# Interpretable Multi-Source Data Fusion Through Latent Variable Gaussian
Process

Sandipp Krishnan Ravi1,∗
Probabilistic Design and Material Informatics Group
GE Aerospace Research
Niskayuna, NY
<EMAIL_ADDRESS>
&Yigitcan Comlek1
Integrated Design Automation Laboratory
Northwestern University
Evanston, IL
<EMAIL_ADDRESS>
&Wei Chen∗
Integrated Design Automation Laboratory
Northwestern University
Evanston, IL
<EMAIL_ADDRESS>
&Arjun Pathak
Metallurgy Group
GE Aerospace Research
Niskayuna, NY
&Vipul Gupta
Metallurgy Group
GE Aerospace Research
Niskayuna, NY
&Rajnikant Umretiya
Metallurgy Group
GE Vernova Research
Niskayuna, NY
&Andrew Hoffman
Metallurgy Group
GE Vernova Research
Niskayuna, NY
&Ghanshyam Pilania
Probabilistic Design and Material Informatics Group
GE Aeropace Research
Niskayuna, NY
&Piyush Pandita
Probabilistic Design and Material Informatics Group
GE Aerospace Research
Niskayuna, NY
&Sayan Ghosh
Probabilistic Design and Material Informatics Group
GE Aerospace Research
Niskayuna, NY
&Nathaniel Mckeever
Materials Characterization
GE Aerospace Research
Niskayuna, NY
&Liping Wang
Probabilistic Design and Material Informatics Group
GE Aerospace Research
Niskayuna, NY

###### Abstract

With the advent of artificial intelligence (AI) and machine learning (ML),
various domains of science and engineering communities have leveraged data-
driven surrogates to model complex systems from numerous sources of
information (data). The proliferation has led to significant reduction in cost
and time involved in development of superior systems designed to perform
specific functionalities. A high proposition of such surrogates is built
extensively fusing multiple sources of data, may it be published papers,
patents, open repositories, or other resources. However, not much attention
has been paid to the differences in quality and comprehensiveness of the known
and unknown underlying physical parameters of the information sources that
could have downstream implications during system optimization. Towards
resolving this issue, a multi-source data fusion framework based on Latent
Variable Gaussian Process (LVGP) is proposed. The individual data sources are
tagged as a characteristic categorical variable that are mapped into a
physically interpretable latent space, allowing the development of source-
aware data fusion modeling. Additionally, a dissimilarity metric based on the
latent variables of LVGP is introduced to study and understand the differences
in the sources of data. The proposed approach is demonstrated on and analyzed
through two mathematical (representative parabola problem, 2D Ackley function)
and two materials science (design of FeCrAl and SmCoFe alloys) case studies.
From the case studies, it is observed that compared to using single-source and
source unaware ML models, the proposed multi-source data fusion framework can
provide better predictions for sparse-data problems, interpretability
regarding the sources, and enhanced modeling capabilities by taking advantage
of the correlations and relationships among different sources.

11footnotetext: Equal Contribution**footnotetext: Corresponding Author

_K_ eywords multi-source modeling, experimental data, probabilistic machine
learning, data fusion, interpretable machine learning

##  1 Introduction

Numerous domains of engineering have benefited from the advent of artificial
intelligence (AI) and machine learning (ML). AI$\backslash$ML frameworks,
mostly in the form of predictive models, play a pivotal role in representing
complex engineering systems and analyzing intricate relationships within data
[1]. Their utilization can be further extended to optimizing designs,
predicting system behavior, and informing downstream decision-making tasks.
Essentially, AI$\backslash$ML frameworks drive innovation across diverse
engineering disciplines. With the increase demand and availability of
information (data), an emerging area of research for the AI$\backslash$ML
community is the fusion of data collected from numerous sources into modeling
[2]. This phenomenon of forming a collective intelligence through numerous
sources of information is also known as data fusion. The concept of data
fusion aims to provide a unified point of view regarding a system of interest
through integration of information available from different sources [3]. In
most cases, data from different sources are collected and a single ML model
that aims to capture the relationship between the design variables and the
output of interest is built without considering the possible underlying
influence of the information sources. In other words, due to the diverse
nature of data collection, each information source could contain different
underlying procedures and processes that are not reported and incorporated
into the ML modeling. The sources of variability between the information
sources encompass a wide range of factors, including but not limited to
measurement errors, confidential information, and various other contributors.
Therefore, a major challenge in data fusion exists for the incorporation of
the details and unknown parameters associated with each information source.

A noteworthy and crucial application of data fusion, along with the challenge
regarding the incorporation of information source details is evident in the
field of material science [4]. In the context of materials engineering, the
impact of AI$\backslash$ML and data fusion frameworks has been remarkable [5,
6, 7, 8, 9]. These data-driven frameworks enabled rapid screening of vast
materials and identification of novel combinations and formulations for the
design and discovery of superior materials with desired properties. However,
even with such proliferated data-driven material discovery efforts, the
availability and quality of data remains an important challenge, especially
for system level experimental materials design and discovery applications
[10]. The challenge of experimental data availability and quality is primarily
rooted in the diverse nature of experimental materials data. Typically, the
first step towards optimizing a material system for a given functionality
starts by accruing data from numerous resources including research literature
(papers, patents) and open-source data repositories [11, 12, 13, 14, 15] to
form a unified view and understanding for the material through data fusion. On
the other hand, unlike computational modeling of material behavior, there
exists no particular standardization for manufacturing and experimentally
synthesizing material systems. Furthermore, issues related to data quality
persist due to inconsistencies in experimental methodologies, measurement
errors, lack of standardized reporting practices, and confidentiality [16,
17]. Therefore, this situation leads to an unavoidable data quality issue that
has a severe impact on subsequent data-driven modeling and optimization of
these material systems.

To tackle the aforementioned challenges, the need for a multi-source data
fusion that considers source variabilities becomes evident. By integrating
data from diverse sources, multi-source data fusion modeling with
interpretable ML can provide a more comprehensive and robust understanding of
material behaviors, leading to faster discovery of novel materials.
Specifically for experimental data sources, the data-driven ML approach can
consider the inherent complexities and uncertainties associated with each
experimental data resource through extraction of meaningful patterns and
insights from different experimental conditions. This factor effectively
mitigates the data availability and quality challenges. To achieve this, a
Gaussian Process (GP) based multi-source modeling through Latent Variable
Gaussian Process (LVGP) [18] is proposed to fuse numerous information (data)
sources together to obtain a single representative interpretable prediction
model. LVGP is novel ML model that allows incorporation of qualitative
(categorical) variables in addition to quantitative (continuous) variables
into surrogate modeling and has been implemented in variety of fields and
applications such as adaptive materials design [19, 20, 21], non-hierarchical
interpretable multi-fidelity modeling [22], and first-ever mixed-variable
(continuous and categorical) metamodel-based global sensitivity analysis [23].
Therefore, throughout this research, a single multi-source ML model that
incorporates engineering system parameters as numerical (and categorical if
there are any) variables while incorporating each data source as a qualitative
variable is developed and studied. The qualitative information sources are
incorporated into the LVGP model by implicitly mapping each qualitative
variable into a low-dimensional quantitative latent space through statistical
inference techniques. Specifically, as the influence of any qualitative
variable on any quantitative response must be always due to some underlying
(and possibly high-dimensional) quantitative physical variables, the latent
variable approach provides physically meaningful dimension reduction. Here,
the term “physically-meaningful” can be associated with explaining the cause-
effect relationships between the information sources and the system response.
Therefore, the qualitative representation of information sources indirectly
accounts for the variability they bring through known and unknown underlying
processing variables, which in return provides further interpretability
regarding the information sources and their relationships on the system of
interest.

The outline of the paper is as follows: In Section 2, descriptions of the ML
models that are used for data fusion, GP and LVGP, are given and multi-source
data fusion through LVGP is introduced. Next, two exemplary mathematical
representative problems in Section 3 are demonstrated to highlight the
advantages of using LVGP for multi-source information modeling. In Sections 4
and 5, the proposed multi-source model is applied on two alloy systems (FeCrAl
and SmCoFe) to illustrate its benefits for material science research. Finally,
the work is concluded through Section 6.

##  2 Methodology

###  2.1 Review of Gaussian Process Regression

With its powerful and versatile capabilities of capturing nonlinear responses
and providing fast and accurate predictive modeling with uncertainty
quantification, Gaussian processes (GPs) are well-known metamodels that have
been implemented in many engineering applications [24]. GPs are nonparametric
regression models and are based upon Bayesian statistics in combination with
numerous kernel methods to provide data-driven probabilistic predictions.
Overall, GPs provide an immensely powerful tool for engineers seeking to build
fast and accurate predictive models while quantifying and managing
uncertainty.

Considering a typical quantitative (numerical or continuous) m-dimensional
input space with $x=[x_{1},x_{2},...,x_{m}]$ with a response of interest,
$y(x)$, a typical single response Gaussian Process model can be described
using Equation (1),

$y(x)=\mu+K_{x}$ (1)

which consists of prior constant mean, $\mu$, that describes the mean response
at any given point ($x$), and a zero-mean Gaussian Process with a covariance
function, $K_{x}$, which determines the relationship or the correlation
between variables in the model. The covariance function $K_{x}$ can be further
extended to $K(x,x^{{}^{\prime}})=\sigma^{2}\cdot c(x,x^{{}^{\prime}})$ where
the $\sigma^{2}$ represents the prior variance of the GP model and
$c(x,x^{{}^{\prime}})$ describes the correlation function between each point
in the model through the specified correlation function. For the modeling of
the GP model, the implemented Gaussian kernel is defined as

$c(x,x^{{}^{\prime}})=exp(-\sum_{i=1}^{m}(\phi_{i}(x_{i}-x_{i}^{{}^{\prime}})^{2}))$
(2)

where $\phi$ describes the scale between input variables. The selected kernel
specifically describes how similar or correlated the function values are at
different input points. For instance, when input variables are spatially
closer to each other, their covariance results in high values, indicating a
strong correlation between the variables. Conversely, when they are far apart,
the covariance decreases, signifying a weaker correlation. The GP model
parameters, $\mu$, $\sigma^{2}$, and $\phi$ are estimated through the maximum
likelihood estimation (MLE).

Although GPs are powerful ML models for numerous engineering applications,
they tend to perform poorly when qualitative (categorical) variables are
present in the input design space. The main challenge arises from the absence
of a well-defined distance metric that differentiates qualitative variables.
To overcome this vital challenge, Latent Variable Gaussian Process (LVGP) has
been developed [18]. The details of the LVGP model are given in the next
section.

###  2.2 Review of Latent Variable Gaussian Process

Due to the nature of the GP correlation functions, it is not plausible to
directly include qualitative variables into the modelling since the
differences between qualitative variables become unclear without a well-
defined distance metric. Recently, Latent Variable Gaussian Process (LVGP) has
been developed to integrate qualitative variables into GP modeling through
quantitative latent variable representation of these variables to address this
difficulty [18].

Consider a new input space, $\mathbf{b}=[\mathbf{x^{T}},\mathbf{t^{T}}]$ where
both quantitative, $\mathbf{x}=[x_{1},x_{2},..x_{m}]\epsilon R^{m}$, and
qualitative $\mathbf{t}=[t_{1},t_{2},...,t_{q}]\epsilon R^{q}$ input variables
are included in this new domain. Each qualitative variable can have $j$ number
of options (levels), i.e $t_{i}=[l_{1}(t_{i}),l_{2}(t_{i}),...,l_{j}(t_{i})]$
where $i=1,2,...,q$. For a given response of interest, $y(b)$, the influence
of each qualitative variable can be represented by an underlying quantitative
(numerical) space,
$v_{t_{i}}(l_{j})=[v_{1}(l_{1}),v_{2}(l_{1}),...,v_{k}(l_{j})]\epsilon R^{k}$.
This underlying quantitative space can possibly be undiscovered, unknown,
complex, or high-dimensional. As a result, the main idea of LVGP lies in
learning a low-dimensional quantitative latent space to approximate the
original underlying space through statistical inference using the available
input-output data. According to the original LVGP paper, a two-dimensional
latent variable vector is sufficient to express the influence of the
qualitative variables on the response of interest [18]. Therefore, qualitative
variable, $t_{i}$ can be represented as
$\mathbf{z}(t_{i})=[z_{1}(l_{1}),z_{2}(l_{1}),z_{1}(l_{2}),z_{2}(l_{2}),...,z_{1}(l_{j}),z_{2}(l_{j})]$,
where each level $l_{j}(t_{i})$ is represented with a 2D latent vector
$[z_{1}(l_{j}),z_{2}(l_{j})]$ . Thus, the previously defined mixed-variable
input space, $\mathbf{b}=[\mathbf{x}^{T},\mathbf{t}^{T}]$, then becomes
$\mathbf{w}=[\mathbf{x}^{T},\mathbf{z(t)}{{}^{T}}]\epsilon R^{m+2\times q}$.
With the newly defined input space, the previously specified GP correlation
function (Equation 2) is updates as,

$c(w,w^{\prime})=exp(-\sum_{i=1}^{m}(\phi_{i}(x_{i}-x_{i}^{{}^{\prime}})^{2})-\sum_{j=1}^{q}||z_{1,j}-z_{1,j}^{{}^{\prime}}||^{2}_{2}+||z_{2,j}-z_{2,j}^{{}^{\prime}}||^{2}_{2})$
(3)

Consequently, the LVGP model parameters $\mu$, $\sigma^{2}$, and $\phi$, along
with the latent variables, $\mathbf{z}$, are estimated through MLE of the log-
likelihood function

$l(\mu,\sigma,\phi,\mathbf{z})=-\frac{n}{2}ln(2\pi\sigma^{2})-\frac{1}{2}ln|C(\mathbf{\phi},\mathbf{z})|-\frac{1}{2\sigma^{2}}(\mathbf{y}-\mu\mathbf{1})^{T}C(\mathbf{\phi},\mathbf{z})^{-1}(\mathbf{y}-\mu\mathbf{1})$
(4)

where $n$ is the number of available training samples, $C$ is a $n\times n$
correlation matrix with $C_{ij}=c(w_{i},w_{j})$ for $i,j=1,2,...,n$,
$\mathbf{y}=[y_{1},y_{2},...,y_{n}]$ is the observed response vector,
$\mathbf{1}$ is a vector of ones with a size of $n\times 1$. Once all the
model parameters are estimated, the LVGP model can make predictions,
$\hat{y}(\mathbf{w}^{*})$, on any given input $\mathbf{w^{*}}$ with the
quantified prediction uncertainty, $\hat{s^{2}}(\mathbf{w}^{*})$:

$\hat{y}(\mathbf{w}^{*})=\mu+\mathbf{c}(\mathbf{w}^{*})\mathbf{C}^{-1}(\mathbf{y}-\mu\mathbf{1})$
(5)

$\hat{s^{2}}(\mathbf{w}^{*})=\sigma^{2}(\mathbf{c}(\mathbf{w}^{*})-\mathbf{c}(\mathbf{w}^{*})\mathbf{C}^{-1}\mathbf{c}(\mathbf{w}^{*})^{T})$
(6)

where
$c(\mathbf{w}^{*})=[C(\mathbf{w}^{*},\mathbf{w}^{(1)}),C(\mathbf{w}^{*},\mathbf{w}^{(2)}),...,C(\mathbf{w}^{*},\mathbf{w}^{(n)})]$
is the pairwise correlation vector between the new point $\mathbf{w}^{*}$ and
each training point $\mathbf{w}^{(i)},i=1,2,...,n$.

With the introduction of LVGP, the primary contribution of this paper lies in
the modeling of the multiple sources of information as a representative
qualitative (categorical) input in addition to known quantitative (numerical)
inputs through the LVGP model, making the surrogate modeling source-aware and
interpretable. Further details of the multi-source data fusion modeling are
provided in the subsequent section.

###  2.3 Interpretable Multi-Source Data Fusion Through Latent Variable
Gaussian Process

For an engineering system of interest, there could be multiple sources of
information (patents, papers, open-source repositories) that aim to analyze
the system. However, due to the diverse nature of data collection, each
information source could contain different underlying procedures and processes
that are not reported such as experimental errors, confidential information
and so on. In this work, a multi-source modeling technique with Latent
Variable Gaussian Process is proposed to combine all of the information
sources into a single source-aware model that considers not only the
relationships between information sources but also that both the known and
unknown underlying physical parameters associated with each information
sources when making predictions. To achieve this, a given engineering system
of interest is modeled with a new input space,
$[\mathbf{x},\mathbf{t},\mathbf{s}]$, where $\mathbf{x}$ and $\mathbf{t}$ are
the known common numerical ($\mathbf{x}$) and categorical ($\mathbf{t}$)
material parameters between information sources, and $\mathbf{s}$ is the newly
introduced separate categorical variable that represent the information
sources ($\mathbf{s}$), along with the output response of interest,
$\mathbf{y}$. Here, $\mathbf{s}$ is treated as a unique separate qualitative
variable that differentiates the information sources and integrates them into
the multi-source model through the same statistical inference techniques used
to estimate the latent variables of the variable $\mathbf{t}$. For this newly
introduced variable $\mathbf{s}$, each information source is labelled with the
letter $l$. A visual representation of the latent mapping of qualitative
variable ($s$), which is denoted as the information sources where each source
is represented by a level ranging from $l_{1}$ to $l_{j}$ in current case, is
shown in Figure 1. By incorporating information sources as a qualitative
variable, the aim is to learn and extract the unknown physical parameters
underlying each information source through 2D latent spaces using statistical
inference techniques that has been reviewed in Section 2.2.1. It is important
here to note that the spatial relations (distances) between the information
sources in the latent space can show similarity and differences regarding the
influence of the information sources on the system of interest. Similarly,
spatial relationships between latent variables can also imply the
dimensionality of underlying known and unknown physical parameters related to
the information sources. Therefore, for a given information source $j$, it is
incorporated into the LVGP model with it’s unique known common parameters
$x_{j}$ and $t_{j}$ and it’s unique source identifier $l_{j}$. Once the LVGP
is trained, the model can become aware of not only the known and unknown
assumptions and parameters underlying each information source but also the
relationships between information sources. The awareness of the sources
provided by the LVGP can further lead to fast and accurate predictions. Figure
2 shows the proposed multi-source modeling framework along with information
source examples and their unique inputs to the LVGP model. It is important
here to note that although common qualitative variables ($\mathbf{t}$) can be
integrated into the multi-source model with LVGP, case studies with common
quantitative variables ($\mathbf{x}$) are only demonstrated, leading to an
input space with only $\mathbf{x}$ and $\mathbf{s}$ that are fed into the LVGP
model.

Figure 1: Mapping of different information sources (qualitative space) onto
the 2-dimensional quantitative latent space

As previously mentioned, the latent variable approach provides physics-based
dimension reduction that could potentially explain the "cause-effect"
relationship between the inputs (the information sources) and outputs
(response). Therefore, the proposed model can give us physical insights
regarding the information sources by analyzing the spatial locations of the
variables (sources) in the latent space. Here, the locations of the latent
variables could suggest that the variables that are closer in the latent space
can have a similar impact on the response. Moreover, for the scenarios where
no prior knowledge exists regarding information sources, the LVGP model can
help us extract and reveal the relationships between the sources. To
understand the underlying similarities and differences between information
sources, a dissimilarity metric ($D$) that quantifies the dissimilarities
between the sources through a distance metric is introduced. This
dissimilarity metric is measured through the Euclidean distance of the data
source $(z^{j})$ from the reference data source $(z^{*})$ in the latent space.
The metric is given in Equation 7. For a given LVGP model, the latent
variables are constrained to be between $z=[-3,3]$ as this was the typical
practice in previous applications of LVGP and the reference data source will
be placed at $z^{*}=[0,0]$ in the latent space. Therefore, the most dissimilar
information source from the reference source is going to be placed at [$\pm
3,\pm 3$], making $z^{max}=[\pm 3,\pm 3]$. Based on this, the maximum distance
between the reference and given information source can be
$||z^{max}-z^{*}||=3\sqrt{2}$. Thus, a normalized unit of dissimilarity ($D$)
can be calculated through Equation 8. With this quantified metric, the
similarities and differences between information sources can be understood,
which could be further used in downstream tasks such as anomaly detection and
fault identification of sources.

Figure 2: Schematic of multi-source data fusion through LVGP

$D(z^{j})=\frac{||z^{j}-z^{*}||}{||z^{max}-z^{*}||}$ (7)

$D(z^{j})=\frac{||z^{j}-z^{*}||}{3\sqrt{2}}$ (8)

##  3 Representative Mathematical Problems

Before the proposed methodology is applied into material science case studies,
it is first implemented on two representative mathematical problems to provide
a more in-depth analysis and understanding. It is demonstrated that when
different information sources are identified and specified to the ML models,
more accurate models and response surfaces can be obtained, which could be
significantly beneficial for model prediction and design optimization
purposes.

###  3.1 Parabola Problem

In the representative parabola problem, four (4) different mathematical
equations are used as different information sources. All of the four different
information sources are created using

$y=(x+x_{shift}-a)\times(x+x_{shift}-b)+y_{shift}$ (9)

where different parameters, $a,b,x_{shift}$ and, $y_{shift}$ are employed for
each of the information sources. For each source, multiple training data
points are created between the one dimensional input space,
$x\epsilon[-10,10]$. The first information source, which is denoted as the
Ground Source, can be considered as the source of data which is most detailed
or in-house generated (and replicable) data. Replicating its cost and access,
the Ground Source only contains 3 training samples. However, the other three
information sources, which can be considered as incomplete or incomprehensive
and thus relatively cheap, have 10 training samples each for building the
machine learning models. The remaining parabolas are referred to as Perturbed
Sources. Supplementary Table 1 shows the parameters used for each of the
information sources and details regarding the number of available samples for
training and testing the ML models. Furthermore, Figure 3a illustrates the
parabolas obtained for each of the information sources. The Perturbed Source 2
is a copy of Ground Source with a $y_{shift}$, Perturbed Source 1 is another
copy of Ground Source with a $x_{shift}$, whereas Perturbed Source 3 is a copy
of Ground Source with shifts in both directions. The parameters of the
perturbed sources are selected carefully to display different resemblance
levels to the Ground Source, which will be important for the latent space
obtained from the LVGP model and the dissimilarity metric ($D$), as seen later
in the manuscript. It can be observed from the Figure 3a that the mathematical
and visual resemblance to the Ground Source is in the order of Perturbed
Source 2, Perturbed Source 1, and Perturbed Source 3.

Next, Gaussian Process and Latent Variable Gaussian Process (detailed as GP
and LVGP in the previous sections) are leveraged to model the response surface
of the representative parabola problem ($y$). For the GP model training, only
the quantitative (numerical) input variable ($x$) is used to learn the
response. In this case, the model is unaware of the sources where the data is
generated from. On the other hand, a separate qualitative variable that
denotes different information sources is included into the training of the
LVGP model, in addition to the quantitative variable ($x$), making the
surrogate model aware of the different sources that the data is generated
from. The black points marked on Figure 3a displays the quantiative training
points used to train the models. A well-known ML model predictive performance
metric, Normalized Root Mean Squared Error (NRMSE) is utilized to compare the
training and testing performances of these models. Table 1 shows the
performance results comparing the two ML models. The low NRMSE value for LVGP
clearly shows that incorporating the information source as a separate variable
into data fusion ML models yield a better predictive model. Furthermore, the
parity plot of the two models (Figure 3b& c), comparing the model predictions
and actual values, clearly demonstrates the advantage of incorporating and
differentiating the source information into the modeling. Finally, Figure 3d &
e shows the predictions made solely on the Ground Source, which once again
clearly demonstrates the predictive capabilities of LVGP. In the context of
engineering and scientific applications, considering cases where conducting
experiments are costly or one has only access to a single experimental
resource, obtaining an accurate predictive model for that specific experiment
can be highly beneficial for advanced design and development.

Table 1: Predictive performance comparison between GP and LVGP for the parabola problem  Model Type | Training NRMSE | Testing NRMSE
---|---|---
GP | $0.199$ | $0.207$
LVGP | $5.26\times 10^{-5}$ | $11.66\times 10^{-4}$

The advantages of LVGP are not limited to providing improved prediction
accuracy and the capability of providing uncertainty quantification in the
similar way as other GP models. Figure 3f shows the benefit of using LVGP for
interpreting the differences between the information sources. The distances
between the latent variables of Perturbed Sources to the Ground Source
directly relate to the previously mentioned resemblance level to the Ground
Source. This resemblance is further quantified with the previously introduced
dissimilarity metric $D$. The $D$ values shown in Figure 3f shows the
dissimilarity metric for each information source with respect to the Ground
Source. It can be inferred from the metric that Perturbed Source 2 is the
closest while Perturbed Source 4 is the farthest to the Ground Source. This
result also matches very well with our known physical knowledge regarding the
information sources and confirms the validity and interpretability of both the
latent space and the dissimilarity metric. For this illustrative example,
although the relationships between the information sources is known
beforehand, the latent variables corresponding to the information source not
only revealed the relationship between the sources but also confirmed our
initial physical knowledge about the differences between the information
sources. Next, the approach is demonstrated on another mathematical testing
function used in optimization studies to further demonstrate the importance
and effectiveness of using LVGP as a multi-source data fusion method.

Figure 3: (a) Overview of Parabola Example. (b-c) The parity plot between the
GP and LVGP models. (d-e) Prediction of Parabola on the Ground source GP model
and LVGP model. (f) Latent Space of Parabola Example from LVGP. Ground Source
is denoted as the reference source (*) for dissimilarity metric, $D$,
calculations

###  3.2 2D Ackley Function

The proposed method is next demonstrated on a well-known mathematical function
that is used in optimization studies. Ackley’s function is a customiziable
multi-dimensional function that has been implemented in numerous optimization
applications for testing and validation purposes [25]. Here, 2D Ackley’s
function, Equation 10, is implemented to further demonstrate the proposed
interpretable multi-source data fusion method. In addition to the original 2D
function, three (3) other variations of Ackley’s function are generated from
the original function as different sources of information. Similar to the
parabola problem, the original function is named as the Ground Source and the
remaining 3 functions are named as Perturbed Sources 1, 2, and 3. The
formulations of the functions are found in Supplementary Table 2 and their
surface of the response, $z$, is shown in Figure 4a.

$z=-a*exp(-b(\sqrt{\frac{1}{2}(x^{2}-y^{2})}))-exp(\frac{1}{2}(cos(cx)+cos(cy)))+a+exp(1)$
(10)

Identical to the previous case study, two ML models are built (GP and LVGP) to
demonstrate the effectiveness of the proposed approach. First, a GP model is
built by only using the quantitative variables, $x$ and $y$ to predict the
response, $z$. Then, in addition to the two quantitative variables, a
qualitative variable is added to differentiate the information sources for the
LVGP model. This approach is the exact same approach taken for the previous
parabola problem and will be implemented for the remainder of this work.
During the training, 5-fold cross validation (CV) technique is implemented and
tested on a separate testing set. Both the training and testing sets are
created by sampling uniformly from the space $[x,y]\epsilon[-5,5]$. The number
of training and testing samples employed to build and validate the models are
available in Supplementary Table 2. Next, a comparison study between the
predictive performance of both models using the NMRSE metric is carried out.
Table 2 (Model Types: GP and LVGP) shows the performance metrics for both
models. From the results, it can be clearly observed once again that
incorporating different sources as a separate variable significantly improves
the predictive performance of the model. The model was also able to capture
the relationships between the response surfaces and provide interpretability
through the latent variables clearly. The latent space obtained from the LVGP,
Figure 4b, matches very well with the response surfaces, Figure 4a.
Furthermore, the dissimilarity metric calculated using the Ground Source as
the reference source, $D$, agree with the response surfaces and what is known
regarding the similarity between the functions, demonstrating the efficacy of
the method and the metric (Figure 4b).

A possible follow-up question one might ask as a response to the demonstrated
results could be to build separate GP models to learn the information sources
and predict their responses individually to achieve better accuracies. To
answer this question, a separate GP model is built only using the Ground
Source and compared with the previously built LVGP on a separate testing set.
For ease of notation, this single-source model is denoted as GP-GS. The
trained GP-GS model only contained 20 samples from the Ground Source whereas
the LVGP model contained all 170 samples from the available sources. The same
testing set from the Ground Source is used and the prediction performances are
compared. Table 2 (Model Types: LVGP and GP-GS) shows the prediction
performances between the GP-GS and LVGP models, along with the parity plot in
Figure 4c. In addition to achieving lower NRMSE values for accurate modeling,
the parity plot demonstrates that the prediction uncertainty provided by the
LVGP model is much less compared to the GP model. This result reveals an
interesting phenomenon. Compared to the model that is only trained on the
original single-source of information, gathering further information from
other sources and explicitly denoting information sources into the data fusion
modeling results in a more confident predictive model that provides lower
prediction uncertainty on the original single-source of information.

Table 2: Predictive performance comparison between GP, LVGP, and GP-GS models on the 2D Ackley function  Model Type |  Mean Training NRMSE |  Mean CV NRMSE |  Testing NRMSE
---|---|---|---
GP |  $0.253$ |  $0.289$ |  $0.254$
|  |  |
LVGP |  $0.013$ |  $0.029$ |  $0.024$
|  |  |
GP-GS |  — |  — |  $0.089$

The conclusions from this study can be significant for many reasons and can be
applied to many engineering and scientific applications, specifically for
materials science. This is because many researchers only have access to or
control over their own experiments/simulations and rely on building a
surrogate model using their own data to guide their next property evaluations.
On the other hand, due to the high costs of property evaluations, data
curation for better predictive surrogate models could be expensive and limited
to only a few samples. However, by incorporating more data from different
available sources (open-sources, papers, patents) and explicitly
differentiating each data source as an input to their models, researchers have
the potential to obtain better and confident predictive model, which in return
has the potential to lead to novel material candidate discovery at a faster
rate. In the following two sections, Section 4 and Section 5, the advantages
of using multi-source data fusion through LVGP on two materials design cases
are demonstrated.

Figure 4: (a) Response Surface of the Ground Source and Perturbed Sources for
the 2D Ackley’s function. (b) The latent variables obtained from the LVGP
model. Ground Source is denoted as the reference source (*) for dissimilarity
metric, $D$, calculations. (c) The parity plot between the GP-GS and LVGP
models tested on the Ground Source data only

##  4 Modeling Thermal Aging Behavior of FeCrAl Alloys

###  4.1 Overview

FeCrAl (Iron-Chromium-Aluminum) alloys have been garnering a lot of attention
for their superior hydro-thermal corrosion resistance at normal operating
conditions ($\sim$300∘C) and high temperature ($\sim$1000 ∘C) steam oxidation
resistance [26, 27, 28, 29, 30, 31, 32, 33]. The key application driving the
exploration and development of the alloy system is for the protective cladding
for nuclear fuels. Their uniqueness lies in their ability of providing
oxidation resistance across a wide range of operating temperatures, usually
ranging from 300∘C to 1200∘C. A schematic of the oxidation phenomenon is shown
in Figure 5. At low operating temperatures (at around 400∘C), Chromium reacts
with environmental oxygen and forms a protective oxide layer serving as a
natural barrier protecting the alloy from environment. As the operating
temperature increases (to around 1200∘C), the Chromium oxide (Cr2O3) layer
decomposes. The decomposition is subsequently and simultaneously followed by
the formation of an Aluminum oxide (Al2O3) layer. The Aluminum oxide layer
serves as the protective natural barrier against the environment. The whole
phenomenon of oxide formation in pure FeCrAl alloys are primarily driven by
Chromium content, Aluminum content in the alloy and operating temperatures.

Despite their superior oxidation resistance, there is one major hindering
aspect of FeCrAl alloys that warrants further study. FeCrAl alloys undergo
aging embrittlement which makes them vulnerable to mechanical perturbation and
failure. The embrittlement is majorly caused by the Cr-rich
$\alpha^{{}^{\prime}}$-phase precipitation in the Fe-rich matrix. The
formation of such $\alpha^{{}^{\prime}}$ phases increases the overall hardness
of the alloy and makes them brittle.

The thermal age hardening phenomenon is driven by the composition of the
FeCrAl alloy, temperature, and duration of the thermal aging process. The
hardness change of the alloy at different stages of thermal aging process is
measured and used as an indication of the embrittlement. With the nuance of
thermal aging phenomenon in FeCrAl alloys, material designers have the task of
optimizing the alloy composition for maximum oxidation resistance and minimum
embrittlement. The levers that material designers have at their disposal is
the composition of the FeCrAl alloys including dopants that can suppress the
formation of $\alpha^{{}^{\prime}}$ phases.

Figure 5: Schematic of the oxidation behavior of the FeCrAl alloy

To demonstrate the nuances, impact, and need of multi-source data fusion,
thermal aging behavior of FeCrAl alloys is used as the first material science
case study. In this work, all sources of data are experimental. Recently,
there have been several efforts towards leveraging AI$\backslash$ML techniques
for modeling and optimizing both the oxidation and thermal aging behavior of
FeCrAl alloys [7, 34, 35, 36]. However, there are still restrictions about the
quality of the data from multiple sources and the unaccounted effects of
underlying unknown physical parameters that is present in the data generation
process during the experimentation. In this study, the aim is to address these
challenges using the proposed multi-source ML model.

###  4.2 Data Description

As with all material discovery efforts towards developing a generalizable
multi-source data fusion model, the very first step is to gather data from
research literature (papers, patents, etc.) and open-source databases. From an
extensive search, seven well-cited and verified external sources of data were
identified and collected. Additionally, there was also elaborate in-house
testing conducted with GE Research to understand the embrittlement behavior of
FeCrAl alloys. The sources of data are summarized through Table 3. The in-
house generated samples are tagged as ’GE Data’. The rest of the collected
samples are denoted by the name of the respective first authors of the paper.
A total of 380 data points has been collected.

Table 3: Sources of information for FeCrAl alloys  Name of Data Source |  Number of Data Samples |  Source of Data
---|---|---
GE Data |  $178$ |  Ravi, Sandipp Krishnan, et al. "Elucidating precipitation in FeCrAl alloys through explainable AI: A case study." Computational Materials Science 230 (2023): 112440. [7]
Kobayashi |  $33$ |  Kobayashi, Satoru, and Takayuki Takasugi. "Mapping of 475 C embrittlement in ferritic Fe–Cr–Al alloys." Scripta Materialia 63.11 (2010): 1104-1107. [27]
Han |  $55$ |  Han, Wentuo, et al. "Effect of Cr/Al contents on the 475ºC age-hardening in oxide dispersion strengthened ferritic steels." Nuclear Materials and Energy 9 (2016): 610-615. [32]
Kim |  $4$ |  Kim, Hyunmyung, et al. "400 C aging embrittlement of FeCrAl alloys: Microstructure and fracture behavior." Materials Science and Engineering: A 743 (2019): 159-167. [37]
Ejenstam |  $12$ |  Ejenstam, Jesper, et al. "Microstructural stability of Fe–Cr–Al alloys at 450–550 C." Journal of Nuclear Materials 457 (2015): 291-297. [38]
Yang |  $64$ |  Yang, Z., et al. "Aluminum suppression of $\alpha^{{}^{\prime}}$ precipitate in model Fe–Cr–Al alloys during long-term aging at 475 C." Materials Science and Engineering: A 772 (2020): 138714. [39]
Capdevila |  $16$ |  Capdevila, Carlos, et al. "Phase separation in PM 2000™ Fe-base ODS alloy: Experimental study at the atomic level." Materials Science and Engineering: A 490.1-2 (2008): 277-288. and Capdevila, Carlos, Michael K. Miller, and Jesús Chao. "Phase separation kinetics in a Fe–Cr–Al alloy." Acta materialia 60.12 (2012): 4673-4684. [29, 40]
Pinkas |  $18$ |  Pinkas, Malki, et al. "Sensitivity of thermo-electric power measurements to $\alpha$ – $\alpha^{{}^{\prime}}$ phase separation in Cr-rich oxide dispersion strengthened steels." Journal of Materials Science 50 (2015): 4629-4635. [41]
Total |  $380$ |

###  4.3 Model Development

To contrast and compare multi-source modeling capabilities, GP and LVGP models
are trained on the collected data. The inputs for the GP model are the
composition of the alloy constituents (Iron, Aluminum, Chromium, Molybdenum,
Zirconium, Manganese, Silicon, Tungsten, Yttrium and Titanium) defined as
weight percentages (wt%), temperature, and duration of the thermal aging
process, totaling up to twelve (12) unique inputs. The output that is used to
quantify the embrittlement behavior is represented through the measured
hardness change (HV). On the other side, the inputs for the LVGP model contain
an additional qualitative input that parameterizes each source of data as a
distinct categorical level (option) along with all the other 12 numerical
inputs used for the GP model. The output for the LVGP is identical to the one
used for the GP model.

###  4.4 Results and Discussion

Within this study, (1) the generalizability of the models, and (2) the
performances of the models with respect to the experimental in-house
information source that can be controlled is investigated. The main goal of
(1) is to observe the usefulness of the models when one has control to
multiple sources of information to perform further experimentations, whereas
the main goal of (2) is to observe which of the models is more reliable when
one has further control over a source of information. Out of all 380 available
samples, 344 samples are used to observe the generalizability of models
through cross validation (CV) and set aside 36 GE Data points as a separate
testing set to demonstrate the prediction capabilities of the models with
respect to the information source that can be control and perform further
experiments on. For the remaining 344 samples, 5-fold cross validation (around
275 training and 69 validation samples per fold) is employed to evaluate the
generalizability of both models. For CV tasks in LVGP, stratified sampling is
used to make sure datasets used for training and testing represent the overall
distribution of the information sources. Then, both models are trained using
the 344 samples and tested on the separate 36 GE testing data. The training,
validation and testing samples were held identical for both models. The
results of this study are shown in Section 4.4.1. In addition to comparing two
ML models on the multi-source data, LVGP is further compared with a separate
GP model trained only on GE Data. For ease of notation, this GP model is
denoted as GP-GE. GP-GE was only trained on GE Data samples available in the
remaining 344 sample set (142 samples). Both LVGP and GP-GE are tested on the
same separate 36 GE Data samples. The results of this study are shown in
Section 4.4.2.

Furthermore, the LVGP model is investigated to extract physical insights and
to obtain interpretability regarding the sources of information and the system
of interest through the latent variables. The results from this examination
are highlighted in Section 4.4.3. Finally, based on insights obtained from the
latent space, a targeted source selection through data filtering mechanism is
implemented to improve the predictive capabilities of multi-source data fusion
in Section 4.4.4.

####  4.4.1 Multi-Source Data Fusion Comparison: LVGP and GP

The predictive capability comparison between multi-source data fusion through
GP and LVGP models are summarized in Table 4 (Model Types: GP and LVGP) and
Panels A and B of Figure 7. From the summarized prediction metric, NRMSE
(Table 4), we observe that the although GP model performs slightly better in
training, we observe a clear improvement in the cross validation (CV), which
contains the samples that the model has not seen during training. The parity
plots from the CV study are shown in Supplementary Figure 1. The CV results
demonstrate that incorporating information sources as a separate variable
through LVGP can lead to a more comprehensive and generalizable prediction
model. Looking further at the testing set which contains separate 36 GE Data
samples, LVGP performs significantly better compared to the GP model,
demonstrating that explicit categorization of the sources provides more
accurate source-specific predictions. Furthermore, the 45-degree parity plot
between the GP and LVGP model across the training and testing data (first row
of Figure 6) demonstrate that in addition to delivering accurate predictions,
LVGP provides lower prediction uncertainties, which can be inferred from the
error bars.

Table 4: Predictive performance comparison between GP, LVGP, GP-GE, and LVGP-Target models on FeCrAl alloys  Model Type |  Mean Training NRMSE |  Mean CV NRMSE |  Testing NRMSE
---|---|---|---
GP |  $0.060$ |  $0.105$ |  $0.088$
|  |  |
LVGP |  $0.064$ |  $0.100$ |  $0.066$
|  |  |
GP-GE |  — |  — |  $0.057$
|  |  |
LVGP-Target |  — |  — |  $0.053$

To strengthen our case of using multi-source modeling through LVGP, response
surface plots from both the GP and LVGP models (with GE Data as reference) are
also provided through Panels A and B of Figure 6. The second and third row
depict the mean and standard deviation of the hardness changes across
variation of the Cr (12%-24%) and Al (0%-7%) and the fourth row depicts the
combined mean and standard deviation in a 3D representation. The model
validity is assessed by focusing on the mean and variation behavior of the
response surface. It is noted that the response surface of the hardness
changes are highly undulated with sharp peaks and valleys with the GP model.
On the other hand, using the GE source as ground truth for prediction, the
response surface is much smoother and more well-behaved with the LVGP model.
One can argue that the unsmooth behavior is caused by the varying level of
noises coming from different sources in the data set when fitting all of them
to one single GP model. On the other hand, LVGP has the capability of
smoothing out the curves from multiple sources after the introduction of the
latent variables. From a material science perspective, it is very unlikely
that phenomenon driving the hardness changes across the composition are highly
transient and acute. This important observation highlights the need of
accounting for known and unknown underlying physical parameters when data from
numerous sources are combined for data fusion. The samples collected from
different papers might have had differences in manufacturing process, thermal
aging procedure, and testing technique that are not accounted for in a single
source model. Furthermore, the downstream tasks of optimization, sensitivity
analysis, uncertainty quantification would also be severely affected by such
distinctions and sharp undulations of the surrogate models, thus making the
case of employing multi-source modeling through LVGP once again.

####  4.4.2 Comparison of the Multi-Source LVGP with the Single-Source GP-GE

A possible argument to achieve more accurate ML models is to build models only
using the data source that one has control over, which would be the GE Data in
the current study. To demonstrate the advantages of using multi-source data
fusion through LVGP over a single-source model for the specified source of
interest, the prediction performances of LVGP and the GP model trained on the
GE Data only, which was previously denoted as GP-GE, are compared. Table 4
(Model Types: LVGP and GP-GE) and Panels B and C of Figure 6 demonstrate the
prediction metrics and the response surface obtained from the models. Due to
the differences in the data used to train models (multi-source vs single-
source), testing accuracy of the models are studied on the common 36 GE Data
testing set. The NMRSE prediction metrics from Table 4 demonstrate that
incorporating other available sources into the modeling yield very similar
predictive performances, with slightly better results towards GP-GE model.
However, comparing the 45-degree parity plots between the two models (first
row of Figure 6), we observe that the prediction uncertainties provided by the
LVGP are much smaller compared to the prediction uncertainties provided by the
GP-GE model. A possible explanation for this phenomenon could be that the
knowledge incorporated from other sources contributes to the learning of the
material system and therefore increases the confidence in the model when
making predictions. Furthermore, the response surfaces between the two models
show high similarity, indicating that the LVGP was able to capture the
response surface of the specific source even when other data sources exist.
The final important observation is that the predictions provided by the LVGP
model possess larger and lower maximum values with lower uncertainties at
different locations. This result could be significant and advantageous for
materials design applications as it can give a multi-step head start to the
design optimization task through more confident and accurate predictions.

Figure 6: Predictive performance comparison between (a) GP, (b) LVGP, and (c)
GP-GE models. The first row shows the parity plots of the models with respect
to their respective training and GE Data testing sets. The x and y axis show
the true and predicted hardness change (HV), respectively, and error bars
demonstrate the prediction uncertainty associated with each prediction.
Second, third and fourth row show the Mean, Standard Deviation and 3D Response
Surfaces of the models at 12%<Cr<24%, 0%<Al<7%, 88% <Fe<69%, Mo = 0%, Zr = 0%,
Mn = 0%, Si = 0%, W = 0%, Y = 0%, Ti = 0%, Temperature = 475∘C and Duration =
5000 hours, respectively

For research transparency, it is important to note that without further
experimental testing and validation in the highlighted compositional space, it
cannot be stated which model is a true and generalizable model. The goal of
the current work is to illustrate the existence of data discrepancies that
could harm multi-source modeling and how they can be integrated into modeling
using LVGP. However, the current research study demonstrates that using LVGP
as a multi-source data fusion model can be advantageous due to providing (1)
smoother response surfaces closer to real physical material behavior, and (2)
accurate and confident predictions which can be highly beneficial for
downstream tasks such as material optimization, sensitivity analysis, and
uncertainty quantification.

####  4.4.3 Interpretability from the Latent Variables

In addition to providing smoother response surfaces closer to the physical
behavior, the LVGP model provides interpretability into different sources
through the latent space and the quantified dissimilarity metric, $D$. The
obtained latent variables are shown in shown in Figure 7a. As previously
highlighted, the spatial relations (distances) between the information sources
can reveal certain information and provide further interpretability and
insights on the sources. For analysis purposes, the GE Data source is used as
reference source and compared with rest of the sources as GE Data source can
be re-manufactured, optimized, validated, and tested further. It is observed
from the latent space that the placement of the sources varies significantly
with respect to the reference source. Specifically, the top three sources of
data that are dissimilar (higher $D$ value) from the GE Data source are Yang,
Kobayashi, and Pinkas, whereas the top three sources of data that are similar
(lower $D$ values) to the GE Data source are Han, Kim, and Ejenstam.

The spatial differences in the latent space can be attributed to differences
in the input space, variations in the output space, or underlying fundamental
differences in the input-output relationships that cannot be captured from the
data. To investigate and interpret the latent spaces, a deeper study on the
input and output (property) space is conducted. First, a principal component
analysis (PCA) is performed to observe the differences in the input space.
Figure 7c shows that the two principal components of the GE Data source cover
and extend beyond the principal components of the other sources, indicating
that the neither the closeness or the remoteness of the sources to the GE Data
source in the latent space can be attributed to the differences in the input
space. Looking further into the output distributions (Figure 7d), it is seen
that the GE Data once again covers and extends beyond the properties of other
available sources, demonstrating that the differences in the latent space are
not due to the variabilities in the output space. As a result, the most
impactful observation made from the latent space of data sources is that there
are some fundamental differences between the data sources that are not
explicitly known. As mentioned earlier, these possible differences for the
current case can be due to differences in the manufacturing process of the
alloy, testing methods, measuring procedure, underlying phenomenon, and more.
Thus, it can be concluded that important underlying differences do exist
between the information sources and LVGP is not only able to capture these
variabilities but also learns the relationships between the sources.

A further validation study is done by separating the GE Data into two
different sources as GE Data 1 and GE Data 2 and retraining the LVGP model to
observe the latent space again. The aim of this validation study is to observe
the new latent space when the same underlying experimental procedures are
implemented for two resources along with other sources that have different
unknown underlying procedures. The new constructed latent space is shown in
Figure 7b. A significant observation from the validation study is that the
latent variables of GE Data 1 and GE Data 2 are located adjacent to each
other. This result was expected since the same experimental procedures are
implemented for both sources. However, the results also once again
demonstrated and validated that the LVGP model is able to capture the
differences in the known and unknown underlying parameters that influence the
material system and learn the relationships between the sources. This
remarkable capability provided by the LVGP can be highly important for further
downstream tasks including but not limited to model validation, dimension
reduction, sensitivity analysis, and more. It is important to highlight that
without using the LVGP approach proposed in this paper, it would be difficult
to learn and assess the relationships between the data sources.

Figure 7: (a) The latent Space of FeCrAl sources from LVGP. GE Data is denoted
as the reference source (*) for dissimilarity metric, $D$, calculations. (b)
The latent Space obtained from the interpretation validation study. GE Data 1
is denoted as the reference source (*). (c) Two principal components of the
input space and (d) output distribution of HV with respect to information
sources

####  4.4.4 Enhancing Modeling Capabilities Through Targeted Source Selection
Based On Latent Variables

It is generally acknowledged in the ML community that the prediction
performances of the models improve when more data is available. Thus, there
has been great efforts of data collection and curation in the ML community.
Although this notion of more the data better the model is generally true,
there could be cases where more data can actually hurt the ML models and
result in poor prediction results. This phenomenon can be especially true for
ML applications in materials science where the fundamentals of the generated
data, specifically for experimental data, are usually either unknown to the
model builders or not disclosed by the data generators due to many reasons
such as errors, confidentiality and so on. Therefore, a major challenge still
exists regarding the quality, and applicability of the available data. Thus,
during the ML model building process, researchers are more inclined to trust
and use the data they either generate themselves or they know the fundamentals
of. However, this comes with two challenges. First, generating materials data
whether experimental or simulation can be expensive and time consuming, which
could result in poor ML models further down the line due to low amounts of
available training samples. As a result, researchers also rely on other data
sources to build their models, in addition to using the data they generate
themselves. However, as mentioned previously, it is not always plausible to
know the quality and the fundamentals of the data generation process, meaning
that there could be notable differences and discrepancies in the data sources
which could significantly influence the performance of the ML models.

So far, it has been demonstrated that the LVGP modeling brings numerous
advantages to multi-source data fusion compared to some of the alternative
approaches. One of the key advantages of the LVGP comes from the physically
interpretable latent spaces that can possibly explain the relationships and
differences between different sources of information. As a result, it is
proposed that the latent space provided by the LVGP model can act as both a
validation and an anomaly detection mechanism to remove the data samples from
the sources that could potentially harm the ML model that will be used for
further optimization tasks on the source of information that one has control
over. For the data sources available in the FeCrAl alloy study, the authors
generated and only have further control over the GE Data source. Therefore,
the further material design optimization tasks can only take place using the
experimental procedures available at GE Research. However, to improve the
quality of the ML modelling, numerous data sources have also been collected
and used for LVGP model building. Looking at the latent spaces from LVGP model
in Figure 7a, it is observed that some data sources, Yang, Kobayashi, and
Pinkas to be specific, are located at a much further point (higher D values)
compared to the other sources when GE Data is selected as the reference point.
As discussed before, this result can indicate that the experimental procedures
taken place by those sources can be different than the ones that are
implemented for GE Data. However, this does not necessarily state that there
are fundamental errors or data quality issues regarding the three data sources
but could potentially indicate that the sources closer to the GE Data can
contribute to model accuracy whereas others can harm the ML model for
predictions on GE Data. Therefore, to investigate the influence of targeted
source selection on the separate testing GE Data set, the three sources that
are further from the GE Data (Yang, Kobayashi, and Pinkas), is removed from
the training sets and the LVGP model is re-trained. For ease of notation, the
new LVGP model is denoted as LVGP-Target. Table 4 shows the comparison between
the prediction accuracy made by the three models, LVGP, GP-GE, and LVGP-Target
on the separate 36 GE Data testing set. Although both LVGP and GP-GE have
already demonstrated remarkable accuracy results on the GE Data, LVGP-Target
provides notable improvement in accuracy, specifically reducing NRMSE by
almost 20% and 7% in comparison to LVGP and GP-GE models, respectively. The
parity plot of the predictions made by the LVGP-Target model are shown in
Supplementary Figure 2. Considering that the evaluation of properties through
experimentation can be relatively expensive, the notable increase in accuracy
provided by the targeted source selection through LVGP can provide notable
cost and time savings for further downstream tasks. Overall, in addition to
providing accurate data fusion modeling, LVGP can be employed as a validation
and data filtering mechanism to improve multi-source modeling.

##  5 Modeling Magnetic Behavior of SmCoFe Alloys

###  5.1 Overview

Electric and hybrid electric transportation has been garnering a lot of
attention in recent times. One of the key focuses has been developing more
denser and powerful motors to provide the propulsion capacity needed. To
increase the power density and capacity of the motor, the housed permanent
magnet plays an essential role. With the direct increase in magnetic
characteristics of fundamental magnetic material, high operational envelope of
motors can be achieved. An overview of performance of magnetic alloy systems
is given through [42]. The work identifies alloys that have higher magnetic
properties measured through saturation magnetization and that have higher
operating temperatures measured through Curie temperature. From the array of
magnetic materials, it was observed that SmCo (Samarium-Cobalt) alloy, also
referred as SmCoFe, exhibit relatively high stability at high temperatures
which is essential to withstand the hot and harsh environments of aircraft
engines. Hence the research community has favored the exploration of SmCoFe
systems for aerospace application.

There have been numerous efforts that seek to identify the composition and the
unique dopants that can improve the energy product envelope of SmCoFe systems
catering to hybrid electric flight applications. Similar to our previously
presented FeCrAl case study, with the advent of AI$\backslash$ML, data-driven
approaches have been extensively explored and used for optimization and
modeling. To demonstrate another key nuance and the differentiation of multi-
source modeling, magnetic behavior of SmCoFe alloys is used as the second
material science case study to address the restrictions about the quality of
data from multiple sources and unaccounted effects of known and unknown
underlying physical parameters.

###  5.2 Data Description

As with the previous materials case study, the first step is to collect data
from open research literature such as papers, patents, and other sources of
information that are restricted to SmCoFe alloy system. From an extensive
search, three primary sources of information were identified. Unfortunately,
the in-house data testing for SmCoFe systems is not readily available but
currently on-going. Table 5 summarizes the sources of data. A total of 71 data
points were collected. One distinction from the previous case study is that
the data from the one of the patents is further differentiated into two
information sources (Patent T2 and Patent T4). The differentiation was
necessitated by additional changes to the post-processing parameters that were
implemented.

Table 5: Sources of information for SmCoFe alloys  Name of Data Source |  Number of Data Samples |  Source of Data
---|---|---
Patent T2 |  $20$ |  Table 2 from Fujiwara, Teruhiko, Hiroaki Machida, and Hideyuki Yoshikawa. "Rare earth-cobalt permanent magnet." U.S. Patent Application No. 14/643,875. [43]
Patent T4 |  $21$ |  Table 4 from Fujiwara, Teruhiko, Hiroaki Machida, and Hideyuki Yoshikawa. "Rare earth-cobalt permanent magnet." U.S. Patent Application No. 14/643,875. [43]
Prop of SmCeCoFeCu |  $30$ |  Senno, Harufumi, and Yoshio, Tawara. "Permanent-magnet properties of Sm-Ce-Co-Fe-Cu alloys with compositions between 1-5 and 2-17." IEEE Transactions on Magnetics 10.2 (1974): 313-317. [44]
Total |  $71$ |

###  5.3 Model Development

Similar to the FeCrAl model development approach, GP and LVGP models are
trained on the collected data. The common inputs for the GP and LVGP models
are the composition of the alloy constituents (Samarium, Zirconium, Copper,
Iron, Cerium, Cobalt) defined as the weight percentages (wt%) and the testing
temperature of the alloy, leading to seven (7) unique input parameters. The
energy density and power capacity of the magnet are experimentally measured
and quantified through energy product (EP). As with the previous case study,
the inputs for the LVGP contain an additional qualitative input characterizing
each source of data as distinct categorical level along all the other 7 inputs
used for the GP model. The outputs for both the models remain the same.

###  5.4 Results and Discussion

Two separate studies are conducted as previously done with FeCrAl study.
Within the first study, generalizability, and the prediction capability of the
models through one source that could be controlled are investigate. In this
case, although there are no in-house experiments that could be controlled,
Prop of SmCeCoFeCu is selected as in-house experiments (reference information
source) as it is the source with the highest number of available samples.
First, generalizability of GP and LVGP is compared through 5-fold cross
validation using the 65 samples out of 71 available samples, and testing the
models over a 6 sample separate testing set that are selected from the Prop of
SmCeCoFeCu source. For CV tasks in LVGP, stratified sampling is used to make
sure datasets used for training and testing represent the overall distribution
of the information sources. The training dataset and testing dataset remain
identical for both models. The results of this study are highlighted in
Section 5.4.1.

In the second study, a separate single-source GP model is built using only the
Prop of SmCeCoFeCu source and compared with the multi-source LVGP model. The
performances of the models are compared according to the same 6 separate
testing set. For ease of notation, this single source GP model is named as GP-
Prop. The results of this study are demonstrated in Section 5.4.2.

####  5.4.1 Multi-Source Data Fusion Comparision: LVGP and GP

Table 6 summarizes the performance of the GP and LVGP model on the SmCoFe
alloy dataset. From the comparison of the NRMSE across the training and cross
validation datasets, it can be observed that the multi-source LVGP model shows
improved performance over the GP model. Therefore, it can be concluded that
explicitly including the source of data could enable accurate, efficient, and
generalizable models. Looking at the testing NRMSE metric, it is observed that
GP model performs slightly better compared to the LVGP model. The parity plots
from the CV study are also shown in Supplementary Figure 3. Moreover, from
45-degree parity plots, it is observed that both models make similar
predictions on both the testing and training set (Panels A and B of Figure 8).
On the other hand, it can be clearly observed that LVGP provides lower
prediction uncertainties as the predictions have lower confidence bounds. Once
again, this could be highly advantageous for further downstream materials
design optimization studies as the information from other sources can provide
a more informed and confident ML model.

Table 6: Predictive performance comparison between GP, LVGP, and GP-Prop models on SmCoFe Alloys  Model Type |  Mean Training NRMSE |  Mean CV NRMSE |  Testing NRMSE
---|---|---|---
GP |  $0.100$ |  $0.134$ |  $0.160$
|  |  |
LVGP |  $0.102$ |  $0.128$ |  $0.165$
|  |  |
GP-Prop |  — |  — |  $0.326$
Figure 8: Predictive performance comparison between (a) GP, (b) LVGP, and (c)
GP-Prop models. First row shows the parity plots of the models with respect to
their respective training and Prop SmCeCoFeCu testing sets. The x and y axis
show the true and predicted energy potential (EP), respectively, and error
bars demonstrate the prediction uncertainty associated with each prediction.
Second, third and fourth row show the Mean, Standard Deviation and 3D Response
Surfaces of the models at 23%<Sm<26%, 55%<Co<63%, 1.0%<Zr<2.0%, Cu = 8.0%, Fe
= 19.0%, Hf = 0%, Ce = 0% and Temperature = 25∘C, respectively

Extending the investigation of the models, the response surface from both the
models are plotted across the same input space of the Prop SmCeCoFeCu source
through Panels A-B in Figure 8. The response surfaces (with Prop SmCeCoFeCu as
reference due to details and nuances provided in the source) of the models are
smooth and well-behaved. The well-behaved nature of the response surface can
be attributed to the low dissimilarity measure of each source given through
Figure 9. This could indicate that all the sources of information possess high
similarities. It is highlighted that since the experiments for Patent T2 and
Patent T4 have been provided through the same Patent [43], it is plausible to
assume that the underlying unknown physical experimental parameters are the
same for both sources of information and that LVGP was able to identify this
explicitly. Furthermore, in addition to providing better predictions, a case
is still made for the need of multi-source data fusion through LVGP. Within
the highlighted compositional space of the response surface, the location of
maximum energy product from GP and LVGP are different. For the GP model, the
maximum energy product is around 17.75 MGOe with a prediction uncertainty of
around 2.5 MGOe, approximately at Co(wt%) = 60. For LVGP models centered at
Prop SmCeCoFeCu, the maximum energy product is around 18.00 MGOe with an
uncertainty of 2.25 MGOe at approximately Co(wt%) = 63. This nuanced
difference is seemingly tied to the source of the data as datasets and models
are identical except for inclusion of data source. Although the mean
predictions provided by the LVGP are slightly higher compared to the GP model,
the predictions come with much lower uncertainty. Furthermore, for the cases
where the designer has access and control over one source of information for
further design optimization, the optimum candidates provided by the GP model
can be misleading and can misguide the researchers towards wrong optimum
designs since there is no way to explicitly differentiate the information
sources. Therefore, the reference source selection capability present in the
LVGP plays an essential role in plausibly guaranteeing the optimized behavior
as it is tied to the most informative source of data (identical procedures can
be followed). Additionally, within the compositional space there is a
difference of 3% in Co (wt%) for the maximum EP property between both models.
This shift in the composition of a key rare earth element can have a
significant impact in realizing such a superior alloy and downstream cost and
supply chain setup.

Figure 9: The latent space of SmCoFe information sources from the LVGP model.
Prop SmCeCoFeCu is denoted as the reference source (*) for dissimilarity
metric, $D$, calculations

####  5.4.2 Comparision of the Multi-Source LVGP with the Single-Source GP-
Prop

To overcome the challenge of performing optimization in the unknown
information source space, one can argue to build GP models on each information
source individually. To demonstrate the effectiveness of multi-source LVGP, it
is compared against a single-source GP model that is built on the Prop
SmCeCoFeCu source only. The single-source GP model, which is previously
denoted as GP-Prop, uses the entire available training data from the Prop
SmCeCoFeCu source (24 samples) for model building, and later tested on the
separate 6 samples obtained from the Prop SmCeCoFeCu once again. The
prediction metrics and the parity plots on testing data are shown in Table 6
(Model Types: LVGP and GP-Prop) and 9 b&c, respectively. From the NRMSE and
parity plot results, a clear advantage of using LVGP is observed as it
provides much better predictions with lower prediction uncertainties.
Furthermore, looking at the response surfaces of both models, Panels B and C
of Figure 8, a tone stark difference is seen. At Co (wt%)>61 (approximately),
the GP-Prop model reverts to mean predictions with high prediction
uncertainty, which is a common behavior observed in GP models when trying to
obtain prediction for out-of-bounds inputs in the training data. However, when
different sources of information are incorporated with non-overlapping input
space through LVGP, the model is able to perform extrapolation with regard to
the reference source. This phenomenon could be a crucial differentiator when
designing next generation materials with restrictions on data-availability
driven by manufacturing and experimental testing capabilities.

For full transparency purposes, it is once again highlighted that without
further experimental testing and validation in the highlighted compositional
space, it cannot be confidently stated which model is a true and generalizable
model. However, such subtle nuances and variations are observed through known
and unknown underlying physical parameters can only be captured by a multi-
source modeling data fusion framework built through LVGP. Furthermore, the
current research study demonstrates that using LVGP as a multi-source model
can be advantageous, especially when low amount of information exists about
the material system of interest as the model provides smoother response
surfaces closer to real physical material behavior, and accurate predictions
that can be significantly beneficial for downstream tasks.

##  6 Conclusion

AI and ML frameworks have had a strong impact in the science and engineering
communities. On the other hand, with the ever-rising application of data-
driven surrogates build on multiple sources of data that it is gathered either
experimentally or computationally, not much attention has been paid to the
quality, completeness, and comprehensiveness of the excluded and unknown
underlying parameters involved in the data generation process. These
parameters vary from differences in manufacturing process, testing technique,
and more. Through the research done in this paper, an interpretable multi-
source data fusion framework based on Latent Variable Gaussian Process (LVGP)
is proposed and extensively studied. Through LVGP models, a categorical
variable was introduced to represent sources of information during model
training and predictions. A low-dimensional latent space is then learned to
incorporate the categorical source variable into multi-source data fusion.
Through latent variable representation, both the excluded and underlying
unknown physical parameter variations among the different sources are captured
and introduced into the modelling, making the multi-source ML model
interpretable and aware of the differences and similarities between the
information sources. The inclusion of the data sources not only leads to more
accurate models but also enables the study of source specific response
surfaces, material optimization, and sensitivity analysis. Compared to using
single source data for ML, the proposed data fusion method using multi-source
information can provide a better prediction for sparse-data problems by taking
advantage of the information of correlations and relationships among different
data sources. With the quantified uncertainty from LVGP for the specified
information source, the approach can further guide adaptive sampling or active
learning when allocating resources for data collection. Additionally, a novel
dissimilarity metric is also defined through the latent variables of the LVGP
models that can help understand the aggregated differences and similarities in
the information sources. The proposed approach is applied on and analyzed
through two mathematical (representative parabola and 2D Ackley function) and
two materials design (design of FeCrAl alloys and design of SmCoFe alloys)
case studies. Finally, a novel application of the framework would involve
performing materials design and development through autonomous experimentation
studies where multiple sources of experimental data are gathered to guide the
study. It is envisioned that the proposed multi-source framework is not
limited to experimental or computational information sources and can be
further extended to mixture of experimental and computational sources in
numerous engineering and science applications.

## Acknowledgments

This work is supported by the US Department of Energy, National Nuclear
Security Administration, under award number DE-NE0009047. Northwestern
researchers are grateful for the support from NSF 2219489 and NSF 2037026.
This material is based upon work supported by the U.S. Department of Energy’s
Office of Energy Efficiency and Renewable Energy (EERE) under the Advanced
Manufacturing Office, Award Number DE-AC0206H11357. The views expressed herein
do not necessarily represent the views of the U.S. Department of Energy or the
United States Government.

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## Supplementary Information

### Supplementary Figures

Figure 1: Cross Validation performance on FeCrAl Alloy dataset using (a) GP
and (b) LVGP models. The x and y axis show the true and predicted hardness
change (HV), respectively, and error bars demonstrate the prediction
uncertainity associated with each prediction Figure 2: Training and testing
performance on FeCrAl Alloy dataset using the LVGP-Target model. The x and y
axis show the true and predicted hardness change (HV), respectively, and error
bars demonstrate the prediction uncertainity associated with each prediction
Figure 3: Cross Validation performance on SmCoFe Alloy dataset using (a) GP
and (b) LVGP models. The x and y axis show the true and predicted energy
potential (EP), respectively, and error bars demonstrate the prediction
uncertainity associated with each prediction

### Supplementary Tables

Table 1: Sources of Information for the Parabola Problem  Information Source | $a$ | $b$ | $x_{shift}$ | $y_{shift}$ | Number of Training Samples | Number of Testing Samples
---|---|---|---|---|---|---
Ground Source | $1$ | $2$ | $0$ | $0$ | $2$ | $30$
Perturbed Source - 1 | $1$ | $2$ | $8$ | $0$ | $10$ | $30$
Perturbed Source - 2 | $1$ | $2$ | $0$ | $100$ | $10$ | $30$
Perturbed Source - 3 | $1$ | $2$ | $12$ | $120$ | $10$ | $30$
Table 2: Sources of Information for the 2D Ackley Problem  Information Source |  Function Formulation $(a=20,b=0.2,c=2)$ |  Number of Training Samples |  Number of Testing Samples
---|---|---|---
Ground Source |  $z=-a*exp(-b(\sqrt{\frac{1}{2}(x^{2}-y^{2})}))-exp(\frac{1}{2}(cos(cx)+cos(cy)))+a+exp(1)$ |  $20$ |  $100$
Perturbed Source - 1 |  $-a*exp(-b(\sqrt{\frac{1}{2}(x^{2}-y^{2})}))+10$ |  $50$ |  $100$
Perturbed Source - 2 |  $exp(\frac{1}{2}(cos(cx)+cos(xy)))+5$ |  $50$ |  $100$
Perturbed Source - 3 |  $\frac{1}{4}(-a*exp(-b(\sqrt{\frac{1}{2}(x^{2}-y^{2})})))-\frac{3}{4}(exp(\frac{1}{2}(cos(cx)+cos(cy))))+a+exp(1)$ |  $50$ |  $100$