17]. It models
the irregularly sampled time series by applying ODE and RNN interchangeably
through a time series. The states evolve continuously using the ODE model,
while they are updated using the RNN cell at the instances where the
measurements are available. The function $f$ at time $t$ is described in a
neural ODE with initial hidden state $h_{t}$:

$\dot{\mathbf{h}}=f(\mathbf{h}_{t},\theta)$ (3)
$o_{t}=c(\mathbf{h}_{t},\theta_{c})$ (4)

where, $\mathbf{h}\in\mathbb{R}^{m}$ is the hidden state of the data,
$\dot{\mathbf{h}}=\frac{d\mathbf{h}}{dt}$ is the derivative of the hidden
state, $o\in\mathbb{R}^{d}$ is the output of the ODE-RNN.
$f:\mathbb{R}^{m}\to\mathbb{R}^{m}$ and $c:\mathbb{R}^{m}\to\mathbb{R}^{d}$
are the neural ODE operator and output function parameterized by neural
network parameters $\theta$ and $\theta_{c}$, respectively. The hidden states
are modeled using a Neural ODE, where they obey the solution of an ODE as,

$\mathbf{h}_{i}^{{}^{\prime}}=ODESolve(\mathbf{h}_{i-1},(t_{i-1},t_{i}),f)$
(5)

Lets represent the RNN cell function by $v(\cdot)$ with parameters
$\theta_{v}$. At each observation $x_{i}$, the hidden states are updated by an
RNN as,

${\mathbf{h}_{i}}=v(\mathbf{h}_{i}^{{}^{\prime}},x_{i},\theta_{v}).$ (6)

The ODE-RNN approach proposed in [17] provides imputations but fails to
quantify the uncertainty associated with it. Therefore, we offer a novel SDE-
RNN framework, which, in addition to the predictions, is capable of
quantifying the uncertainties in a principled manner.

##  III Proposed SDE-RNN Approach

In this section, we describe the key elements of the proposed SDE-RNN
approach. Our approach involves the modification of ODE-RNN to include a
stochastic differential equation (SDE-RNN) to capture the evolution of the
hidden states (as opposed to ODE). Stochastic differential equation that
governs the dynamics of the hidden states corresponds to,

$d\mathbf{h}_{t}=f(\mathbf{h}_{t},t)dt+g(\mathbf{h}_{t},t)d\mathbf{B}_{t}$ (7)

where, $f$ and $g$ are the drift and diffusion functions, respectively. The
transformations $f$ and $g$ are carried through neural networks.
$\mathbf{B}_{t}$ represents the Brownian motion with the distribution
$d\mathbf{B}_{t}\sim\mathcal{N}(0,\mathbf{Q}\Delta t)$ where $\mathbf{Q}$ is
the diffusion matrix. $g(\mathbf{h}_{t},t)$ denotes the variance of the
Brownian motion and represents the epistemic uncertainty in the hidden state
dynamics. Both the drift and diffusion functions are nonlinear, thanks to the
nonlinear transformation of neural networks $f$ and $g$. Therefore, the SDEs
in (7) is a nonlinear SDE. The hidden states in the SDE-RNN model are the
solution of the SDE in (7), i.e.,

$\mathbf{h}_{i}^{{}^{\prime}}=SDESolve(\mathbf{h}_{i-1},(t_{i-1},t_{i}),f,g)$
(8)

Here, SDESolve represents the SDE Solver. It is important to note that states
are updated based on observations $x_{i}$ at time $t_{i}$ using the RNN
model.Using this SDE-RNN approach, we can quantify the uncertainty in the
hidden and output states, as discussed in the following subsection.

Figure 1: Propagation of uncertainty. $x_{i}$: Input at irregular intervals

###  III-A Uncertainty quantification

We propose a novel uncertainty propagation approach in the SDE-RNN model. The
proposed method quantifies the uncertainty in a holistic manner accounting for
both aleatoric and epistemic, without relying on prior specification of model
parameters and complicated Bayesian inference. The SDE-RNN approach propagates
the uncertainty from the noisy input observations to the final outputs using
the CVRNN model and the SDE model. Here, we refer to the CVRNN model as a
modified form of RNN, that propagates the mean and uncertainty arising from
the previous hidden states and the inputs to the next hidden state. Fig. 1 and
Algorithm 1 illustrate the overall uncertainty propagation in the SDE-RNN
framework. We can initialize the mean and covariance of the hidden state as
zero and then update them continuously. The CVRNN model updates the hidden
states at the time when an observation is available along with its associated
uncertainty. This approach accounts for the uncertainty in both the input
observation and the previous hidden states. SDE model captures the epistemic
uncertainty of the SDE-RNN approach. In the following subsections, we
elaborate the uncertainty propagation in both the CVRNN model and SDE model.

####  III-A1 CVRNN Model

The RNN cell in Eq. (6) updates the hidden states at time instants where
measurements are available. The dynamics of a general RNN cell are modeled by,

${\mathbf{h}_{i}}=v(\mathbf{h}_{i-1},x_{i},\theta_{v}),$ (9)
$o_{i}=c(\mathbf{h}_{i},\theta_{c}).$ (10)

Assume the input $x_{i}$ is corrupted with additive Gaussian noise given as

$\tilde{x}_{i}={x}_{i}+w_{i},$ (11)

where $w_{i}$ = $\mathcal{N}(0,\Sigma_{i}$) with $\Sigma_{i}$ representing the
covariance matrix of the measurement. These noisy observations are fed to the
RNN model, where the hidden states and the outputs become a random variable
corresponding to,

${\tilde{\mathbf{h}}_{i}}=v(\tilde{\mathbf{h}}_{i-1},\tilde{x}_{i},\theta_{v}),$
(12) $\tilde{o}_{i}=c(\tilde{\mathbf{h}}_{i},\theta_{c}).$ (13)

Let $\hat{\mathbf{h}}_{i}=\operatorname{\mathbb{E}}[\tilde{\mathbf{h}_{i}}]$
and $\hat{P_{i}}$ denotes the estimate of the covariance matrix
$P_{i}=\operatorname{\mathbb{E}}\\{(\tilde{\mathbf{h}_{i}}-\hat{\mathbf{h}_{i}})(\tilde{\mathbf{h}_{i}}-\hat{\mathbf{h}_{i}})^{\intercal}\\}$.
Based on (12), the linearization around $\hat{\mathbf{h}}_{i-1}$ implies that,

$\begin{split}{\tilde{\mathbf{\mathbf{h}}}_{i}}&=v(\tilde{\mathbf{h}}_{i-1},\tilde{x}_{i})\\\
&=v(\hat{\mathbf{h}}_{i-1}+\delta h_{i-1},\hat{x}_{i}+w_{i})\\\
&=v(\hat{\mathbf{h}}_{i-1},\hat{x}_{i})+\Delta_{h}v\delta\mathbf{h}_{i-1}+\Delta_{x}vw_{i}+o(\delta\mathbf{h}_{i-1}^{2},w_{i}^{2})\end{split}$
(14)

where, $\Delta_{h}v$ and $\Delta_{x}v$ are calculated at operating points
$(\hat{\mathbf{h}}_{i-1},x_{i})$. According to the transformation of
uncertainty [25] given in Theorem 1, we can evaluate the expected value and
covariance of $\tilde{\mathbf{h}}_{i}$.

###### Theorem 1

Transformation of uncertainty: Consider a recurrent neural network represented
by $v(\cdot)$ with neural network parameters $\theta_{v}$ and hidden states
given by
${\tilde{\mathbf{h}}_{i}}=v(\tilde{\mathbf{h}}_{i-1},\tilde{x}_{i},\theta_{v})$.
The input $\tilde{x}_{i}$ is corrupted with noise $w_{i}$ =
$\mathcal{N}(0,\Sigma_{i})$. The estimation of the expected value and
covariance matrix of the hidden state ${\tilde{\mathbf{h}}_{i}}$ at time $i$
can be recursively calculated as,

$\hat{\mathbf{h}}_{i}=v(\hat{\mathbf{h}}_{i-1},x_{i})$ (15)
$\hat{\mathbf{P}}_{i}=(\Delta_{h}v)\hat{\mathbf{P}}_{i-1}(\Delta_{h}v)^{\intercal}+(\Delta_{x}v)\Sigma_{i}(\Delta_{x}v)^{\intercal}$
(16)

where, the expected value of the previous hidden state
$\operatorname{\mathbb{E}}[\tilde{\mathbf{h}}_{i-1}]=\hat{\mathbf{h}}_{i-1}$
and its associated covariance matrix is ${\hat{\mathbf{P}}_{i-1}}$ are given.

Thus, the mean and covariance can be computed in a CVRNN framework with noisy
input measurement data at time $i$ using (15) and (16). We discuss the SDE
model in the following subsection.

####  III-A2 SDE Model

A neural SDE model aims to capture the epistemic uncertainty with Brownian
motion and propagate it to the next time step. It also aims to capture the
uncertainties arising from the previously obtained hidden states and their co-
variances. A neural SDE can be expressed using the dynamical equation in Eq.
(7). As this SDE is nonlinear, its statistics can be computed by adopting
certain approximations. Linearized approximation of SDEs [26] is a technique
which computes the statistics by linearizing the drift and diffusion function
around a certain point. Using the Taylor series, the drift function
$f(\mathbf{h},t)$ around the mean $\mathbf{m}$ is linearized as,

$f(\mathbf{h},t)\approx
f(\mathbf{m},t)+\mathbf{F}_{h}(\mathbf{h},t)(\mathbf{h}-\mathbf{m})$ (17)

The diffusion function is linearized as,

$g(\mathbf{h},t)\approx
g(\mathbf{m},t)+\mathbf{G}_{h}(\mathbf{h},t)(\mathbf{h}-\mathbf{m})$ (18)

A linearized approximation to estimate the mean and covariance of an SDE (7)
can be obtained by integrating the following differential equations from the
initial conditions $\mathbf{m}(t_{0})=E[\mathbf{h}(t_{0})]$ and
$\mathbf{P}(t_{0})$ = Cov$[\mathbf{h}(t_{0})]$ to the target time t:

$\frac{d\mathbf{m}}{dt}=f(\mathbf{m},t),$ (19)
$\frac{d\mathbf{P}}{dt}=\mathbf{P}\mathbf{F}_{h}^{\intercal}(\mathbf{m},t)+\mathbf{F}_{h}(\mathbf{m},t)\mathbf{P}+\mathbf{G}_{h}(\mathbf{m},t)\mathbf{Q}\mathbf{G}_{h}^{\intercal}(\mathbf{m},t)$
(20)

Thus, (19) and (20) captures the evolution of both the mean and covariance of
the hidden states. After the hidden states are updated, they are transformed
by another neural network to get final predictions (see Fig.1). The estimation
of the mean and the covariance of the output is obtained according to the
transformation of uncertainty (Theorem 1) as,

$\hat{o}_{i}=c(\hat{\mathbf{h}}_{i})$ (21)
$\hat{R}_{i}=(\Delta_{h}c)\hat{\mathbf{P}}_{i}(\Delta_{h}c)^{\intercal}$ (22)

Algorithm 1 illustrates the working mechanism of the proposed SDE-RNN
approach. As discussed earlier, the hidden states $\mathbf{m}_{i}$ are updated
at times $i$ where measurements are present. The availability of the
measurements is indicated by the vector mask $\in\\{0,1\\}$. The imputed
measurements along-with the associated variances are indicated by
$\\{o_{i}\\}_{i=0}^{N}$ and $\\{R_{i}\\}_{i=0}^{N}$ respectively.

Input: Datapoints $\\{x_{i}\\}_{i=1}^{N}$ and the corresponding times
$\\{t_{i}\\}_{i=1}^{N}$, drift function $f$, diffusion function $g$

Initialization: $\mathbf{h}_{0}$, $\mathbf{P}_{0}$

1: for $i=1,..,N$ do

2:
${\mathbf{h}_{i}^{{}^{\prime}},\mathbf{P}_{i}^{{}^{\prime}}}=SDESolve(\mathbf{h}_{i-1},\mathbf{P}_{i-1},(t_{i-1},t_{i}),f,g)$

3:
${\mathbf{h}_{i},\mathbf{P}_{i}}=CVRNN(\mathbf{h}_{i}^{{}^{\prime}},\mathbf{P}_{i}^{{}^{\prime}},x_{i})$

4: $\mathbf{h}_{i}$ = mask $\times$ $\mathbf{h}_{i}$ \+ (1-mask) $\times$
$\mathbf{h}_{i}^{{}^{\prime}}$

5: $\mathbf{P}_{i}$ = mask $\times$ $\mathbf{P}_{i}$ \+ (1-mask) $\times$
$\mathbf{P}_{i}^{{}^{\prime}}$

6: $\\{o_{i},R_{i}\\}_{i=1}^{N}=c(\mathbf{h}_{i},\mathbf{P}_{i})$

7: end for

8: return $\\{o_{i}\\}_{i=0}^{N}$, $\\{R_{i}\\}_{i=0}^{N}$

Algorithm 1 SDE-RNN Approach

###  III-B Uncertainty propagation for GRU

The CVRNN model discussed in the previous section can be any form of the
recurrent neural network; for example, it could be either RNN, Long Short Term
Memory (LSTM), or Gated Recurrent Unit (GRU) model whose hidden states are
given as,

${h_{t}}=v(h_{t-1},x_{t},\theta_{v}),$ (23)

In this section, we will derive the uncertainty propagation for a GRU model by
using the transformation of uncertainty as discussed in Theorem 1. To begin
with, we need to compute the $\Delta_{h}v$ and $\Delta_{x}v$. In a GRU model,
the hidden states are given as,

$h_{t}=z_{t}\circ h_{t-1}+(1-z_{t})\circ h_{t}^{{}^{\prime}}$ (24)
$h_{t}^{{}^{\prime}}=\text{tanh}(W_{in}x_{t}+b_{in}+r_{t}(W_{hn}h_{t-1}+b_{hn}))$
(25) $z_{t}=\sigma(W_{iz}x_{t}+b_{iz}+W_{hz}h_{t-1}+b_{hz})$ (26)
$r_{t}=\sigma(W_{ir}x_{t}+b_{ir}+W_{hr}h_{t-1}+b_{hr})$ (27)

Here, $x_{t}$, $z_{t}$ and $r_{t}$ are the inputs, update and reset gates of
the GRU model, respectively. tanh and $\sigma$ are the Tanh and Sigmoid
activation functions, respectively. $\circ$ represents the Hadamard product.

The gradient of the hidden states (24) at time $t$ computed at operating
points $(\hat{h_{t-1}},x_{t})$ are given as,

$\Delta_{h}v=\frac{\partial z_{t}}{\partial
h_{t-1}}h_{t-1}+z_{t}\frac{\partial h_{t-1}}{\partial
h_{t-1}}+\frac{\partial(1-z_{t})}{\partial
h_{t-1}}h_{t}^{{}^{\prime}}+\frac{\partial h_{t}^{{}^{\prime}}}{\partial
h_{t-1}}(1-z_{t})$ (28)

The derivative of the sigmoid function is denoted as
$\sigma^{{}^{\prime}}(x)=\sigma(x)(1-\sigma(x))$. Each gradient in the (28) is
given as,

$\frac{\partial z_{t}}{\partial
h_{t-1}}=\sigma^{{}^{\prime}}(W_{iz}x_{t}+b_{iz}+W_{hz}h_{t-1}+b_{hz})\circ
W_{hz}$ (29) $\frac{\partial r_{t}}{\partial
h_{t-1}}=diag(r_{t}(1-r_{t}))\circ W_{hr}$ (30)

$\frac{\partial h_{t}^{{}^{\prime}}}{\partial
h_{t-1}}=diag(1-h_{t}^{{}^{\prime}2})\circ\\{\frac{\partial r_{t}}{\partial
h_{t-1}}(W_{hn}h_{t-1}+b_{hn})+W_{hn}r_{t}\\}$ (31)

Similarly, $\Delta_{x}v$ is given as,

$\Delta_{x}v=\frac{\partial z_{t}}{\partial
x_{t}}h_{t-1}+\frac{\partial(1-z_{t})}{\partial
x_{t}}h_{t}^{{}^{\prime}}+\frac{\partial h_{t}^{{}^{\prime}}}{\partial
x_{t}}(1-z_{t})$ (32)

Each gradient in (32) is given as,

$\frac{\partial z_{t}}{\partial x_{t}}=diag(z_{t}(1-z_{t}))\circ W_{iz}$ (33)
$\frac{\partial r_{t}}{\partial x_{t}}=diag(r_{t}(1-r_{t}))\circ W_{ir}$ (34)
$\frac{\partial h_{t}^{{}^{\prime}}}{\partial
x_{t}}=diag((1-h_{t}^{{}^{\prime}2})\circ(W_{in}+\frac{\partial
r_{t}}{\partial x_{t}}(W_{hn}h_{t-1}+b_{hn}))$ (35)

Using Eq. (28) and (32), the covariance matrix of the hidden states $P_{t}$ at
time $t$ is given by (16).

##  IV Experimental results

We validated the proposed SDE-RNN approach for imputations on irregularly
sampled measurements from the power distribution network (IEEE 37 bus test
system). It is important to note that the proposed framework can be applied to
any irregularly sampled time series data.

###  IV-A Data pre-processing

The irregularly sampled measurements are considered from two sensors, namely
the smart meters and SCADA sensors. The smart meter measurements consist of
active and reactive power injection for 24-hrs duration. This 24-hr load
profile consists of a mixture of load profiles, i.e., industrial/commercial
load profiles [27], and residential loads [4]. Reactive power profiles are
obtained by assuming a power factor of $0.9$ lagging. The SCADA measurements
are the voltage magnitude measurements obtained by executing load flows on the
test network. The SCADA measurements are assumed at a subset of node
locations. The aggregated smart meter data are averaged over 15-minute
intervals while the SCADA measurements are sampled at 1-min interval. Gaussian
noise with $0$ mean and standard deviation equal to $10\%$ of the actual power
values is added to the smart meter data to mimic real-world measurement noise.
The smart meter and SCADA measurements constitute our training dataset.

The sensors data is represented as a list of records, where each record
represents the information about the time-series data with the format given
as, record = [measurement type, values, times, mask]. Here, time-series data
at each node of the IEEE 37 bus network represents one record. The measurement
type denotes the sensor type, i.e., $P,Q,$ or $V$. Values
$\in\mathbb{R}^{N\times 1}$ represent the sensor measurements with times
$\in\mathbb{R}^{N}$ as the corresponding time instants. Mask
$\in\mathbb{R}^{N\times 1}$ represents the availability of the corresponding
measurements. The dataset is further normalized between [0,1] intervals and
takes the union of all time points across different nodes in the dataset that
are irregularly sampled.

###  IV-B Model Specifications

We use the GRU cell with hidden size 5 to encode the observations. The drift
function of the SDE is a feedforward neural network with 1 layer and 100
units. The diffusion function is a feedforward neural network with 1 layer,
100 units, and a sigmoid activation function. We consider Ito SDE with
diagonal noise of Brownian motion. Python is used for coding the entire
framework with the support of PyTorch’s torchsde package [11]. We consider the
output neural network as a 1-layer feedforward network with an input size of 5
and an output size of 1. We set the learning rate to 0.01, batch size as 10
and report the loss as mean squared error (MSE). We compare the efficacy of
the proposed SDE-RNN with the classic GRU approach. The classic GRU approach
consists of a GRU model with 3 input features (observations, times, and mask)
and 5 output features. The predictions are obtained by transforming the output
of the GRU cell via another feedforward neural network with 2 layers. The
first layer is a neural network with output features 100 followed by the Tanh
activation function. We then employ a dropout rate of 0.3 after this
activation function followed by another feedforward neural network of input
features 100 and output feature 1.

The uncertainty estimates provided by the SDE-RNN is compared with that of
Monte-Carlo dropout approach in classic RNN using the Expected normalized
calibration error (ENCE) metric [28]. Calibration error is typically used to
evaluate uncertainty estimates since they are not associated with any
groundtruth values. In a well calibrated model, average confidence score
should match with the average model accuracy [29]. Thus, in a classification
setting, calibration implies that whenever a forecaster assigns a probability
of $p$ to an event, that event should occur about $p\%$ of the time. On the
other hand, in a regression setting, calibration signifies that the prediction
$y_{t}$ should fall in a $c\%$ (e.g., $90\%$) confidence interval
approximately $c\%$ (e.g., $90\%$) of the time. The expected normalized
calibration error (ENCE) is based on the similar regression setting that for
each value of uncertainty, measured through the standard deviation $\sigma$,
the expected mistake (measured in MSE) matches the predicted error
$\sigma^{2}$, i.e.,

$E_{x,y}[(\mu(x)-y)^{2}|\sigma(x)^{2}=\sigma^{2}]=\sigma^{2}$ (36)

ENCE metric is evaluated using the binning approach, where we assume that the
number of bins $N$ divides by the number of time points $T$. We divide the
indices of the examples to $N$ bins, $\\{B_{j}\\}_{j=1}^{N}$, such that
$B_{j}=\\{(j-1)\frac{T}{N}+1,...,j\frac{T}{N}\\}$. Each bin represents an
interval in the standard deviation axis: $[min_{t\in
B_{j}}\\{\sigma_{t}\\},max_{t\in B_{j}}\\{\sigma_{t}\\}]$. Expected Normalized
Calibration Error ($ENCE$) is calculated as,

$ENCE=\sqrt{\frac{1}{N}\sum_{j=1}^{N}\frac{|mVAR(j)-RMSE(j)|}{mVAr(j)}}$ (37)

where, the root of the mean variance ($mVAr(j)$) at bin $j$ is,

$mVAr(j)=\sqrt{\frac{1}{|B_{j}|}\sum_{t\in B_{j}}\sigma_{t}^{2}},$ (38)

and root mean squared error ($RMSE(j)$) at bin $j$ is,

$RMSE(j)=\sqrt{\frac{1}{|B_{j}|}\sum_{t\in B_{j}}(y_{t}-\hat{y}_{t})^{2}}.$
(39)

It is expected that the $mVar$ will equal the $RMSE$ for each bin, i.e., the
plot of $RMSE$ as a function of $mVar$ should be an identity function.
However, in reality, the models are not well calibrated and the plot is not an
identity function. Thus, a model is said to provide better uncertainty
estimates if they have a lower ENCE. We compare the uncertainty estimation of
our SDE-RNN approach with the classic RNN in the following subsection.

Figure 2: Uncertainty quantification in SDE-RNN approach- node 1 Figure 3:
Uncertainty quantification in SDE-RNN approach- node 12

###  IV-C Imputation:

In the first experiment, the smart meter measurements are imputed at a 1-min
interval using the proposed SDE-RNN approach. The training and test dataset
contains the available AMI (15-minute) and SCADA (1-minute interval)
measurements. In addition, in the test dataset, we introduce missing data as a
percentage of the total number of time instants (minutes) in the 24-hr
interval (i.e., 1440 time instants). The prediction and uncertainty estimates
provided by the SDE-RNN model are shown in Fig.2 and Fig. 3 for nodes 1 and
12, respectively. Here, $5\%$ of total interpolated points (i.e., 1440 data
points) are observed. It can be observed that the epistemic uncertainty
increases as we move away from the observed values. For instance in Fig. 2,
the sparse region of $10^{th}$-$12^{th}$ hr depicts less confidence compared
to data-rich regions of $5^{th}$-$9^{th}$ hr. The aleatoric uncertainty due to
the sensor noise is well captured by the SDE-RNN model, reflected by the
uncertainty estimates at the observations. The uncertainty estimates at the
observation times is nonzero and present due to the noisy input data, which is
the aleatoric uncertainty part.

In the second experiment, we compare the uncertainty estimates provided by the
SDE-RNN with the classic GRU + MC dropouts approach. We assume the number of
bins is 5. Table I shows the performance of both these approaches for
different percentages of missing data in the test dataset. We provide the test
dataset’s mean squared error (MSE) and ENCE. It is evident from the Table that
the SDE-RNN approach offers better accuracy and uncertainty estimates than the
classic GRU model for all the levels of missingness in the test dataset.

TABLE I: Imputation results for different percentages of missing data in test dataset Metric | MSE | ENCE
---|---|---
| Missing
---
data (%)
40% | 60% | 80% | 40% | 60% | 80%
SDE-RNN | 0.0005 | 0.0008 | 0.0013 | 57.45 | 49.11 | 54.35
Classic-GRU | 0.0506 | 0.0978 | 0.1527 | 534.05 | 419.06 | 457.92

##  V Conclusion and Future work

This paper proposes a neural SDE-RNN framework for imputing the multi-
timescale measurements and quantifying the uncertainty associated with them.
We capture both the aleatoric and epistemic uncertainty and propagate it
across different time instances through neural SDE-RNN modules. Simulation
results (ENCE, MSE) on power distribution (IEEE 37 bus) test system validate
the effectiveness of our approach. As a part of our future work, We aim to
compare the epistemic uncertainty obtained through our model with that of
conventional approaches such as Monte-Carlo dropouts, Deep Ensemble, etc.

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# NiSi: New venue for antiferromagnetic spintronics

P. Ghosh1 J. Guo1,† F. Ye2 T. Heitmann3 S. Kelley4 A. Ernst5 V. Dugaev6 D. K.
Singh1,7,∗ 1Department of Physics and Astronomy, University of Missouri,
Columbia, MO, USA  2Oak Ridge National Laboratory, Oak Ridge, TN, USA
3University of Missouri Research Reactor, Columbia, MO, USA  4Department of
Chemistry, University of Missouri; Columbia, MO, USA  5Institut for
Theoretical Physics, Johannes Kepler University, Linz, Austria  6Department of
Physics and Medical Engineering, Rzeszów University of Technology, Rzeszów,
Poland  7MU Materials Science and Engineering Institute, Columbia, MO, USA
∗email<EMAIL_ADDRESS>
###### Abstract

Envisaging antiferromagnetic spintronics pivots on two key criteria of high
transition temperature and tuning of underlying magnetic order using
straightforward application of magnetic field or electric current. Here, we
show that NiSi metal can provide suitable new platform in this quest. First,
our study unveils high temperature antiferromagnetism in single crystal NiSi
with $T_{N}\geq 700$ K. Antiferromagnetic order in NiSi is accompanied by the
non-centrosymmetric magnetic character with small ferromagnetic component in
a-c plane. Second, we find that NiSi manifests distinct magnetic and
electronic hysteresis responses to field applications due to the disparity in
two moment directions. While magnetic hysteresis is characterized by one-step
switching between ferromagnetic states of uncompensated moment, electronic
behavior is ascribed to metamagnetic switching phenomena between non-collinear
spin configurations. Importantly, the switching behaviors persist to high
temperature. The properties underscore the importance of NiSi in the pursuit
of antiferromagnetic spintronics.

Antiferromagnetism, originally discovered by Louis Neel more than five decades
ago, was explained on the basis of quantum mechanical exchange coupling
between ions with opposite spin polarity.Neel ; Nagamiya  The unconventional
mechanism, at that time, elucidated the origin behind the absence of net
magnetization in antiferromagnetic materials and introduced the concept of
sub-lattice magnetizations. While the zero magnetization in an antiferromagnet
minimizes the stray field between magnetic sites, thus maximizing the packing
density for magnetic random access memory application, strong exchange
coupling (compared to ferromagnetic system) enhances the characteristic
frequency of spin dynamics that is faster and stable against the external
perturbation.Zelezny  These properties are at the core of new push behind
spintronics research.Siddiqui ; Baltz ; Jungwirth ; Yan  Antiferromagnetic
(AFM) metal provides facile platform in this endeavor.Pickett ; Wadley ;
Fukami ; Wu ; Gomonav  An AFM metal tends to exhibit high electrical and
thermal conductivities.Siddiqui  However, unlike ferromagnetic compounds where
the anisotropic magnetoresistance (AMR) is commonly used for reading and
writing information in the form of two magnetic polarities, separated by an
energy barrier, zero net magnetization and stronger spin dynamics in
antiferromagnetic compounds spur weak magnetic field dependence. Thus, the
reading of information using AMR is difficult in an antiferromagnetic metal,
even though the AMR of relativistic origin has been demonstrated in some
cases.Shick

More recently, novel approaches of current-induced staggered relativistic
field and the spin-orbit torque (among others) have provided new mechanisms to
alter the magnetic order parameter in antiferromagnetic metals, such as
CuMnAs.Zelezny ; Wadley ; Siddiqui  The uncoupled applications of magnetic
field at high temperature followed by electric current along certain
crystallographic directions at relatively lower temperature was demonstrated
to manipulate the magnetic state in AFM metal FeRh.Marti  Similarly, the
concept of exchange coupling as a field-tunable parameter in the artificially
engineered composite system of thin ferromagnetic and antiferromagnetic layers
is actively pursued to control the magnetization direction in
antiferromagnetic component.Wang ; Tang ; Duine

Figure 1: Lattice structure of NiSi and spin correlation between Ni atoms.
(a-b) Orthorhombic crystal structure of NiSi. Numerically simulated
diffraction profile for the NiSi lattice. (c-d) AFM structure of Ni-ions and
the simulated scattering profile, exhibiting many distinct peaks arising
solely due to spin correlation between Ni ions, such as (001) and (110). (e-f)
Theoretical calculations within the self-interaction correction model exhibit
large Ni density of states near the Fermi surface (Fig. e) and maxima in total
magnetic energy J(q) between high symmetry points of R and U (Fig. f),
suggesting AFM propagation vector.

Unlike the fully compensated AFM metal where the manipulation of magnetic
state requires the application of a complex process involving electric current
and magnetic field or multiple current sources, a non-compensated AFM metal
with small ferromagnetic component provides a rather accessible platform for
the reading of information using only magnetic field application. As shown in
the case of Mn3Si, the anomalous Hall conductivity and magnetoresistance were
used to deduce information about the field-induced change in magnetic
configuration.Nakatsuji  Metallic AFM compounds, arguably, exhibit large
anomalous Hall effect due to the conjugated operations of time-reversal and
space inversion symmetries.Vries ; Zhang ; Chen ; Nakatsuji  However, there
are two benchmark criteria that an AFM metal must satisfy for spintronic
applications: it must have high Neel temperature, $T_{N}$ $>$ 500 K, for the
resulting device to be practically usable, and also manifest easy assimilation
with the complex integrated circuit.Samarth

Nickel monosilicide (NiSi) fits both criteria. NiSi is a technologically
important electronic material. It is used for making electrical contacts in
field-effect transistors and nanoelectronic devices.Robinson ; Lavoe ; Weber
In contrast to other bi-elemental transition metal systems that are known to
manifest rich magnetic and topological phases,Chubukov ; Nakajima ; Huber
magnetic order was never reported in NiSi or any Ni-Si compound.
Straightforward density functional calculation under the local density
approximation (LDA) is also one of the reasons behind the lack of magnetic
information in this compound, as it does not yield a magnetic ground state in
NiSi.Dahal  Additionally, previous experimental investigations were focused on
powder samples that often averages out finer details. Synthesis of single
crystal specimen of NiSi of high chemical purity solves this conundrum. For
the first time, our research shows that nickel monosilicide (NiSi) not only
possesses uncompensated AFM ground state with a high Neel temperature of
$T_{N}\geq 700$ K but also exhibits remarkable magnetic switching and
electronic properties that can be utilized for the spintronics device designs.

Figure 2: Elastic neutron measurements revealing magnetism in NiSi. (a-b)
Representative scans, exhibiting temperature dependence, across (001) and
(002) peaks. Experimental data are well-described by resolution limited
Gaussian line-shape, suggesting long range magnetic order in NiSi. (c) Plot of
integrated intensity as a function of temperature, showing the onset of
magnetism at $T\geq 700$ K. Inset shows that the integrated intensity also
increases, albeit modestly, at nuclear peak. (d) Detailed mesh scan in [h0l]
crystallographic plane at $T=300$ K. Peaks surrounded by black circles arise
due to a small crystallite in the sample, which is not aligned with the main
crystal specimen; thus not affecting experimental results. Powder rings are
due to aluminum sample can. (e) Numerically simulated scattering profile
showing both lattice and magnetic contributions. Error bar represents one
standard deviation in all data.

NiSi crystallizes in the zinc blend-type orthorhombic structure with lattice
parameters of $a=5.178$ Å, $b=3.331$ Å, $c=5.1624$ Å. Ni ions occupy the
centrosymmetric positions,Connetable  see Fig. 1a. To understand the origin
and nature of magnetism in NiSi, we have performed synergistic theoretical and
experimental researches. Unlike the LDA approximation mechanism, theoretical
calculations using the self-interaction (SI) correction method reveal an AFM
ground state in NiSi.Luders ; Liechtenstein ; Dane  We have performed first
principle study using a self-consistent full-potential Green’s function
method, implemented within the multiple scattering theory. Ni 3d orbitals are
almost fully occupied and not spin-polarized. In our theoretical approach, Ni
3d atoms were self-interaction corrected by varying all possible
configurations and minimizing the total energy as suggested for monoxides. We
find that the Ni 3d ions are occupied in accordance with the Hund’s rule,
similar to NiO. Self-interaction corrected Ni 3d states are located mainly at
8-10 eV below the Fermi level. However, a substantial part of the density of
states is situated at the Fermi level due to the hybridization with the Si sp-
states, see Fig. 1e (also see Fig. S1 in Supporting Information).
Nevertheless, a large exchange splitting and the strong localization of the Ni
3d states favor an antiferromagnetic order of Ni magnetic moments. This is
confirmed by our total energy calculations. The total energy of the
antiferromagnetic stacking per Ni moment is about 40 meV lower than the
ferromagnetic configuration.

To theoretically refine magnetic structure of NiSi, we calculate the exchange
coupling constants using the magnetic force theorem within the ambit of
multiple scattering theorem. Fig. 1f shows the Fourier transformed exchange
constants $J(q)$ as a function of the wave vector q. The maximum $J(q)$
corresponds to the minimum in total energy and the corresponding wave vector,
associated to magnetic ordering, suggests an AFM order in NiSi. According to
our calculations, maximum $J(q)$ is found to be located between the high
symmetry points of $\textbf{{U}}=(1/2,0,1/2)$ and $\textbf{{R}}=(1/2,1/2,1/2)$
in the Brillouin zone. It suggests antiferromagnetic order of Ni spins with
moment size of 1.12 $\mu_{B}$, canted in the direction of q. Detailed elastic
neutron scattering measurements on single crystal NiSi indeed reveal AFM
correlation between Ni-ions with moments symmetrically shifted across the
(111) diagonal plane, see Fig. 1c, as discussed in the following paragraphs.

Study of single crystal sample provides crystallographic dependence of the
underlying magnetism in a material. We synthesized single crystal specimens of
NiSi using the self-flux method via slow cooling. Samples were characterized
using powder and Laue X-ray diffraction methods (see Experimental Section). To
gain insight into possible magnetism in NiSi, we have performed detailed
elastic neutron scattering measurements on a 0.14 g single crystal sample on
CORELLI and TRIAX spectrometers at SNS-ORNL and MURR, respectively (see
Experimental Section for detail about neutron scattering measurements).
Experimental data reveal temperature dependent diffraction peaks as the sample
is cooled from $T\sim$ 700 K. We show the representative scans at two
q-vectors, $q_{1}$ = (001) rlu and $q_{2}$ = (002) rlu, in Fig. 2a-b. $q_{1}$
and $q_{2}$ wave-vectors represent distinct magnetic and lattice positions, as
shown in Fig. 1b and 1d. The integrated intensities as a function of
temperature are plotted in Fig. 2c. While the intensity increases
significantly as temperature decreases at $q_{1}$ vector, indicating the onset
of magnetism at T $\geq$ 700 K, a modest increment is also detected in peak
intensity at $q_{2}$ wave vector. The transition temperature can be possibly
higher as our measurement is limited to $T=700$ K.

To deduce the microscopic nature of Ni spin correlation, a detailed mesh scan,
obtained at T = 300 K, is shown in Fig. 2d (see Fig. S2 for mesh scans in
(0kl) and (hk0) scattering planes). Magnetic peak intensity does not change
beyond 2$\sigma$ below T = 300 K in Fig. 2c. So, we chose this temperature for
spin structure refinement. In addition to distinct magnetic and lattice peaks,
some peaks (such as (h0h) wave vectors) have both lattice and magnetic
contributions. However, magnetic contribution outweighs the lattice at smaller
q value. The spin structure is determined by performing magnetic symmetry
analysis using FullProf Suite.Rodriguez  The best fit to experimental data is
obtained for the magnetic moment arrangement shown in fig. 1c where oppositely
polarized Ni spins, forming AFM configuration, cant out of (111) diagonal
plane (see the section on ’Magnetic Structure Refinement’ in Supporting
Information for more detail). NiSi is not known to exhibit strong uniaxial
anisotropy. Therefore, such a non-collinear spin configuration is not
surprising. Also, the strength of Ni moment varies on two sites: $m_{2}$ =
1.58(05) $\mu_{B}$ and $m_{2}$ = 1.19(04) $\mu_{B}$. It suggests the non-
centrosymmetric character of magnetic unit cell. In addition to the disparity
in ordered moments with non-conforming canting on two Ni sites, analysis of
change in nuclear peak intensity at T = 300 K (Fig. 2c) reveals a non-
compensated moment (ferromagnetic (FM) component) per unit cell of magnitude
$\sim 0.08(02)\mu_{B}$ in a-c plane (nominally), given by the polar angles of
$\theta=38.4^{\circ}$ and $\phi=15.4^{\circ}$. The uncompensated FM moment
accompanies the AFM correlation of Ni ions. The numerically simulated pattern
of combined magnetic and lattice structure factors, shown in Fig. 2e, depicts
excellent agreement with experimental data.

The magnetic correlation of Ni spins can be useful for spintronic
applications, as inferred previously in AFM metals.Siddiqui  The disparity in
antiferromagnetic and the uncompensated ferromagnetic moment directions can
exhibit different responses to magnetic field applications. To explore the
practical implication of this important property of NiSi, we have performed
magnetic and Hall probe measurements for different magnetic field application
directions. For the tuning of FM component, nominally in the a-c plane,
magnetic field is applied along the b-axis. In Fig. 3a, we show the
q-independent magnetic measurements at two characteristic temperatures of T =
350 K and T = 20 K. Magnetic field application perpendicular to the
uncompensated moment in a-c plane will induce maximum torque ($\tau$ = M
$\times$ H), thus can force it to align to field direction as described
schematically in Fig. 3b. This is precisely what we see in magnetic
measurements where highly symmetric hysteresis loops, accompanied by the one-
step switching process, is detected. It is characterized by the sharp
transition between two-magnetic states at a switching field of
$\mu_{0}|$H$|\sim$ 0.08 T. Measurement at low temperature does not affect the
switching field value. The absence of any field-induced enhancement in
magnetization at higher field hints of a magnetization switching process
between the two locally minimum energy FM configurations. AFM material with
high Neel temperature tends to exhibit paramagnetic behavior at higher
field,Baranov  which is not the case here. Rather, the bulk magnetization
saturates to a modest value of 0.67 $\times 10^{-7}$ Am2 at T = 350 K, which
is comparable to the uncompensated FM moment for the measured sample. The one-
step switching transition at moderate field sets the foundation for the
development of magnetic random access memory spintronic devices. Perhaps, the
switching field can be further reduced in the thin film specimen of NiSi where
the magnitude of overall magnetization due to the uncompensated FM moment is
expected to be even smaller.

Figure 3: Magnetic switching in NiSi. (a) Bulk magnetometry study reveals a
sharp transition between two magnetic states at $H\sim 0.08$ T. The switching
occurs due to the field alignment of uncompensated (FM) moment to field
direction, depicted schematically in (b). Figure 4: Magneto-electronic
hysteresis revealed in Hall probe measurements on NiSi. (a-c) Hall data for
field application along c-direction, perpendicular to the a-b plane of the
sample. Several magneto-electronic hysteresis loops, characterized by sharp
jump in Rxy, centered at non-zero field values develop at high temperature. As
temperature reduces, hysteresis loops converge and eventually disappear. (d-f)
Similar behavior, albeit weakly, is observed in the longitudinal resistance
Rxx. (g-l) No magneto-electronic hysteresis is detected in inplane field
measurements. Insets in fig. f and fig. l show measurement schematics.

Unlike the magnetometry study, Hall probe measurements of NiSi unveil strong
magneto-electronic hysteresis curves centered around non-zero magnetic field
values; thus, hinting the role of antiferromagnetic spin correlation. Hall
measurements were performed for both perpendicular and inplane magnetic field
applications with respect to the a-b plane of NiSi. In Fig. 4a-f, we show the
plots of Hall resistance Rxy and longitudinal resistance Rxx as a function of
perpendicular magnetic field at representative temperatures (also see fig.
S5). The finite value of Rxy = 85 $\times 10^{-6}\>\Omega$ at H = 0 T suggests
AHE in NiSi, which can be ascribed to the uncompensated FM moment. Since the
magnetic moment per lattice unit cell is not zero, the time inversion symmetry
of the system is broken explicitly. It is a necessary condition for non-zero
AHE.Tatara ; Chang  Besides the uncompensated moment, the non-collinear
antiferromagnetic ordering of Ni atoms may also be contributing to AHE via the
accumulation of finite Berry curvature, as inferred previously in other AFM
materials.Chang ; Surgers ; Smejkal  In Addition to AHE, the Hall data reveal
several interesting trends: first, the system exhibits magneto-electronic
hysteresis in both Rxy and Rxx. We observe two sets of hysteresis at T = 250
K: one centered at low field of $\mu_{0}$H $\sim$ 1.2 T, characterized by the
gradual enhancement in Hall resistance as a function of the forward field
sweep from $\mu_{0}$H = $-$7 T to $+$7 T. When the field sweep direction is
reversed, then the enhancement in Rxy switches from positive to negative
field, thus developing a magneto-electronic hysteresis. Second set of
electronic hysteresis develops at higher field, marked by the sharp
metamagnetic-type transitions at ${\mu_{0}}|$H$|>$ 3 T. At T = 250 K, two sets
of sharp jumps are observed at higher field application. The occurrence of
magneto-electronic hysteresis centered at non-zero field values suggest the
AFM origin of the phenomena. An electronic hysteresis centered around
$\textit{H}=0$ T can be arising either due to FM or AFM magnetic order (as in
the case of Mn3Si), but the non-zero field center unequivocally suggests the
AFM origin.Taguchi

Second, the field-induced asymmetric jump in Rxy (Rxx) gradually diminishes as
temperature reduces. For example, the measurement at T = 100 K reveals the
occurrence of electronic hysteresis only at higher field. At much lower
temperature e.g. T = 20 K, the hysteresis behavior completely disappears.
Rather, linear magnetic field dependences are observed in both the Rxy and Rxx
data, indicating ordinary Hall effect. These observations suggest the
prominent role of thermal fluctuation in the electronic hysteresis
development. Unlike the electronic behavior in perpendicular field, no
magneto-electronic hysteresis is observed in Rxy or Rxx measurements in the
inplane field, see Fig. 4g-l. In the latter case, both current and field are
applied in the same direction that nullify the field effect on electric charge
carriers.

Figure 5: Schematic depiction of field-induced metamagnetic transition. (a)
Magnetic field application along c-axis induces torque on Ni moments, thus
forces it to align along the field-direction. It results in quasi-
ferromagnetic arrangement of Ni moments along c-direction, but remains
antiferromagnetic along the a-direction. (b) Qualitative plot, showing energy
of different magnetic configurations as a function of external field along the
c-axis.

In principle, the magneto-electronic hysteresis in perpendicular field can be
arising due to the cumulative effects of Berry curvature due to the non-
collinear spin configuration, spin re-orientation and AHE.Smejkal ; Taguchi
However, a straightforward energetic analysis for magnetic field application
along c-direction (perpendicular to a-b plane of the sample) can provide
qualitative insight into the phenomena. We can consider a model where a chain
of localized moments parallel to the a-axis in (111) diagonal plane undergoes
a transition from AFM to FM state, see Fig. 5a. We write the total energy of
the chain as,

$\textit{E}=\sum_{i}\big{(}J_{a}M^{a}_{i}M^{a}_{i+1}+J_{c}M^{c}_{i}M^{c}_{i+1}-g{\bf
M}_{i}\cdot{\bf B}\big{)}$ (1)

where Mi is the total moment of chain i and Ja, Jc are AFM exchange constants
along a- and c-directions, respectively (similar analysis can be performed for
AFM order of total chain moment along b-direction, while magnetic field is
applied along c-direction). Assuming that Ja $>$ Jc $>0$, we find that the
ground state of the system corresponds to AFM ordering of the chain moments
$\textit{M}_{i}$ along the a-direction. The energy of this state is denoted by
$E^{a}_{\uparrow\downarrow}$. Similarly, $E^{c}_{\uparrow\downarrow}$
corresponds to the AFM ordering of $\textit{M}_{i}$ along the c-direction. Due
to the AFM ordering of chain moments, the total energy
$E^{a}_{\uparrow\downarrow}$ in this configuration does not depend on magnetic
field. On the other hand, energies $E^{a}_{\uparrow\uparrow}$ and
$E^{c}_{\uparrow\uparrow}$, corresponding to the FM ordering of
$\textit{M}_{i}$ along a- and c-directions, respectively, exhibit linear
dependences on magnetic field. The qualitative plots of the energies of
different magnetic configurations as a function of magnetic field are shown in
Fig. 5b. We find that for magnetic field application along c-direction, a
transition to FM ordering of $\textit{M}_{i}$ along c-direction with minimum
energy occurs for $B_{c}>B_{cr}$. Interestingly, in-plane field application
does not lead to any reconstruction of magnetic moment, which is consistent
with experimental observation in Fig. 4g-l. Since the transition between
different energy states of the system requires overcoming a potential barrier
(see Figure S6-S7), this can lead to the hysteresis in magnetization dependent
properties, such as Hall resistance. However, at low temperature, the
probability of overcoming the potential barrier is smaller. Hence, the
hysteretic behavior disappears.

Finally, we discuss the experimental results. The synergistic study using
first principle calculations and detailed experimental investigations of
single crystal specimen has revealed non-compensated antiferromagnetic ground
state in NiSi. In a highly surprising observation, the antiferromagnetic
ground state is onset at high temperature of $T\geq$ 700 K. There are very few
AFM metals with $T_{N}\geq$ 500 K, in particular with non-compensated
magnetization. Magnetic character of NiSi is also confirmed from bulk
magnetization and the Hall probe measurements. There are both fundamental and
practical implications of the finding. First of all, our study strongly
complements the existing as well as the ongoing efforts on the elucidation of
novel magnetic and topological ground states in bi-elemental transition metal
intermetallic compounds, such as MnSi and FeSi. Second, the system manifests
significant anomalous Hall conductivity at high temperature. It further
validates the argument that an antiferromagnetic metal can also exhibit
anomalous Hall effect.Chen  Although, the magnetic structure in NiSi has small
FM component in the form of uncompensated moment, but the anomalous Hall
conductivity is quite large. Therefore, it may not be entirely arising due to
the latter contribution. Third, the magneto-electronic and magnetic properties
at high temperature in NiSi can have implication for spintronics research. It
exhibits a one-step switching transition, owed to the response of non-
compensated moment to moderate magnetic field. The phenomenon is in stark
resemblance with ferromagnetic materials of spintronic importance.

Even though the uncompensated moment accompanies the AFM correlation of Ni
ions, it behaves independently in perpendicular field application. This is
because the AFM and uncompensated FM moments point in different directions.
Thus, they respond differently to magnetic field applications. The switching
property, which persists well above the room temperature and thus fulfills one
of the key criteria for practical applications, can be employed for the MRAM
device design. Similarly, the magneto-electronic hysteresis, attributed to
sharp jumps in magnetoresistance at moderate field values, can be further
explored for application in magnetic logic devices, such as memristor devices
that exploits the field-induced history of magnetoresistance. Marti  Although,
the phenomenon is qualitatively explained in terms of metamagnetic transition
between the field-induced arrangement of Ni moments, further theoretical
understanding of the origin of magneto-electronic hysteresis in NiSi is highly
desirable.

Figure 6: (a) Powder X-ray Diffraction pattern of NiSi. Rietveld analysis of
the powder diffraction data is performed using FULLPROF Suite. All detected
peaks are identified with the orthorhombic lattice structure of space group
pnma. (b-d) Single crystal X-ray diffraction pattern (obtained using a
standard Rigaku single crystal diffractometer at SNS, ORNL) of the reciprocal
lattice of NiSi at $T$ = 250K projected along the a, b, and c- direction,
respectively. All detected peaks are identified with the orthorhombic lattice
structure of space group pnma.

Experimental Section

Synthesis of single crystal NiSi \- Single crystal samples of Nickel Silicide
were synthesized by implementing slow cooling method with high purity powders
of Ni (99.996%) and Si (99.999%) as starting materials from Alfa Aesar. The
two powders are mixed in their stochiometric ratio (1:1) and finely ground
using an agate mortar and pestle.Dahal  This mixture is then packed into a
latex balloon and twisted back and forth from the outside in order to fill-in
possible air gaps along the length of the balloon. Thereafter, the powder-
packed balloon is evacuated and sealed before putting it in the hydrostatic
press at a pressure of $\sim$ 65 MPa for 20 minutes. Finally, it is inserted
in a one-side sealed quartz tube (inner diameter $=$ 7 mm) followed by
evacuating the tube with the sample and sealing it on the other side. A
programmable furnace is utilized to enable better control of the rate of
heating and cooling. The sample is sintered at 900∘C for 24 hours in the
programmable furnace. Subsequently, the sample is re-sintered at 1050∘C for 24
hours followed by cooling it at 0.35∘C/min. The resulting specimen is $\sim$
9.1mm(length) $\times\sim$ 7mm(diameter). The sample is characterized using
X-ray diffraction, which confirms high chemical purity of the specimen, see
Fig.6 below. FullProf refinement of XRD data confirms the zinc blend-type
orthorhombic structure of NiSi with lattice parameters of a = 5.178 $\AA$, b =
3.331 $\AA$, c = 5.1624 $\AA$.

Neutron scattering measurements \- Elastic neutron scattering measurements
were performed on single crystal sample of NiSi on two different
spectrometers: thermal triple axis spectrometer TRIAX and elastic diffuse
scattering time-of-flight spectrometer CORELLI at the University of Missouri
Research Reactor and at Spallation Neutron Source at the Oak Ridge National
Laboratory, respectively. Measurements on TRIAX were performed on 1.2 g single
crystal sample. Experimental data were collected between $T$ 5 K and $T$ = 300
K. Elastic measurements were performed at the fixed final energy of 14.7 meV
using pyrolytic graphite (PG) monochromator. The measurements employed a flat
PG analyzer with collimator sequence of PG filter - 60’ - 60’ - sample - 40’ -
PG filter - 40’. The sample was mounted at the end of the cold finger of a
closed cycle refrigerator with a base temperature of T $\sim$ 5 K.

Higher temperature data were obtained on CORELLI spectrometer. The single
crystal sample was installed inside an aluminum thermal shield and heated in a
cryo-furnace in a wide temperature range from $T$ = 300 K to $T$ = 700 K. The
sample was exposed to a white beam of incident energy 10 - 200 meV and a large
volume of reciprocal space consisting multiple Brillouin zones were explored
by rotating the sample by 360o in 1o step. Experimental data were collected at
several temperatures of $T$ = 300, 550, 630 and 700 K. Data were treated with
the cross-correlation method which reconstructs the elastic scattering signal.

Magnetic measurements \- Bulk magnetization measurements were performed in a
SQUID magnetometer with a base temperature of $T\sim$ 5 K. Magnetic field was
applied along the $c$-axis of the sample.

Hall probe measurements \- Hall measurements were performed on a $\sim$ 2.0
sq.mm NiSi specimen in van der Pauw configuration. Longitudinal resistance
(Rxx) and Hall resistance (Rxy) are measured utilizing a Closed cycle
refrigerator (CCR) system and Resistance Bridge. Measurements were performed
for magnetic field applications perpendicular as well as in-plane (parallel to
a-b plane) to the sample. Data were collected at several temperatures. At each
temperature, experimental data were collected for magnetic field sweep from
$H$ = 7 Tesla to -7 Tesla and from -7 Tesla to 7 Tesla. In addition to Fig. 4,
additional data at $T$ = 150 K and $T$ = 50 K for perpendicular magnetic field
application to the a-b plane of the sample are shown in Fig. S5.

Acknowledgements

PG and JG made equal contributions to this research. DKS thankfully
acknowledges support from the U.S. Department of Energy, Office of Basic
Energy Sciences under Grant No. DE-SC0014461. The research conducted at ORNL’s
Spallation Neutron Source is supported by the Office of Basic Energy Sciences
at DOE.

Keywords

Antiferromagnetic metal, Transition metal intermetallics, Spintronics, Neutron
scattering

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