# Probabilistic Physics-integrated Neural Differentiable Modeling for
Isothermal Chemical Vapor Infiltration Process

Deepak Akhare  Zeping Chen  Richard Gulotty  Tengfei Luo  Jian-Xun Wang
Department of Aerospace and Mechanical Engineering, University of Notre Dame,
Notre Dame, IN, USA  Center for Sustainable Energy (ND Energy), University of
Notre Dame, Notre Dame, IN, USA  Department of Chemical and Biomolecular
Engineering, University of Notre Dame, Notre Dame, IN, USA  Lucy Family
Institute for Data & Society, University of Notre Dame, Notre Dame, IN, USA
Honeywell International Inc., Phoenix, AZ, USA

###### Abstract

Chemical vapor infiltration (CVI) is a widely adopted manufacturing technique
used in producing carbon-carbon and carbon-silicon carbide composites. These
materials are especially valued in the aerospace and automotive industries for
their robust strength and lightweight characteristics. The densification
process during CVI critically influences the final performance, quality, and
consistency of these composite materials. Experimentally optimizing the CVI
processes is challenging due to long experimental time and large optimization
space. To address these challenges, this work takes a modeling-centric
approach. Due to the complexities and limited experimental data of the
isothermal CVI densification process, we have developed a data-driven
predictive model using the physics-integrated neural differentiable (PiNDiff)
modeling framework. An uncertainty quantification feature has been embedded
within the PiNDiff method, bolstering the model’s reliability and robustness.
Through comprehensive numerical experiments involving both synthetic and real-
world manufacturing data, the proposed method showcases its capability in
modeling densification during the CVI process. This research highlights the
potential of the PiNDiff framework as an instrumental tool for advancing our
understanding, simulation, and optimization of the CVI manufacturing process,
particularly when faced with sparse data and an incomplete description of the
underlying physics.

###### keywords:

Hybrid model , Differentiable Programming , Neural Networks , Scientific
Machine Learning , Operator Learning

††journal: Elsevier

##  1 Introduction

Carbon-carbon (C/C) and carbon-silicon carbide (C/SiC) composites are
recognized as vital materials for applications exposed to extreme thermal
conditions, notably in contexts such as airplane brake discs, reentry vehicle
heat shields, and rocket engine nozzles. These composites are distinguished by
their exceptional thermal stability and superior mechanical properties under
high temperatures. Their fabrication is largely achieved via the isothermal
chemical vapor infiltration (I-CVI) process, which is known for its precision
in defining final properties, its adaptability in accommodating complex
geometries, and its ability to yield final products that are both lightweight
and thermally robust. Since the performance, consistency, and quality of these
composites are largely influenced by the manufacturing procedures, optimizing
the I-CVI process is of great importance [1, 2]. However, this is not a
trivial task and presents significant challenges, given the long duration of
I-CVI, often lasting months, rendering conventional trial-and-error approaches
infeasible [3].

The inherently time-consuming nature of the I-CVI process necessitates the
creation of a predictive model that enables efficient computer-based
simulations. This strategic shift offers the potential for markedly reduced
turnover times, allowing for a comprehensive exploration of varied
manufacturing conditions. A reliable and efficient computer-based predictive
model can enable us to effectively optimize the I-CVI process, exerting finer
control over its intricacies, with the aim of reducing production cycles and
ensuring resultant materials meet targeted properties. Although several
numerical models, particularly those focusing on Carbon and SiC deposits, have
been developed for the I-CVI process in the literature [4, 5, 6, 7, 8], they
face inherent challenges. These models often rely on multiple assumptions due
to unresolved/unknown physical phenomena, which compromise their accuracy and
reliability. Most of the time, these models are tailored to specific
processes, limiting their adaptability to other new process designs. Given the
multifaceted physical interactions of I-CVI and its nuanced mechanical and
chemical mechanisms, a purely physics-based modeling approach appears
impractical and non-scalable. Moreover, the development and simulation of
these models demand substantial computational resources, restricting their
immediate applicability for optimization and uncertainty assessments.

To effectively simulate and optimize the I-CVI process, the creation of a new
modeling framework that is both generally applicable and scalable is
indispensable. The growing availability of data offers promising avenues for
machine learning (ML)-based, data-driven modeling techniques to make
substantial advancements in this regard. Deep neural networks (DNNs),
cornerstones of ML, have long been recognized for their efficacy in diverse
fields, ranging from speech recognition and image analysis to natural language
processing. Harnessing the rapid advancements in computational resources and
algorithmic developments, along with a vast accumulation of experimental and
simulation data, DNNs have been established as essential tools in the domain
of scientific modeling and simulation. Their impact is notably evident in the
field of scientific machine learning (SciML), which is progressively being
adopted in predictive composite manufacturing [9, 10, 11]. Through SciML,
researchers have been able to identify previously unknown constitutive
relationships of composite materials and have optimized the speed and
efficiency of multiscale simulations [10]. Wang et al. [9], for instance,
utilized ML techniques to analyze patterns within additive manufacturing
datasets and developed models that describe process-structure-property
relationships across different parameters. Huang et al. [11] highlighted the
advantages of their data-driven model, which was trained on a previously
published experimental dataset, in predicting the mechanical characteristics
of carbon nanotube-reinforced cement composites, especially when compared to
traditional response surface methods. Nguyen et al. [12] investigated the
impact of curing on stress development and tensile transverse failure response
by integrating a thermo-chemo-mechanical finite element model with a DNN-based
constitutive model. This allowed them to understand matrix mechanical property
changes in relation to temperature and curing levels. Kopal et al.[13]
employed regression neural networks to predict the curing properties of rubber
blends enriched with carbon black, considering both blend types and curing
temperatures. Baek et al.[14] employed graph neural networks to study the
effects of nanoparticle distribution patterns and agglomeration phenomena in
polymers. Li et al.[15] designed a data-driven DNN model to learn the
nonlinear interdependencies between I-CVI process parameters and the resulting
physical properties of C/C composites, which can be used for process
optimization.

While these pure data-driven DNN methods have been successful across diverse
applications in composite manufacturing modeling, there is a noticeable gap in
the literature concerning their use for modeling the CVI process. One of the
primary obstacles is the substantial data requirement intrinsic to these
models. Collecting sufficient data to effectively train and utilize a purely
data-driven DNN model for CVI optimization can be extremely challenging, if
not infeasible. Another limitation is the restricted generalizability of these
models, where failures were often observed beyond the training regimes [10],
rendering them less suited for complex CVI processes that operate under a
spectrum of manufacturing conditions. The physics-informed deep learning
(PIDL) strategy, leveraging physics principles to enhance the design,
training, or inference of DNNs, presents a promising potential solution to
these challenges. It is worth noting that the PIDL approach has garnered
attention and demonstrated considerable potential across various domains,
including solid mechanics [16, 17, 18], turbulent flows [19, 20, 21],
materials [22], heat transfer [23, 24, 25, 26, 27, 28, 29] , and biomechanics
[30, 31, 32]. A notable PIDL method is the Physics-informed neural network
(PINN) [33, 34, 35, 36, 37, 24], wherein the governing equations are
incorporated into the loss function to regularize the training process,
thereby reducing the required labeled data. However, the introduction of
nonlinearity into the loss function can pose significant challenges in
optimization [38], and the physics-based loss function requires complete
governing equations of the underlying physics, which are not available for the
intricate I-CVI processes.

To integrate incomplete physics with deep learning, the hybrid differentiable
neural modeling emerges as a notable alternative PIDL method, which fuses
physics-derived mathematical models with the robust learning capability of
neural networks, ensuring efficient learning even with limited data.
Differentiable programming (DP) serves as a cornerstone in this framework,
allowing for the joint optimization of both DNNs and physics-centric
components within a unified training environment. The development of
differentiable physics solvers and hybrid neural models has recently gained
traction, exemplifying their adaptability across various scientific fields
[39, 40, 41, 42, 43, 44, 45, 46]. Notably, Akhare et al. [44] introduced a
physics-integrated neural differentiable (PiNDiff) framework, developed for
the curing process of composites. This approach seamlessly integrates
partially-known physics into neural networks while preserving the mathematical
integrity of governing equations through DP. Akhare and co-workers
demonstrated the effectiveness of the PiNDiff method in capturing the
interplay between heat transfer and curing dynamics [44]. Driven by its
potential, our objective in this work is to employ the PiNDiff framework to
develop a predictive model for the I-CVI process in fabricating C/C
composites, highlighting its ability to work with partially-known physics and
sparse indirect measurements. Extending from its original formulation, our
study broadens the PiNDiff’s scope to the I-CVI process, governed by both
hyperbolic and elliptic partial differential equations (PDEs). A primary
thrust of this effort is to distill the existing physics-based model into a
more concise PDE system that serves as the physics-based backbone of the
PiNDiff model. However, this simplification, along with the potential over-
parameterization of neural operator components, may introduce errors in
predictions. To address this issue, we extend the PiNDiff framework by
integrating uncertainty quantification (UQ) features, utilizing Deep Ensemble
(DeepEn) techniques [47]. Consequently, our probabilistic PiNDiff I-CVI model
is not only able to learn and predict I-CVI processes with limited training
data but also to gauge the confidence of its predictions. This development
facilitates a more comprehensive evaluation of the model’s reliability,
thereby guiding decision-making processes more adeptly. The rest of the paper
is organized as follows: the overall methodology of the PiNDiff I-CVI model is
introduced in Section 2. Numerical experiments of the PiNDiff I-CVI model and
its comparison with the experimental data are presented in Section 3. Finally,
Section 4 concludes the paper.

##  2 Methodology

###  2.1 Isothermal Chemical Vapor Infiltration (I-CVI) process: an overview

CVI stands as a pivotal process in the production of composite materials,
notably C/C and C/SiC composites. The core principle behind this process is
the infiltration of a porous preform using reactive gases, which undergo
chemical reactions to deposit solid material into the interstitial spaces of
the preform’s structure (known as the matrix), thereby resulting in a
reinforced composite. The isothermal CVI variant, known as I-CVI,
distinguishes itself by maintaining a constant temperature throughout the
infiltration process. A typical I-CVI procedure unfolds as follows:

  * •

Preform Design and Creation: Using materials such as carbon fibers, a porous
preform is meticulously crafted. Its structure is designed to embody the
desired shape, serving as the foundational framework for the final composite.

  * •

Reactor Chamber Setup: The crafted preform is situated within a specialized
reactor chamber for the I-CVI procedure. This chamber is not only constructed
to endure high temperatures but also engineered to regulate gas flow during
infiltration.

  * •

Gas Selection and Introduction: Depending on the intended composition of the
resulting composite, specific reactive gases, such as hydrocarbons or silanes,
are channeled into the chamber. These gases contain the essential elements
required for the desired chemical reactions and material deposition.

  * •

Chemical Reactions & Material Deposition: As these gases permeate the preform,
they undergo thermal reactions, especially on the surface of the preform. Such
chemical reactions facilitate the progressive deposition of solid
material–commonly carbon or a carbon-silicon carbide mix–into the matrix of
the preform layer by layer. This deposition methodically fills the preform’s
voids, thereby reinforcing its structure.

  * •

Regulation of Temperature and Duration: The I-CVI process is generally
executed at a consistently elevated temperature, frequently between $900^{o}$
C to $1200^{o}$ C. Both the temperature and the process duration are
meticulously regulated to guarantee accurate deposition and desired material
characteristics. Depending on the intended composite thickness and properties,
the procedure’s duration can range from a few hours to multiple days.

  * •

Post-Process Refinement/Machining: Following the main I-CVI procedure, the
produced composite might necessitate further refinements, such as surface
treatments or precise machining, to fine-tune its properties and dimensions.

The I-CVI process involves complex multiphysical and multichemical phenomena
operating over multiple spatial and temporal scales. These include fluid
dynamics, homogeneous gas-phase reactions, as well as intricate heterogeneous
reactions leading to surface deposition. A comprehensive schematic
illustrating the dominant chemical reactions for hydrocarbon deposition, along
with the varying scales of these multiphysics phenomena, is presented in Fig.
1.

Figure 1: A schematic illustrating the I-CVI process. Depicted are the
dominant chemical reactions specific to hydrocarbon deposition, integrated
with visual representations of the varying scales of multiphysical and
multichemical phenomena inherent in the process.

###  2.2 Physics-based modeling for I-CVI process and challenges

The I-CVI process encompasses multiple interconnected physical phenomena,
including gas flow, diffusion, and both homogeneous and heterogeneous
reactions, which can be mathematically modeled as a set of partial
differential equations (PDEs) describing the mass and momentum conservation
[48, 49]:

  *   * •

Mass balance (reaction-diffusion): This equation portrays the interplay
between the reaction and diffusion of gas-phase species,

$\frac{\partial(\varepsilon_{k}\rho_{k})}{\partial
t}+\nabla\cdot(\varepsilon_{k}\rho_{k}\textbf{u})=D_{k}\nabla^{2}(\varepsilon_{k}\rho_{k})+\dot{\omega}_{k}\
\ k=1,...,N_{g}.$ (1a)

  * •

Mass balance (deposition): This equation captures the dynamics of species that
contribute to deposition,

$\frac{\partial(\varepsilon_{k}\rho_{k})}{\partial
t}+\nabla\cdot(\varepsilon_{k}\rho_{k}\textbf{u}_{k})=D_{k}\nabla^{2}(\varepsilon_{k}\rho_{k})+\dot{\omega}_{k}\
\ k=1,...,N_{s}.$ (1b)

  * •

Momentum balance for gas species: This encapsulates the conservation of
momentum for gas-phase species:

$\frac{\partial(\rho_{g}\textbf{u})}{\partial
t}+\nabla\cdot(\rho_{g}\textbf{u}\textbf{u})=\nabla.\boldsymbol{\sigma}+\rho_{g}\textbf{g}+\textbf{F}.$
(1c)

In the above equations, $N=N_{g}+N_{s}$ is the total number of species, where
$N_{g}$ is the number of species in the gas phase and $N_{s}$ is the number of
species resulting in deposition; $\varepsilon_{k}$ represents the volume
fraction of the $k$-th species, $\rho_{k}$ and $\rho_{g}$ is the average
intrinsic mass density of the $k$-th species and gas, $D_{k}$ is the diffusion
coefficient of the $k$-th species, and $\dot{\omega}_{k}$ is the production
rate of the $k$-th species; u represents the gas velocity,
$\boldsymbol{\sigma}$ denotes the total stress tensor, g is the external body
force per unit mass, and F is the macro-scale momentum force.

While the I-CVI process can be described by an extensive set of physics
principles, using numerical techniques to simulate these intricate phenomena
poses substantial challenges in practice. At the forefront of these challenges
is the need for a comprehensive understanding of the dominant reactions and
key parameters, such as the collision integral, crucial for determining the
diffusion coefficient $D_{k}$ for all species. Such detailed information,
however, often remains elusive in many real-world applications. Moreover,
translating these foundational physics concepts into predictive simulations
demands numerically solving N+1 sets of mass and momentum PDEs to capture the
behavior of N species, thereby compounding the complexity. A noteworthy caveat
is that the most rigorous physics-based models often fall short in capturing
the actual measurements, possibly due to an incomplete understanding of the
underlying physics. For instance, Kang et al. [50] compared the experimental
results of SiC densification with predictions from physics-based numerical
models that incorporated a variety of experimentally established reaction
equations. The models, however, failed to accurately match the experimental
data. On successfully constructing a numerical model, one is still confronted
with its intractable computational costs, making it infeasible for tasks such
as optimization and UQ. Additionally, these models are often tailored for a
specific precursor gas, limiting their applicability to that particular
substance and requiring reconfiguration for alternative precursor gases.

###  2.3 Probabilistic physics-integrated neural differentiable modeling for
I-CVI process

To address the challenges highlighted above, the physics-integrated neural
differentiable (PiNDiff) modeling framework, as proposed by Akhare et al.
[44], emerges as an innovative approach for constructing an effective
predictive surrogate model for the I-CVI process. This strategy leverages both
the established knowledge of physics and the insights learned from sparse,
indirect data. Central to the PiNDiff modeling framework is its seamless
fusion of deep learning capabilities with foundational physics principles,
resulting in a hybrid neural solver. Within the PiNDiff module, trainable DNNs
are utilized to learn unresolved/undetermined physics, while the non-trainable
networks are pre-determined by the PDE operators from the partially known
physics. Notably, the entire architecture of this hybrid neural solver is
fully differentiable, facilitating its holistic training and optimization. To
account for data scarcity and inherent model-form error, the PiNDiff framework
is extended in this work to equip it with UQ capability.

####  2.3.1 PiNDiff model architecture design

To construct a PiNDiff surrogate model for the I-CVI process, the major
foundational physics can be distilled based on a set of assumptions to
streamline the modeling process. Specifically, (a) the flow outside the
fibrous preform is considered to have negligible influence on the diffusion
process or the concentration of the precursor gas at the boundary of the
preform; (b) flow within the fibrous preform is assumed to be negligible,
suggesting that the transportation of the precursor gas is predominantly
through diffusion; and (c) instead of simulating multiple species, an
aggregate representation is used, introducing an effective species responsible
for both diffusion and deposition processes [6]. Operating under these
assumptions allows for a more simplified neural modeling strategy.
Specifically, the flow outside the preform becomes irrelevant to the model,
restricting the domain of interest solely to the preform itself. Additionally,
the foundational physics is substantially reduced to two PDEs: one for
reaction-diffusion and another for deposition. By invoking a quasi-steady-
state approximation, the time derivative term in the reaction-diffusion
equation can be eliminated, further simplifying the governing equations [6].
For the deposition reactions, a first-order rate expression, given by
$\dot{\omega}=-M_{d}KS_{v}C$, is adopted [6, 50]. Finally, the two PDEs can be
derived as follows (see derivation details in B),

Reaction-Diffusion mass balance equation: $D_{eff}\nabla^{2}(C)=KS_{v}C,$ (2a)
Deposition mass balance equation:
$\rho_{d}\frac{d\varepsilon}{dt}=-qM_{d}KS_{v}C.$ (2b)

In the above equations, $C=C(\mathbf{x},t)$ denotes the effective molarity
field (mol m-3) of all reactive gases, $\varepsilon=\varepsilon(\mathbf{x},t)$
is the porosity of the preform, $q$ represent a constant stichometric
coefficient, $M_{d}$ is the molar mass (kg mol-1), and $\rho_{d}$ is the
density (kg m-3) of the deposited solid (carbon or SiC).
$D_{eff}=D_{eff}(\mathbf{x},t)$ represents the effective diffusion coefficient
field, $K=K(\mathbf{x},t)$ is the deposition reaction rate, and
$S_{v}=S_{v}(\mathbf{x},t)$ corresponds to the surface-to-volume ratio.

Figure 2: Extracted foundational physics of the I-CVI process with neural
operator approximations.

Deriving constitutive relations for the spatio-temporal functions
$D_{eff}(\mathbf{x},t)$, $K(\mathbf{x},t)$, and $S_{v}(\mathbf{x},t)$ poses
significant challenging. Specifically, the models for $D_{eff}$ and $K$ are
sensitive to the choice of precursor gas, while $S_{v}$ is influenced by the
architecture of fibers and pores. Given these intricacies, we propose to learn
these constitutive relations from sparse measurements within the PiNDiff
framework. Namely, these constitutive relations are essentially unknown
mappings between different spatio-temporal functions, which can be
approximated by trainable neural operators. Specifically, the effective
diffusion coefficient field $D_{eff}(\mathbf{x},t)$ and deposition rate field
$K(\mathbf{x},t)$ are modeled as,

$\displaystyle D_{eff}(\mathbf{x},t)$
$\displaystyle\approx\mathcal{D}_{nn}\big{[}\varepsilon(\mathbf{x},t),T(\mathbf{x},t),P(\mathbf{x},t);\boldsymbol{\theta}_{D_{eff}}\big{]},$
(3a) $\displaystyle K(\mathbf{x},t)$
$\displaystyle\approx\mathcal{K}_{nn}\big{[}T(\mathbf{x},t);\boldsymbol{\theta}_{K}\big{]},$
(3b)

where $\mathcal{D}_{nn}$ and $\mathcal{K}_{nn}$ are neural operators with
trainable parameters $\boldsymbol{\theta}_{D_{eff}}$ and
$\boldsymbol{\theta}_{K}$, respectively. As for the effective surface area
$S_{v}(\mathbf{x},t)$, an analytical model, $\hat{S}_{v}(\mathbf{x},t)$, is
leveraged as the base model. To accommodate a broader range of fiber
configurations, a trainable neural operator $\mathcal{S}_{nn}$ with trainable
parameters $\boldsymbol{\theta}_{S_{v}}$, is introduced to augment and
generalize the base model as follows:

$\displaystyle S_{v}(\mathbf{x},t)$
$\displaystyle\approx\hat{S}_{v}(\mathbf{x},t)(1+\tilde{S}_{v}(\mathbf{x},t)),$
(4a) $\displaystyle\tilde{S}_{v}(\mathbf{x},t)$
$\displaystyle=\mathcal{S}_{nn}[\hat{S}_{v}(\mathbf{x},t),\varepsilon(\mathbf{x},t);\boldsymbol{\theta}_{S_{v}}],$
(4b) $\displaystyle\hat{S}_{v}(\mathbf{x},t)$
$\displaystyle=2\frac{\varepsilon}{\varepsilon_{0}}\frac{1-\varepsilon_{0}}{r_{f}}$
(4c)

where $\varepsilon_{0}$ is the initial porosity of the preform and $r_{f}$ is
the radius of fibre filament (m).

In Fig. 3, the PiNDiff I-CVI model is presented, revealing a seamless
integration of known physics with trainable neural operators. These operators
are strategically interconnected, drawing inspiration from the foundational
principles delineated by the simplified governing PDEs in Eq.2b. Within this
architectural framework, the DNNs operate in a pointwise fashion, effectively
learning continuous functions/operators. This design ensures both mesh and
domain independence, enhancing the model’s adaptability and versatility.

(a) Auto-regressive PiNDiff prediction model for I-CVI process model

(b) one-step PiNDiff module for I-CVI process model

Figure 3: (a) The overview of the auto-regressive learning architecture of the
PiNDiff model and (b) Zoom-in view of the PiNDiff module for one time-step
prediction (fu: function).

As the building blocks, the PiNDiff I-CVI module (Fig. 3b) operates as a
sequential neural predictor, capturing the evolution of molarity ($C$),
porosity ($\varepsilon$), and composite density ($\rho_{c}$) fields from time
step $t$ to $t+1$. This module incorporates trainable neural operators
designed to approximate the spatiotemporal fields, namely
$D_{eff}(\mathbf{x},t)$, $K(\mathbf{x},t)$, and $S_{v}(\mathbf{x},t)$. These
operators are interlinked by the discretized governing PDEs with finite
difference methods. A unique aspect of this design is the harmonious merger of
neural networks and numerical PDEs via differentiable programming, ensuring
efficient gradient propagation throughout the model (detailed in the next
subsection). Upon domain-wide integration, the model is capable of predicting
pertinent variables, notably the mass or weight of the materials involved. The
ability to predict such metrics is of significant importance, especially given
that monitoring material mass/weight is a common practice in manufacturing
processes. Leveraging the PiNDiff I-CVI module, we can construct sequential
neural networks using auto-regressive techniques, adeptly capturing the
system’s temporal dynamics, as illustrated in Fig. 3a, highlighting the
structured sequential learning for the I-CVI process.

###  2.4 Auto-regressive training of PiNDiff model via differentiable
programming

To ensure its robust long-term forecast capabilities, the PiNDiff I-CVI model
undergoes auto-regressive training throughout the entire rollout sequence,
which is achieved by differentiable programming ($\partial$P) [39], a
generalized concept of deep learning. Namely, the entire computer program is
architected for end-to-end differentiability. In the construction of the
PiNDiff I-CVI model, we utilize the automatic differentiation (AD) engine
provided by JAX [51] to propagate gradients through the hybrid model, enabling
the optimization of all trainable parameters, denoted as
$\boldsymbol{\theta}=\\{\boldsymbol{\theta}_{D_{eff}},\boldsymbol{\theta}_{K},\boldsymbol{\theta}_{S_{v}}\\}$,
through stochastic gradient descent techniques. Notably, in contrast to
traditional deep learning paradigms, this differentiable framework offers the
flexibility to train the model using indirect labels, even if the state
variables of interest are not directly observable.

In the context of I-CVI, experimentally monitoring the states such as porosity
$\varepsilon(\mathbf{x},t)$ and molarity $C(\mathbf{x},t)$ during the
manufacturing process is impractical. Meanwhile, hidden physics such as
effective diffusion, reaction, and deposition mechanisms are not observable at
all. Nonetheless, acquiring measurements for the mass of either the entire
sample or its segments is considerably more feasible. Consequently, mass
measurements ($\tilde{\mathbf{m}}_{t}$) at a few different time step $t$
serves as indirect labeled data for training the PiNDiff I-CVI model. The
associated loss function, $\mathcal{L}$, is then defined as,

$\mathcal{L}(\boldsymbol{\theta})=\sum_{t=0}^{t_{n}}\Big{\lVert}\mathcal{M}\big{(}{\textbf{v}_{t}(\bar{\textbf{x}}|\boldsymbol{\theta})}\big{)}-\mathbf{\hat{m}}_{t}\Big{\lVert}_{L_{2}}+\beta_{1}\big{\lVert}\boldsymbol{\theta}\big{\lVert}_{L_{2}}+\beta_{2}\sum_{t=0}^{t_{n}-1}\Big{\lVert}\textbf{v}_{t+1}(\bar{\textbf{x}}|\boldsymbol{\theta})-\textbf{v}_{t}(\bar{\textbf{x}}|\boldsymbol{\theta})\Big{\lVert}_{L_{2}}$
$+\beta_{3}\Big{\lVert}D_{eff}(\bar{\textbf{x}}|\boldsymbol{\theta})\Big{(}\nabla^{2}C(\bar{\textbf{x}})-C(\bar{\textbf{x}})K(\bar{\textbf{x}}|\boldsymbol{\theta})S_{v}(\bar{\textbf{x}}|\boldsymbol{\theta})\Big{)}\Big{\lVert}_{L_{2}},$
(5)

where $\mathcal{M}:\mathbf{v}\to\mathbf{m}$ is the state-to-observable map,
which maps from the state variables
$\mathbf{v}=[\varepsilon(\mathbf{x},t),\rho_{c}(\mathbf{x},t),C(\mathbf{x},t)]$
to the observables (e.g, mass) of the composite sample,
$\lVert\cdot\lVert_{L_{2}}$ represents the L2 norm, and
$\bar{\textbf{x}}=\\{\textbf{x},t\\}$. In the loss function, there are four
different components play distinct roles: The first component quantifies the
deviation between the model’s rollout predictions and the experimental labels
of mass across the entire temporal sequences; the second component acts as a
regularization term, promoting sparsity; the third component seeks to impose
trajectory smoothness; while the last loss component serves to ensure
efficient convergence of the elliptic solver and also guide the model to avoid
parameter regions where the elliptic solver may become unstable, stabilizing
the training process. The coefficients $\beta_{1}$, $\beta_{2}$, and
$\beta_{3}$ are regularization term weights, and their magnitudes are
typically maintained at low values to preserve the fundamental nature of
regularization. The objective of the PiNDiff model training is to minimize the
total loss function $\mathcal{L}(\boldsymbol{\theta})$, seeking the optimal
parameter set $\boldsymbol{\theta}^{*}$ that most accurately aligns model
predictions with experimental observations, subject to the described
constraints and regularizations.

###  2.5 Uncertainty quantification

The PiNDiff I-CVI model fuses a physics-derived model, sparse measurements,
and neural operators, aiming to capture the complex spatio-temporal behaviors
of the I-CVI process. Yet, challenges arise from incomplete physics knowledge,
the potential over-parameterization of DNNs, and the ever-present issue of
data scarcity. These factors can compromise the model’s prediction
reliability, especially when the extent of missing physics is pronounced. In
light of these challenges, UQ becomes a critical step, ensuring a more
reliable and robust prediction framework for the I-CVI process.

In order to address the uncertainties inherent in PiNDiff model predictions
based on training dataset $\mathcal{D}$, it is essential to determine the
posterior distribution over the model parameters, denoted as
$p(\boldsymbol{\theta}|\mathcal{D})$. Employing gradient descent leads to a
singular realization of $p(\boldsymbol{\theta}|\mathcal{D})$, resulting in the
predicted state $\textbf{v}(\bar{\textbf{x}}|\boldsymbol{\theta})$ being one
realization of the model output, modeled as random variables
$\textbf{V}(\bar{\textbf{x}}|\boldsymbol{\theta})$. Namely, the inherent model
prediction uncertainty can be quantified by the probability distribution
$p(\mathbf{V}|\bar{\mathbf{x}},\mathcal{D})$, which can be obtained using the
Bayesian Model Averaging (BMA),

$p(\mathbf{V}|\tilde{\mathbf{x}},\mathcal{D})=\int
p(\mathbf{V}|\tilde{\mathbf{x}},\boldsymbol{\theta})p(\boldsymbol{\theta}|\mathcal{D})d\boldsymbol{\theta},$
(6)

and approximated using Monte Carlo integration,

$p(\mathbf{V}|\tilde{\mathbf{x}},\mathcal{D})\approx\frac{1}{M}\sum_{j=1}^{M}p(\mathbf{V}|\tilde{\mathbf{x}},\boldsymbol{\theta}^{(j)}),\quad\boldsymbol{{\theta}}^{(j)}\sim
p(\boldsymbol{\theta}|\mathcal{D}),$ (7)

where posterior distribution $p(\boldsymbol{\theta}|\mathcal{D})$ of trainable
parameters is theoretically computed using Bayes’ theorem,

$p(\boldsymbol{\theta}|\mathcal{D})=\frac{p(\mathcal{D|\boldsymbol{\theta}})p(\boldsymbol{\theta})}{\int
p(\mathcal{D}|\boldsymbol{\theta})p(\boldsymbol{\theta})d\boldsymbol{\theta}}.$
(8)

In this context, $p(\boldsymbol{\theta})$ represents the prior distribution
over $\boldsymbol{\theta}$ and $p(\mathcal{D|\boldsymbol{\theta}})$ is the
joint likelihood of the dataset. Obtaining the posterior, as indicated by Eq.
8, either analytically or through traditional Bayesian sampling techniques, is
computationally intractable [52]. As a solution, we employ the DeepEn method
[47] here to tackle this challenge. The increasing appeal of ensemble-based
techniques lies in their capacity to explore the multi-modal posterior
landscape and their straightforward implementation [53]. To effectively
approximate parameter distribution and gauge model uncertainty, multiple
PiNDiff model instances are trained in parallel, each with a distinct
initialization. This approach enables capturing multiple local maximum a
posteriori (MAP) estimates in the posterior distribution for the parameter
$\boldsymbol{\theta}$, each corresponding to distinct local minima within the
landscape of the loss function $\mathcal{L}(\boldsymbol{\theta})$.

During the inference stage, predictions for each variable v are derived using
MAP samples $\\{\boldsymbol{\theta}^{(j)}\\}_{j=1}^{M}$ obtained through
DeepEn training. As a result, the mean and variance of the model predictions
are calculated as,

$\displaystyle\mathbb{E}(\textbf{V}|\bar{\textbf{x}},\mathcal{D})$
$\displaystyle\approx\frac{1}{M}\sum^{M}_{j=1}\textbf{v}(\bar{\textbf{x}}|\boldsymbol{\theta}^{(j)}),$
(9a)
$\displaystyle\mathbb{V}\mathrm{ar}(\textbf{V}|\bar{\textbf{x}},\mathcal{D})$
$\displaystyle\approx\frac{1}{M}\sum^{M}_{j=1}(\textbf{v}(\bar{\textbf{x}}|\boldsymbol{\theta}^{(j)})-\mathbb{E}(\textbf{V}|\bar{\textbf{x}},\mathcal{D}))^{2}.$
(9b)

In this context, $\mathbb{E}(\textbf{V}|\bar{\textbf{x}},\mathcal{D})$ is the
finalized model prediction, and thrice the value of variance
$\mathbb{V}\mathrm{ar}(\textbf{V}|\bar{\textbf{x}},\mathcal{D})$ (considering
only its diagonal component) is used to define the confidence interval. These
estimations of mean and variance equip us with a more robust predictive
capability and facilitate quantifying the uncertainties inherent in the
model’s predictions.

####  2.5.1 Synthetic data generation

To assess the effectiveness of the proposed I-CVI PiNDiff model, a series of
virtual experiments have been conducted as preliminary steps prior to using
the real-world data. This necessitates the generation of synthetic data for
model training, where ground truth is available for validation. To this end, a
physics-based numerical model is developed, drawing inspiration from the
formulation presented by Wei et al. [6]. Gaussian noise is added to the
synthetic data to emulate real-world conditions. Specifically, the general
model of C/SiC deposition for synthetic data generation is based on Eq. 2b.
For the carbon deposition in the I-CVI cycle, the “true” parameter values for
data generation are defined as follows: $M_{d}=0.01199,\text{kgmol}^{-1}$,
$\rho_{d}=2260,\text{kgm}^{-3}$, $q=1$, and
$K=k_{0}\exp\left(\frac{-E_{r}}{RT}\right)$ with $k_{0}=2.62\text{(m/s)}$ and
$E_{r}=1.46\times 10^{5},\text{Jmol}^{-1}$. The effective diffusion
coefficient is determined as $D_{eff}=\frac{\varepsilon
D_{K}D_{AB}}{\tau_{0}(D_{K}+D_{AB})}$, where $D_{AB}$ is the binary diffusion
coefficient, $\tau_{0}$ is the torosity, and the Knudsen diffusion coefficient
is given as $D_{K}=\frac{2}{3}\left(\frac{8RT}{\pi M}\right)^{1/2}r$; $r$
represents the characteristic pore radius in meters. For the purpose of
demonstration, the binary diffusion coefficient is simplified as
$D_{AB}=10^{-5}\frac{T^{1.75}}{P}$ and torosity as $\tau_{0}=6.78$. The true
model form of surface-to-volume ratio $S_{v}$ is defined as,

$S_{v,true}=\frac{\varepsilon}{\varepsilon_{0}}\frac{1-\varepsilon_{0}}{r_{f}}\Big{[}1-\Big{(}\frac{\varepsilon_{0}}{1-\varepsilon_{0}}\Big{)}ln\Big{(}\frac{\varepsilon}{\varepsilon_{0}}\Big{)}\Big{]}^{1/2},$
(10)

where $\varepsilon_{0}$ is the initial porosity, and for synthetic data
generation, $\varepsilon_{0}=0.6$. Given that the C/C composite preforms
examined in this investigation exhibit axisymmetry, our simulations are
performed within a 2-dimensional axisymmetric domain, assuming no variation of
physical properties in the azimuthal plane. The resulting global variables,
such as mass, are obtained by integrating over the entire 3-dimensional
domain.

##  3 Results

The PiNDiff I-CVI model, enhanced with a probabilistic learning approach, is
evaluated using both synthetic and real experiment datasets from C/C composite
manufacturing processes. With the synthetic dataset, a comprehensive set of
numerical experiments is conducted to investigate the model’s performance in
inference, prediction accuracy, and its capability to quantify uncertainty
across diverse training settings. In evaluating its real-world applicability,
the model also undergoes validation with an experimental dataset sourced from
Benzinger and Huttinger’s work [54, 55]. This exercise allows an exploration
of the model’s effectiveness in forecasting the densification process,
especially under varied partial pressures, some of which deviate from those
seen during model training. Extending the model’s evaluative scope, the
PiNDiff approach is employed to model the scenarios involving multiple I-CVI
cycles, mirroring the manufacturing protocols employed by Honeywell Inc. when
producing C/C composites for airplane brake systems. Both synthetic and real
multi-CVI datasets from Honeywell are used for this assessment.

###  3.1 Inference of unknown physics

The PiNDiff I-CVI model incorporates three different neural networks,
$\mathcal{D}_{nn}(\cdot;\boldsymbol{\theta}_{D_{eff}})$,
$\mathcal{K}_{nn}(\cdot;\boldsymbol{\theta}_{K})$, and
$\mathcal{S}_{nn}(\cdot;\boldsymbol{\theta}_{S_{v}})$, to capture the unknown
operators $D_{eff}(\mathbf{x},t)$, $K(\mathbf{x},t)$, and
$S_{v}(\mathbf{x},t)$, respectively, as detailed in Section 2. To evaluate the
model’s capability in inferring these unknown fields from indirect mass
measurements, we conducted a set of three experiments using synthetic data. In
each experiment, one of the operators –either $D_{eff}$, $K$, or $S_{v}$ – was
treated as unknown and to be inferred with its corresponding neural network.
The representations for the other two operators were provided based on the
“ground truth” model presented in section 2.5.1. Due to the inherent
difficulties in collecting experimental data, the training dataset exclusively
comprises measurements of the preform’s mass taken at varied time intervals
throughout the I-CVI cycle. Namely, only these sparse mass data points are
used for model training (blue dots in the first column of Fig. 4).

Figure 4: Prediction and inference results of the trained PiNDiff I-CVI model
with quantified uncertainty. The operators $D_{eff}$, $K$, and $S_{v}$ are
trained independently.

In Fig 4, the PiNDiff model’s predicted mass growth (represented by the solid
line) aligns well with the synthetic data (indicated by dots) across all three
cases. The trained model is also able to accurately capture the spatio-
temporal dynamics of both the unknown parameters ($D_{eff}$, $S_{v}$, and $K$)
and state variables (porosity and molarity), as evident by examining two
critical locations on the preform: it’s center (green) and corner or surface
(red). The close match between the model’s predicted mean (solid line) and the
ground truth (dashed line) underscores the PiNDiff model’s capability to infer
unobserved states and parameters from limited, indirect observations.
Furthermore, the model provides a quantification of the uncertainty associated
with each prediction. These uncertainties are notably low, given that only one
of the functional representations is assumed to be unknown.

Figure 5: Prediction and inference results of the trained PiNDiff I-CVI model
with quantified uncertainty. All three operators $D_{eff}$, $K$, and $S_{v}$
are trained simultaneously.

An extended investigation was conducted where the functional forms of all
three operators, $D_{eff}$, $K$, and $S_{v}$, were treated as unknown and
trained using the same dataset. The results of this exploration are
illustrated in Fig. 5. Although the predictions for mass remain accurate, the
inferred mean values for the hidden states and parameters are slightly deviate
more than those observed in previous experiments. This discrepancy arises due
to the increased complexity introduced by a diminished extent of known
physics. However, the model’s uncertainty in its predictions is elevated,
indicating a reduced confidence in its predictions. This highlights the
PiNDiff model’s ability to reasonably signal prediction confidence through UQ
when faced with different levels of unknown physics of the system. Moving
forward, all subsequent analyses will operate under the assumption that
$D_{eff}$, $K$, and $S_{v}$ are unknown, mirroring real-world scenarios.

###  3.2 Generalizability with respect to operating conditions

To assess the model’s generalizability across different input parameters, we
trained it using synthetic data generated from a diverse range of operating
conditions. Subsequently, it was validated against different sets of operating
conditions not encountered during training. This approach provides insight
into the model’s capability to handle and predict the I-CVI process for new
input scenarios beyond its training conditions.

####  3.2.1 Case 1: Training using segmented sample mass data

To facilitate the I-CVI model in learning spatial dynamics, the preform sample
was segmented into three non-uniform sections along the z-direction, as
visualized in the top-left panel of Fig. 6. The training was performed using
mass data of these segmented sections over time, as shown by blue dots. These
noisy data were generated under nine varying conditions, involving three
different temperatures (1200 K, 1250 K, and 1300 K) and three different
partial pressures (800 Pa, 1600 Pa, and 3200 Pa), as illustrated in Fig. 7.

Figure 6: Prediction and inference results of the trained PiNDiff I-CVI model
with quantified uncertainty. The model is trained with synthetic data of mass
for three pieces.

The model’s excellent predictive capability is evident in the bottom-left
segment of Fig. 6, where its estimations closely match the projected mass
evolution across a span of 350 hours for each piece. Additionally, the model
adeptily infers the spatio-temporal trajectories of porosity $\varepsilon$,
molarity $C$, effective diffusion coefficient $D_{eff}$, and effective
deposition rate $KSv$. As shown in Fig. 6, the confidence interval provided by
the model adequately encompasses the ground truth for the entire duration,
indicating that the proposed PiNDiff model can effectively uncover the hidden
states/parameters from indirect data with quantified uncertainty.

Figure 7: PiNDiff predictions on porosity with uncertainty for unseen partial
pressure and temperature. The model is trained on synthetic mass data for
three pieces.

To assess the generalizability of the trained I-CVI model, we test it to a
range of unseen operating conditions, spanning different temperatures $T$ and
partial pressures $P$. This evaluation covers both interpolation and
extrapolation domains within this $T$-$P$ parameter space. The results,
particularly the predicted porosity across these testing conditions, are
visualized in Fig. 7. For reference, conditions encountered during training
are marked by blue squares within the $T$-$P$ space. Note that the dynamics of
predicted porosity at the two distinct locations are displayed solely for the
testing conditions. For the operating conditions within the interpolation
domain, the model demonstrates a higher level of accuracy – its predictions
aligned well with the ground truth and are accompanied by a relatively
confined uncertainty range. Remarkably, even though the model’s training was
based on the dataset obtained from just nine operating conditions (blue
squares), it still demonstrates strong capability in extrapolation. As shown
in Fig. 7, for the temperatures and partial pressures beyond the training
range, the model not only delivers accurate predictions but also maintains a
credible uncertainty envelope that encompasses the ground truth. As we move
further from the training conditions, the confidence intervals of model
predictions increase, indicating a higher uncertainty. It is clear that the
extrapolative predictions are naturally paired with a more expansive
uncertainty envelope compared to those from the interpolation domain,
demonstrating that the model is able to account for potentially diminished
prediction fidelity in these unfamiliar zones.

####  3.2.2 Case 2: Training using total mass data of entire sample

Obtaining detailed measurement data by segmenting multiple C/C composite
samples during the manufacturing process can be labor-intensive and often
impractical. Building upon our earlier study, here we focus on assessing the
model’s ability to generalize within the input operating parameter space using
only the total mass data from the entire unsegmented composite sample. This
implies that our training dataset lacks detailed spatial information,
presenting a substantial challenge to the model’s capacity for spatial field
inference and prediction. Specifically, the training of the PiNDiff model, in
this case, is exclusively based on total mass data. To enhance the model’s
training, a richer set of conditions was included, including five temperatures
(1200 K, 1225 K, 1250 K, 1275 K, and 1300 K) and three partial pressures (800
Pa, 1600 Pa, and 3200 Pa), leading to a total of 18 training conditions, as
illustrated by blue squares in Fig. 9. This broader spectrum was deemed
necessary due to the absence of spatial information in the training set.

Figure 8: Prediction and inference results of the trained PiNDiff I-CVI model
with quantified uncertainty. Model is trained with synthetic data of total
mass for the entire composite sample.

The prediction and inference results of one training condition are shown in
Fig. 8. As displayed in the bottom-left of Fig. 8, the trained model can
accurately trace the mass trajectory over a 350-hour interval with minimum
uncertainty. However, the inferred spatio-temporal dynamics for $D_{eff}$ and
$KSv$ are notably deviated from the ground truth, primarily attributed to the
limited and spatially non-informative data. Nonetheless, the estimated
uncertainty of the prediction signals its inaccuracies, and the confidence
interval still envelopes the ground truth. Despite the less accurate
inferences for $D_{eff}$ and $KS_{v}$, the spatio-temporal behavior of the
porosity and molarity remains plausible, which can be credited to the PiNDiff
model’s inherent capability of maintaining the mathematical structure of the
physics-derived models. Furthermore, the model’s uncertainty predictions for
molarity and porosity are both reasonable and inclusive of the ground truth.
Hence, even when faced with limited and less descriptive data, the model
remains resilient, with its quantified uncertainty adeptly capturing
variations in data quality.

Figure 9: PiNDiff predictions on porosity with uncertainty for unseen partial
pressure and temperature. The model is trained on synthetic total mass data.

The PiNDiff model, once trained, is tested under operating conditions that
were not part of the training set, spanning both interpolation and
extrapolation regions within the $T$-$P$ parameter space. Fig. 9 presents the
results for the predicted porosity under these unseen conditions. Similar to
the previous experiment, the model continues to provide reliable spatio-
temporal porosity predictions for both interpolated and extrapolated unseen
operating conditions, and its uncertainty reasonably covers the ground truth.
Compared to the previous cases with detailed mass data of segmented samples,
the mean prediction of porosity in this scenario tends to be less accurate and
the uncertainty margin also increases when training is reliant on total mass
measurements of the entire sample.

###  3.3 Training on real experimental data from literature

Previous studies primarily evaluate the PiNDiff I-CVI model using synthetic
datasets. In this section, we shift the focus to its performance on real-world
data. In particular, the training and testing of the PiNDiff model are based
on real experimental data, as detailed in Benzinger and Huttinger’s work [54,
55]. Their seminal experiments concentrated on the I-CVI densification
process, maintaining methane at a constant temperature of $1100^{o}C$, while
adjusting total pressures to either 20 kPa [54] or 100 kPa[55]. Within these
setups, methane’s partial pressure was modulated and varied during the
experiments. The selected substrate for these experiments was a cylindrical,
porous alumina ceramic, with dimensions of 20 mm in height and 16 mm in
diameter and a total porosity of 23%.

Figure 10: PiNDiff predictions with uncertainty on Benzinger and Huttinger’s
experimental mass data at various partial pressure [54, 55].

During the experiment, the mass gain over time is measured given different
partial pressures, and thus, two datasets at different total pressures are
available. These sparse mass measurement data are plotted as triangles in Fig.
10. For both cases, the PiNDiff I-CVI model is trained with mass data from
three different partial pressures and subsequently undergoes testing on the
other four partial pressures absent from the training set. Notably, two of
these test scenarios were in the extrapolation zone, falling outside the
training condition range. Separate trainings are carried out for each
experimental dataset, corresponding to the 20 kPa and 100 kPa total pressures.
Comparative analysis between the PiNDiff I-CVI model’s predictions and the
real measurements across both training and testing scenarios can be viewed in
Fig. 10. The model’s predictions exhibited excellent alignment with the real
data for both training and testing partial pressures, attesting the learning
and predictive capability of the proposed model. Moreover, it is important to
note that the model’s uncertainty predictions are higher in regions where the
training data are absent. And, during testing phases, higher prediction
uncertainty is observed for elevated pressure values, suggesting diminished
confidence in the model’s predictions in those specific regions.

###  3.4 Modeling densification process via multiple I-CVI cycles

Previous results have demonstrated the PiNDiff I-CVI model’s proficiency in
learning unknown physics operators across various operating parameters.
Consequently, the model holds great potential for predictive and optimization
tasks. In industrial manufacturing settings, however, gathering experimental
data under varied operational conditions can be prohibitively costly and time-
intensive. The routine extraction of composites for mass data collection
throughout the process also poses practical challenges. Therefore, in large-
scale applications, such constraints might result in very limited data across
the parameter space, thereby restricting the model’s training and predictive
capabilities under different conditions. However, with substantial
knowledge—like the porosity field derived from the PiNDiff I-CVI model—it
becomes feasible to postulate parameters for subsequent experiments,
facilitating optimization. In this section, we focus on evaluating the model’s
inference capability in a more realistic manufacturing setting but under a
single operational condition.

Figure 11: Multi-I-CVI manufacturing process with intermediate machining.

During the I-CVI process, the concentration of the precursor reactant gas
diminishes as it diffuses deeper into the preform, leading to a decelerated
deposition rate towards the preform’s core. As a result, deposition on the
preform’s external surface is faster compared to its interior. This
preferential exterior deposition causes the outer pores to seal prematurely
before the interior is adequately filled, obstructing subsequent gas diffusion
and impeding the densification process. In response to this challenge,
industrial entities such as Honeywell have adopted a technique of machining
that removes the outer layer to expose these pores inside again. Consequently,
we need a series of multiple I-CVI cycles performed consecutively, along with
the machining process, to ensure enhanced densification. Figure 11 provides a
schematic representation of three such interlinked I-CVI and machining cycles.
This cyclic methodology is adopted by Honeywell in their C/C composite brake
manufacturing protocol. Typically, during the I-CVI phases, the brakes gain
mass, which is then reduced during the machining stages. Data harvested from
Honeywell’s iterative I-CVI brake production serves as the training data for
our PiNDiff I-CVI model in inferring porosity distribution. Nevertheless, a
challenge arises in Honeywell’s real manufacturing cases as the inferred
quantities lack accompanying validation data. To address this, initial cross-
validation of the model is undertaken using synthetic data of multi-
CVI/machining cycles, synthesized in alignment with established industry
practices.

The proposed PiNDiff I-CVI model is modified to accommodate multiple I-CVI
cycles interspersed with machining phases. In the machining interval,
attributes like geometry, computational grid, and foundational physical
parameters (e.g., porosity from the previous cycle) are re-calibrated in line
with the post-machining preform specifications. Synthetic mass data is
collected at the end of each cycle.

Figure 12: Prediction and inference results of the trained PiNDiff model with
quantified uncertainty for multi-I-CVI and machining cycles. Model is trained
with synthetic data of mass for three pieces.

Initially, the model is trained using synthetic data, aiming to assess its
inference capability for a given operating condition. The data used for
training include mass measurements of three unevenly segmented portions cut
along the z-axis of the brake sample (blue dots in Fig. 12). After undergoing
500 epochs of training, the PiNDiff model can accurately forecast the
incremental brake mass gain throughout the manufacturing cycles, as
illustrated in Fig. 12. Moreover, the model also accurately infers the spatio-
temporal dynamics of key hidden parameters and states such as $D_{eff}$,
$KSv$, molarity, and porosity, with its predictions aligning closely with the
ground truth, as evinced by their confinement within the confidence intervals.
In line with the previous results, the uncertainty is lower as the training
data contains spatial information. While the model’s predictions might warrant
caution under varied operational scenarios, the derived porosity remains
crucial, offering substantive insights into the composite’s characteristics,
and thereby facilitating the subsequent optimization endeavors.

Figure 13: Prediction and inference results of the trained PiNDiff model with
quantified uncertainty for multi-I-CVI and machining cycles. The model is
trained with synthetic data of the total mass for an entire brake.

The PiNDiff model’s efficacy is subsequently examined using the total mass as
synthetic data rather than multiple cuts. Illustrated in Fig. 13, the results
follow a similar trend to the previous case. After a training of 500 epochs,
the model accurately predicts the mass trajectory throughout the fabrication
process. Additionally, the hidden parameters and states like $D_{eff}$, $KSv$,
molarity, and porosity are also estimated by the model, where the mean values
agree with the ground truth reasonably well. It’s noteworthy that, in this
setup, the model’s uncertainty bounds are more expansive than those in the
previous scenario that utilizes localized mass data. The increased uncertainty
arises from the comparatively diminished spatial information in the present
training dataset.

Lastly, we evaluate the PiNDiff model for learning and predicting multiple CVI
cycles using the real measurement data from Honeywell’s C/C brake
manufacturing processes. The results are presented in Fig. 14. The trained
model successfully captures the evolution of the densification process
throughout the manufacturing phase. Crucially, during its training, the model
offers spatio-temporal forecasts of porosity, an indispensable parameter for
optimization endeavors. While no supplementary experimental data exists for
direct validation of inferred quantities, the model’s performance with
synthetic data suggests a substantial degree of reliability in its
predictions. Such capabilities of the PiNDiff I-CVI model hold significant
potential for enhancing optimization strategies of CVI processes and informed
decision-making in the industry.

Figure 14: Prediction and inference results of the trained PiNDiff model with
quantified uncertainty, based on the Honeywell manufacturing dataset.
(Specific scales have been omitted due to export-control restrictions).

##  4 Conclusion

This study presents the development and assessment of a physics-integrated
neural differentiable (PiNDiff) model for I-CVI processes. By seamlessly
integrating the partially known physics into a deep learning framework via
differentiable programming, the PiNDiff model has established itself as an
effective modeling framework for the I-CVI process, showing competency even
when reliant on limited, indirect training datasets.

The PiNDiff model has been thoroughly evaluated through a series of
experiments and tests, confirming its effectiveness and capabilities. Even
when trained on limited and indirect synthetic data, the model exhibits
promising performance, accurately predicting mass deposition and inferring the
spatio-temporal fields of porosity, diffusion coefficient, deposition rate,
and molarity. A salient feature of the model is its intrinsic capability to
encapsulate and quantify uncertainties, particularly when confronted with
incomplete physics and scarce data. Furthermore, the PiNDiff model
demonstrates its capability to generalize across different operational
conditions by accurately predicting the spatio-temporal dynamics of porosity
under unseen temperatures and partial pressures. The model exhibits excellent
performance in both interpolation and extrapolation scenarios, with lower
model uncertainty observed in interpolation cases and higher uncertainty in
extrapolation cases. This showcases the model’s versatility and potential to
effectively predict the I-CVI process under a wide range of conditions while
accounting for its prediction confidence.

By leveraging experimental data, notably from studies by Benzinger and
Huttinger, along with data obtained from Honeywell’s manufacturing processes,
the PiNDiff model reaffirmed its effectiveness and ability in predictive
modeling of the I-CVI processes. Notably, the model successfully captures the
higher uncertainty in regions where data was absent during training. The
results highlight the model’s potential to support process optimization and
decision-making tasks, emphasizing its practical applicability in real-world
scenarios.

To conclude, the PiNDiff model proves to be a valuable tool for understanding,
simulating, and predicting the I-CVI densification process. Its ability to
integrate partially-known physics with deep learning enables effective
predictive modeling using sparse and indirect measurements. While further
comprehensive validation with more experimental data is desirable, the PiNDiff
I-CVI model offers a promising approach for advancing the manufacturing of
carbon-based composites to enhance their performance and quality.

## Acknowledgment

The authors would like to acknowledge the funds from the Air Force Office of
Scientific Research (AFOSR), United States of America, under award number
FA9550-22-1-0065. JXW would also like to acknowledge the funding support from
the Office of Naval Research under award number N00014-23-1-2071 and the
National Science Foundation under award number OAC-2047127 in supporting this
study.

## Compliance with Ethical Standards

Conflict of Interest: The authors declare that they have no conflict of
interest.

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##  Appendix A Nomenclature

$\varepsilon_{k}$ | volume fraction of species $k$
---|---
$\varepsilon_{g}$ | volume fraction of gas
$\varepsilon$ | porosity
$\rho_{k}$ | density of species $k$ [$kg/m^{3}$]
$\textbf{u}_{k}$ | velocity vector of species $k$ [$m/s$]
$\dot{\omega}_{k}$ | production rate of species $k$ [$kg/m^{3}s$]
$D_{k,eff}$ | effective diffusion coefficient of species $k$ [$kg/m^{3}s$]
$C$ | Morality of gas [$mol/m^{3}$]
$M$ | Molar mass of depositing species [$mol/m^{3}$]
$S_{v}$ | Effective infiltration area [$m^{2}/m^{3}$]
$*_{g}$ | gas
$*_{s}$ | solid
Table 1: Caption

##  Appendix B Derivation

The governing equations to simulate the I-CVI process for each species are
given as [49]

$\frac{\partial(\varepsilon_{k}\rho_{k})}{\partial
t}+\nabla.(\varepsilon_{k}\rho_{k}\textbf{u}_{k})=D_{k}\nabla^{2}(\varepsilon_{k}\rho_{k})+\dot{\omega}_{k}\
\ k=1,...,N$ (11) Gas $\frac{\partial(\varepsilon_{k}\rho_{k})}{\partial
t}+\nabla.(\varepsilon_{k}\rho_{k}\textbf{u}_{k})=D_{k}\nabla^{2}(\varepsilon_{k}\rho_{k})+\dot{\omega}_{k}\
\ k=1,...,N_{g}$ (12a) Solid
$\frac{\partial(\varepsilon_{k}\rho_{k})}{\partial
t}+\nabla.(\varepsilon_{k}\rho_{k}\textbf{u}_{k})=D_{k}\nabla^{2}(\varepsilon_{k}\rho_{k})+\dot{\omega}_{k}\
\ k=1,...,N_{s}$ (12b)

Summing over all gaseous and solid species results in

$\frac{\partial(\varepsilon_{g}\rho_{g})}{\partial
t}+\nabla.(\varepsilon_{g}\rho_{g}\textbf{u}_{g})=D_{eff,g}\nabla^{2}(\varepsilon_{g}\rho_{g})+\dot{\omega}_{g}$
(13a)

$\frac{\partial(\varepsilon_{s}\rho_{s})}{\partial
t}+\nabla.(\varepsilon_{s}\rho_{s}\textbf{u}_{s})=D_{eff,s}\nabla^{2}(\varepsilon_{s}\rho_{s})+\dot{\omega}_{s}$
(13b)

where,

$\varepsilon_{g}\rho_{g}=\sum_{k=1}^{N_{g}}\varepsilon_{k}\rho_{k}\ \
\varepsilon_{s}\rho_{s}=\sum_{k=1}^{N_{s}}\varepsilon_{k}\rho_{k}$ (14a)
$\varepsilon_{g}\rho_{g}\textbf{u}_{g}=\sum_{k=1}^{N_{g}}\varepsilon_{k}\rho_{k}\textbf{u}_{k}\
\
\varepsilon_{s}\rho_{s}\textbf{u}_{s}=\sum_{k=1}^{N_{s}}\varepsilon_{k}\rho_{k}\textbf{u}_{k}$
(14b) $\dot{\omega}_{g}=\sum_{k=1}^{N_{g}}\dot{\omega}_{k}\ \
\dot{\omega}_{s}=\sum_{k=1}^{N_{s}}\dot{\omega}_{k}$ (14c)
$\dot{\omega}_{g}+\dot{\omega}_{s}=\sum_{k=1}^{N}\dot{\omega}_{k}=0$ (14d)
$\varepsilon_{g}+\varepsilon_{s}=\sum_{k=1}^{N}\varepsilon_{k}=1$ (14e)

We only consider the simulation inside perform domain with the following
assumptions

  1. 1.

For stationary solid matrix: $\textbf{u}_{s}=0$,

  2. 2.

No diffusion by solid species: $D_{k,s}=0$,

  3. 3.

negligible flow and diffusion dominant inside the porous substrate[49]:
$\textbf{u}_{g}=0$,

  4. 4.

gas is in a quasi-steady state,
$\frac{\partial(\varepsilon_{g}\rho_{g})}{\partial t}\approx 0$

After applying the above assumptions, the equation simplifies to

$D_{eff,g}\nabla^{2}(\varepsilon_{g}\rho_{g})=-\dot{\omega}_{g}$ (15a)
$\frac{\partial(\varepsilon_{s}\rho_{s})}{\partial t}=\dot{\omega}_{s}$ (15b)

Here we redefine $\varepsilon_{g}\rho_{g}=MC$, where $C$ is molarity and $M$
is the molar mass, use the equation $\varepsilon_{s}=1-\varepsilon_{g}$, and
assume the reaction to be first-order $\dot{\omega}_{s}=MKCS_{v}$. Therefore
the governing equations result in the following form.

$D_{g,eff}\nabla^{2}(C)=KCS_{v}$ (16)
$\rho_{s}\frac{d\varepsilon_{g}}{dt}=-MKCS_{v}$ (17)

##  Appendix C I-CVI Solver

Function  _PiNDIff-I-
CVI(_$C_{0}(\mathbf{x}),\varepsilon_{0}(\mathbf{x}),T,P,C_{BC},\boldsymbol{\theta}=(\boldsymbol{\theta}_{D_{eff}},\boldsymbol{\theta}_{K},\boldsymbol{\theta}_{S_{v}})$_)_:

$t\leftarrow 0$

while  _$t <t\\_max$_ do

$D_{eff}\leftarrow\mathcal{D}_{nn}\big{[}\varepsilon_{t},T,P;\boldsymbol{\theta}_{D_{eff}}\big{]}$
$\triangleright$ Compute Diffusion coefficient

$K\leftarrow\mathcal{K}_{nn}\big{[}T;\boldsymbol{\theta}_{K}\big{]}$
$\triangleright$ Compute reaction deposition rate

$\tilde{S}_{v}\leftarrow\mathcal{S}_{nn}[\hat{S}_{v},\varepsilon_{t};\boldsymbol{\theta}_{S_{v}}]$
$\triangleright$ Compute S2V ratio correction

$S_{v}\leftarrow\hat{S}_{v}(1+\tilde{S}_{v})$ $\triangleright$ Compute S2V
ratio

$C_{t}\leftarrow\text{Elliptic\\_Molarity\\_Solver}(D_{eff},K,S_{v},C_{BC})$
$\triangleright$ Compute Molarity

$\varepsilon_{t+1}\leftarrow\varepsilon_{t}+\Delta
t\frac{M_{d}}{\rho_{d}}C_{t}KS_{v}$ $\triangleright$ Euler time stepping, (can
use RK4)

end while

return
$\big{(}\\{C(\mathbf{x})\\}_{t=0}^{t\\_max},\\{\varepsilon(\mathbf{x})\\}_{t=0}^{t\\_max}\big{)}$
$\triangleright$ Predicted time series

Algorithm 1 An algorithm for PiNDIff I-CVI solver

To solve the elliptic PDE Eq.16, boundary values for molarity are needed.
Dirichlet boundary conditions for molarity are applied, and these boundary
values are computed based on the partial pressure $P_{r}$ as

$C_{BC}=\text{mole fraction}\times\text{total
moles}=\frac{P_{r}}{P}\times\frac{P}{RT}=\frac{P_{r}}{RT}.$ (18)

The point-Jacobi method is employed to iteratively solve Eq.16. While the ODE
Eq. 17, needs the initial porosity $\varepsilon_{0}$ that depends on the
preform. The spatiotemporal porosity and molarity
$\big{(}\\{C(\mathbf{x})\\}_{t=0}^{t\\_max},\\{\varepsilon(\mathbf{x})\\}_{t=0}^{t\\_max}\big{)}$
is obtained by stepping in time using Euler or RK4 methods.

##  Appendix D Hyper-parameters for PiNDiff I-CVI model

In the PiNDiff model, the following learning setting is used for the neural
network trainable parameters,

  * •

Initial learning rate = $10^{-3}$

  * •

Optimizer = Adam

  * •

Scheduler = cosine_decay_schedule($\alpha=10^{-2}$),