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$Id:XXX-08-000.tex626982011-06-2100:28:58Zalverson$ CR-2011/097

# Centrality and $\PT$ dependence of charged particle $\RAA$ in PbPb
collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 2.76 TeV

Andre S. Yoon for the CMS collaboration

###### Abstract

The transverse momentum ($p_{T}$) spectra of charged particles is measured by
CMS as a function of collision centrality in PbPb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV. The results are compared to a pp reference
spectrum, constructed by interpolation between $\sqrt{s}$ = 0.9 and 7 TeV
measurements. The nuclear modification factor ($R_{AA}$) is constructed by
dividing the PbPb $p_{T}$ spectrum, normalized to the number of binary
collisions ($N_{coll}$), by the pp spectrum. Measured $R_{AA}$ in 0–5%
centrality bin is compared to several theoretical predictions.

Presented at QM2011: Quark Matter 2011

##  0.1 Introduction

The charged particle $\PT$ spectrum is an important observable for studying
the properties of the hot, dense medium produced in the collisions of heavy
nuclei. In particular, the modification of the $\PT$ spectrum compared to
nucleon-nucleon collisions at the same energy can shed light on the detailed
mechanism by which hard partons lose energy traversing the medium [1]. The
modification is typically expressed in terms of

$\RAA(\PT)=\frac{d^{2}N_{\mathrm{ch}}^{\mathrm{AA}}/d\PT
d\eta}{\left<T_{AA}\right>d^{2}\sigma_{\mathrm{ch}}^{\mathrm{NN}}/d\PT
d\eta},$ (1)

where $N_{\mathrm{ch}}^{\mathrm{AA}}$ and $\sigma_{\mathrm{ch}}^{\mathrm{NN}}$
represent the charged particle yield in nucleus-nucleus collisions and the
cross section in nucleon-nucleon collisions, respectively. The nuclear overlap
function $\left<T_{AA}\right>$ is the ratio of the number of binary nucleon-
nucleon collisions $\left<N_{coll}\right>$ calculated from a Glauber model of
the nuclear collision geometry. At RHIC, the factor of five suppression seen
in $\RAA$ was an early indication of strong final-state medium effects on
particle production. In this contribution, measured charged particle spectra
and $\RAA$ in bins of centrality for the $\PT$ range of 1–100 $\GeVc$ [2] are
presented.

##  0.2 Experimental Methods

This measurement is based on a data sample corresponding to an integrated
luminosity of 7 $\,\mu\text{b}^{\text{$-$1}}$ collected by the CMS experiment
[3] in 2.76 TeV PbPb collisions. A minimum bias event sample was collected
based on a trigger requiring a coincidence between signals in the +$z$ and
–$z$ sides of either the forward hadronic calorimeters (HF) or the beam
scintillator counters (BSC). To ensure a pure sample of inelastic hadronic
collision events, additional offline selections were performed. These include
a beam halo veto based on the BSC timing, an offline HF coincidence of at
least three towers ($E>3$ GeV) on each side of the interaction point, a
reconstructed vertex consisting of at least two pixel tracks of $\PT>75$
$\MeVc$, and a rejection of beam-scraping events based on the compatibility of
pixel cluster shapes with the reconstructed primary vertex. The collision
event centrality is determined on the total energy deposited in both HF
calorimeters.

In this analysis, data recorded by single-jet triggers with uncorrected
transverse energy thresholds of $E_{T}=35$ GeV (Jet35U) and 50 GeV (Jet50U)
are included in order to extend the statistical reach of the $\PT$ spectra.
The use of the jet-triggered data, which contributes dominantly to the
high-$\PT$ region ($\PT>50$ $\GeVc$), also allows to keep low
misidentification rates of the selected tracks being enforced by an implicit
track-calorimeter matching. The jet-energy thresholds in the trigger are
applied after subtracting the underlying event energy but without correcting
for calorimeter response. $E_{T}$ distributions of the most energetic
reconstructed jet with $|\eta|<2$, referred to as the leading jet, are
normalized per minimum bias event. Following the procedure introduced in the
analogous measurement of the charged particle spectra in 0.9 and 7 TeV pp
collisions [4], the spectra are calculated separately in three ranges of
leading-jet $E_{T}$, each corresponding to a fully efficient trigger path, and
then combined to obtain the final result.

Figure 1:  (Left) Interpolated 2.76 TeV charged particle differential
transverse momentum cross section with ratios of combined interpolation to
various predictions and interpolation. (Right) Invariant differential yield in
bins of collision event centrality (symbols), compared to a pp reference
spectrum, scaled by the corresponding number of binary nucleon-nucleon
collisions (dashed lines). The systematic uncertainties on the PbPb
differential yields as a function of $\PT$ for the 0–5% and 5–90% are shown in
the bottom panel.

##  0.3 Results

The interpolated differential cross section in pp at 2.76 TeV [4] is shown in
the upper panel of Fig. 1, and its ratio with respect to various pythia tunes.
Also shown in the lower panel of Fig. 1 is the ratio of the predicted 2.76 TeV
cross section to that found by simply scaling the CMS measured 7 TeV result by
the expected 2.75 TeV to 7 TeV ratio from NLO calculations [5] and the
interpolation used in the recent ALICE publication [6].

The charged particle invariant differential yields are shown for the six
centrality bins in Fig. 1 and compared to the corresponding quantity taken
from the interpolated pp reference spectrum [4], normalized by the number of
binary collisions. The points have been scaled by the factors given in the
figure for easier viewing.

$\RAA$ is presented as a function of transverse momentum in Fig. 2 for each of
the six centrality bins. The shaded areas (boxes) around the points show the
systematic uncertainties including those from the interpolated pp reference
spectrum. An additional systematic uncertainty from the $T_{AA}$
normalization, common to all points is displayed as the shaded band around
unity in each plot. In the most-peripheral events (70–90%), a moderate
suppression of about two ($\RAA=0.5$) is observed at low $\PT$ with $\RAA$
rising gently with increasing transverse momentum. The suppression is
increasingly pronounced in the more-central collisions, as expected from the
longer average path lengths traversed by hard-scattered partons as they lose
energy via jet quenching. In the 0–5% bin, $\RAA$ reaches a minimum value of
0.13 around 6–7 GeVc. At higher $\PT$, the value of $\RAA$ rises and levels
off above 40 GeVc at a value of approximately 0.5.

Figure 2: (Left) $\RAA$ (filled circles) as a function of $\PT$ for six
centrality intervals. The error bars represent the statistical uncertainties,
and the yellow boxes the $\PT$-dependent systematic uncertainties on the
$\RAA$ measurements. (Right) Comparison to the measurements of $\RAA$ in
central heavy ion collisions at three different center-of-mass energies as a
function of $\PT$ for neutral pions and charged hadrons and to several
theoretical predictions [7, 8, 9].

Measured $\RAA$ in the most central PbPb events (0–5 %) is compared to a
number of model predictions, both for the LHC design energy of
$\sqrt{s_{{}_{\mathrm{NN}}}}$ = 5.5 TeV (PQM [7] and GLV [8]) and for the
actual 2010 collision energy of $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 2.76 TeV (ASW,
YaJEM, and an elastic scattering energy-loss model with parameterized escape
probability [9]). While most models predict the generally rising behavior that
is observed in the data at high $\PT$, the magnitude of the predicted slope
varies greatly depending on the details of the jet quenching implementation.

##  0.4 Summary

In this contribution, measurements of the charged particle transverse momentum
spectra have been presented for 2.76 TeV PbPb collisions in bins of collision
centrality. The results have been normalized to an interpolated 2.76 TeV
reference $\PT$ spectrum to construct $\RAA$. A dramatic suppression (2–6
$\GeVc$) and a significant rise in the higher $\PT$ region with leveling-off
around 40 $\GeVc$ of the charged particle spectrum have been observed for the
most-central PbPb events. The new CMS measurement should help constrain the
quenching parameters used in various models and further the understanding of
the energy-loss mechanism.

## References

  * [1] D. d’Enterria, “Jet quenching”,  Landolt-Boernstein, Springer-Verlag Vol. 1-23A (2010) 99,  arXiv:0902.2011.
  * [2] CMS Collaboration, “Nuclear modification factor for charged particle production at high $p_{T}$ in PbPb collisions a $\sqrt{s_{{}_{NN}}}=2.76$ TeV”, CMS Physics Analysis Summary CMS-PAS-HIN-10-005 (2011).
  * [3] CMS Collaboration, “The CMS experiment at the CERN LHC”,  JINST 3 (2008) S08004.  doi:10.1088/1748-0221/3/08/S08004.
  * [4] CMS Collaboration, “Charged particle transverse momentum spectra in pp collisions at $\sqrt{s}$ = 0.9 and 7 TeV”,  arXiv:arXiv/1104.3547.
  * [5] F. Arleo, D. d’Enterria, and A. S. Yoon, “Single-inclusive production of large-pT charged particles in hadronic collisions at TeV energies and perturbative QCD predictions”,  JHEP 06 (2010) 035,  arXiv:1003.2963.  doi:10.1007/JHEP06(2010)035.
  * [6] ALICE Collaboration, “Suppression of Charged Particle Production at Large Transverse Momentum in Central Pb–Pb Collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV”,  Phys. Lett. B696 (2011) 30,  arXiv:1012.1004.  doi:10.1016/j.physletb.2010.12.020.
  * [7] A. Dainese, C. Loizides, and G. Paic, “Leading-particle suppression in high energy nucleus-nucleus collisions”,  Eur. Phys. J. C38 (2005) 461,  arXiv:hep-ph/0406201.  doi:10.1140/epjc/s2004-02077-x.
  * [8] I. Vitev and M. Gyulassy, “High $p_{T}$ tomography of $d$ \+ Au and Au+Au at SPS, RHIC, and LHC”,  Phys. Rev. Lett. 89 (2002) 252301,  arXiv:hep-ph/0209161.  doi:10.1103/PhysRevLett.89.252301.
  * [9] T. Renk et al., “Systematics of the charged-hadron $p_{T}$ spectrum and the nuclear suppression factor in heavy-ion collisions from $\sqrt{s}=200$ GeV to $\sqrt{s}=2.76$ TeV”,  arXiv:arXiv/1103.5308.


# Predicting Fatigue Crack Growth via Path Slicing and Re-Weighting

Yingjie Zhao  Yong Liu  Zhiping Xu

###### Abstract

Predicting potential risks associated with the fatigue of key structural
components is crucial in engineering design. However, fatigue often involves
entangled complexities of material microstructures and service conditions,
making diagnosis and prognosis of fatigue damage challenging. We report a
statistical learning framework to predict the growth of fatigue cracks and the
life-to-failure of the components under loading conditions with uncertainties.
Digital libraries of fatigue crack patterns and the remaining life are
constructed by high-fidelity physical simulations. Dimensionality reduction
and neural network architectures are then used to learn the history dependence
and nonlinearity of fatigue crack growth. Path-slicing and re-weighting
techniques are introduced to handle the statistical noises and rare events.
The predicted fatigue crack patterns are self-updated and self-corrected by
the evolving crack patterns. The end-to-end approach is validated by
representative examples with fatigue cracks in plates, which showcase the
digital-twin scenario in real-time structural health monitoring and fatigue
life prediction for maintenance management decision-making.

[1]organization=Applied Mechanics Laboratory, Department of Engineering
Mechanics, Tsinghua University, city=Beijing, postcode=100084, country=China

[2]Lead contact. Correspondence<EMAIL_ADDRESS>(Z.X.)

## Introduction

Fatigue life prediction (FLP) is of critical importance for structural
integrity design in, for example, aerospace and nuclear engineering [1]. After
fatigue initiation with accumulated damage, fatigue cracks grow and can be
monitored after the size reaches a detection threshold. In practice, periodic
inspection is commonly arranged to identify flaws in the structural
components. The information is fed into fracture mechanics analysis (FMA)
where the remaining life can be calculated from empirical rules of fatigue
crack growth (FCG). Predictive maintenance schemes could further reduce the
life-cycle cost and increase system safety, which have been actively explored
in recent studies [2].

However, the physics governing fatigue is entangled with the microstructural
evolution of materials and the profiles of loading conditions [3, 4]. The
microscopic processes of fatigue may involve plasticity, fracture, and phase
transitions, which are defined by chemical compositions, and atomic-level and
microscopic structures [5]. As a result, material fatigue, like fluid
turbulence, becomes a complex system process of the microscopic components
with their interaction spanning across multiple space and time scales [6]. The
nonlinearity and heterogeneity embedded in the mathematical or data-driven
models make predicting behaviors of these systems challenging [7]. Fingerprint
features of materials and structural components as well as the history or path
dependence of FCG make their responses susceptible to statistical noises and
rare events resulting from intrinsic or extrinsic sources [8, 9].
Uncertainties thus exist in the microstructure-sensitive constitutive
relations of materials and the loading conditions in experimental tests or
under specific service conditions, respectively, which can alter the processes
of material damage and FCG [10].

Material responses during fatigue can be characterized by statistical data of
the fatigue life or FCG rates obtained from experimental tests [11, 12]. With
this knowledge, recent developments in data science and machine learning
techniques allow engineers to take a data-driven approach to structural health
monitoring (SHM) and FLP [13]. However, a practical solution that tackles the
history dependence, statistical noises, and rare events has not been
established yet [14, 15]. With digital libraries constructed from high-
fidelity physical simulations, neural networks with specially designed
architectures can extract the characteristic features and make predictions
[16, 17]. The history dependence can be handled by the long short-term memory
(LSTM) network by introducing gating and memory functions [18]. Surrogate
models trained by machine learning techniques can reduce the computational
costs significantly from physical modeling. Prediction in a real-time or
digital-twin paradigm can thus be achieved, where the interaction with
physical sensor networks (PSNs) can be included to update the model
parameters, quantify the uncertainties, and evaluate the models based on the
Bayesian theory [19, 20].

In this work, we develop a statistical learning framework to predict the
growth of fatigue cracks and the life-to-failure of structural components.
Digital libraries are constructed by finite element analysis (FEA).
Variational autoencoder (VAE) is used to learn the latent representation of
the fatigue cracks, followed by LSTM and feedforward neural network (FNN) for
the history dependence and the life prediction of FCG, respectively. Path
slicing and re-weighting techniques are introduced to address uncertainties in
the service conditions. Fully resolved fatigue crack paths and accurate
prediction of the remaining life are demonstrated by examples.

## Results

### The statistical learning framework.

In engineering, FLP can be achieved through techniques that can be expressed
into $4$ levels (Figure 1). Empirical models for the relations between the FCG
rate (${\rm d}a/{\rm d}N$) and the amplitude of stress intensity factors
(SIFs, $\Delta K$) can be fed into analytical solutions of SIF under specific
loading conditions and sample geometries [21]. FEA calculates the evolution of
SIFs as fatigue cracks grow with high accuracy. The surrogate models
constructed using, for example, statistical learning methods can reduce the
computational costs. Digital twins with accurate and fast algorithms of FLP
and interaction with the physical systems can be implemented for real-time
monitoring and prognosis. Our FLP framework includes modules of data
generation, model training, and applications (Figure 2, see Experimental
Procedures for details). The extended finite element method (XFEM) is utilized
for modeling and constructing digital libraries. Statistical noises and rare
events are considered through the loading amplitudes with Gaussian
distributions in FCG modeling using XFEM. The produced datasets contain
fatigue cracks and their corresponding residual life. We combine VAE, LSTM,
and FNN in model training, which is used for SHM and FLP for the structural
components in the application module. A path-slicing technique is used based
on the datasets produced by FEA for the history dependence and statistical
noises, and a re-weighing technique is introduced in the training process to
signify the impact of rare events on the model parameters (Figure 3A).

### Dimensionality reduction of the crack patterns.

Crack patterns or fractured surfaces modeled by XFEM are stored as voxel data.
These data characterize the competition between the material resistance to
fracture and the driving force of FCG. For simple structures, the mapping
between the far-field loading conditions and the crack-tip driving forces can
be captured by FMA. As a result, dimensionality reduction of the crack
patterns is performed to extract crack surfaces to improve the learning
efficiency in downstream tasks. We use VAE to learn the latent representations
of fatigue cracks in reduced dimensions. The density distributions of fatigue
crack patterns spanned in the space of the two primary features are summarized
in Figure 3B. The latent representations of fatigue cracks tend to follow the
Gaussian distribution owing to the competition between the reconstruction loss
and the Kullback-Leibler (KL) divergence in VAE (see Experimental Procedures
for details) [22].

### Path slicing and re-weighting for history dependence, statistical noises,
and rare events.

We introduce a path-slicing technique to address the complexity of the loading
profiles and the effects of statistical noises on FCG (Figure 3A). The domain
of interest is discretized to $N_{\rm s}$ regions. As FCG proceeds across
their boundaries and reaches the next region, the tension and shear loads are
re-sampled from the Gaussian distribution (Figure 3A). Structural components
in service may also experience rare events such as catastrophic failure of the
components and unexpected external impact. Rare events are identified from the
fatigue crack patterns in the pre-constructed digital libraries by
unsupervised clustering in the latent space (shown as red data points in the
space spanned by two major features in Figure 3C) [23]. Their weights are
boosted in downstream training to improve the performance of prediction, where
an enrichment factor $\lambda$ is defined for the updated loss function

${\rm Loss}={\rm MSE}(\mathbf{z},\mathbf{\widehat{z}})+\lambda{\rm
MSE}(\mathbf{z}_{\rm rare},\mathbf{\widehat{z}}_{\rm rare}),$ (1)

where $\mathbf{z}$ and $\mathbf{\widehat{z}}$ are the ground truth and
predictions of the latent vectors, respectively. The subscript indicates the
rare records of fatigue cracks.

### Prediction and correction in digital twins.

To demonstrate the capability of our framework, we consider a flat plate
subjected to loads with statistical noises and rare events as a representative
example (Figure 4). Figures 4A-4C show the representative fatigue crack paths
from FLP. The results show that without path slicing or re-weighting, the
deflection of fatigue cracks cannot be correctly predicted. FCG along the
direction of the embryo crack is predicted without necessary changes induced
by the uncertainties in the loading conditions (Figure 4A). In contrast, path
slicing solves the problem by constantly updating the model prediction based
on observation in experiments or physical modeling (in our work) at time
$t_{\rm obs}$, correctly predicting crack deflection (Figure 4B). With re-
weighting further implemented, the process of FCG and the remaining life are
accurately predicted (Figure 4C).

For quantitative assessment of the framework, the root mean square error
(RMSE) and structural similarity (SSIM) are evaluated between the predicted
crack path and the ground truth as a function of the observation time, $t_{\rm
obs}$ (Figure 4D, see Experimental Procedures for details). RMSE offers a
local measure for the deviation of fatigue crack paths, showing that combining
path slicing and re-weighting effectively minimizes the errors by addressing
the history dependence and noisy features in the loading conditions as well as
the presence of rare events. SSIM measures the global similarity from the mean
and variance of the voxel values, showing that the predictions of fatigue
crack paths are improved as $t_{\rm obs}$ increases Figures 5C and 5D show the
error in life prediction as well as the accuracy of representative paths and
all the path samples in the test set, respectively, which comprises $20\%$ of
data in the digital libraries. As $t_{\rm obs}$ increases, the performance of
models equipped with path slicing and re-weighting is continuously improved.

In practice, the loading conditions can change continuously in service, the
complexity of which is quantified using symbolic aggregate approximation (SAX)
(Figures 6A-6C, see Experimental Procedures for details) [24]. We find that
path slicing and re-weighting can significantly reduce the data complexity
actually needed in the statistical learning framework. Specifically for the
plate example, with only $200$ loading-profile samples in the training set,
path slicing with $N_{\rm s}=5$ can accurately predict FCG in the test set
with $200$ samples, even for the loading conditions with statistical noises
and rare events (Figures 6B and 6C). It should be noted that the performance
of prediction depends on the size of training sets and the choice of $N_{\rm
s}$. Their values are determined for structural components for specific
geometries, sizes, and loading conditions, and can be guided by the analysis
of the loading-profile complexity.

## Discussion and Conclusion

To predict FCG and the life-to-failure in real time, our digital framework
needs only the information of fracture crack morphologies, which can be
integrated with PSNs and FMA to assess the model performance [25]. In addition
to crack patterns captured by high-speed cameras, PSN data such as local
strain measured by strain gauges, elastic waves detected by acoustic emission
sensors, and electrical impedance measured by piezoelectric sensors can be
used to identify or infer the features of fatigue cracks and loading
conditions [26]. These data can be fed into FMA to calculate the stress fields
at the crack tips based on analytical models or FEA [25]. FCG is then
predicted by using, for example, the Paris-Erdogan equation, and can be used
to evaluate the accuracy of our statistical learning framework. In practice,
implementing digital twins for FLP still faces challenges in addition to real-
time monitoring of fatigue crack patterns. Firstly, the use of sensors on
critical structural components such as turbine blades in the engines may not
be feasible in a harsh environment. Even for structural components where
sensors can be installed, the driving force of FCG at the crack tip needs to
be calculated from limited data from the PSNs. Data-driven approaches were
recently proposed to address this issue. Strain fields are predicted from data
collected from digital image correlation (DIC) and a limited number of strain
gauge sensors using deep neural networks and experimental libraries [26].
Secondly, FMA using FEA, although offering higher accuracy compared to
analytical models, demands a considerable amount of time for numerical
calculations, especially for structural components with complex geometries,
crack-tip morphologies, and load conditions (Figure 6D). The applications of
our model in FLP can be extended straightforwardly to engineering components
such as the turbine blades with a 3D geometry, where the surface cracks are
represented by using the point-cloud representation [27, 28]. However, the
digital models need to be updated by the observed physical states typically at
computing time scale within milliseconds. Pre-calculated digital libraries and
surrogate models mitigate this burden, although including extremely rare
events that deviate far from the Gaussian distribution (e.g., bird strikes on
aircraft) remains difficult. Our end-to-end, self-updated model thus offers
FLP from the evolution of fatigue crack patterns and can interact with PSNs
for performance. The demonstrated accuracy and efficiency by considering the
history dependence, statistical noises, and rare events can be integrated into
digital twins of aerospace and nuclear power applications.

## Experimental procedures

### Resource availability

#### Lead contact

Further information and requests for resources should be directed to and will
be fulfilled by the lead contact, Zhiping Xu (xuzp@tsinghua.edu.cn).

#### Materials availability

This study did not generate new unique materials.

#### Data and code availability

All data needed to evaluate the conclusions are present in the paper.
Additional data related to this paper may be requested from the lead contact.
The code used for this study is available at https://github.com/zhaoyj21/FCG.

### Modeling fatigue crack growth

To construct the digital libraries of fatigue crack patterns, we consider a
linear elastic material for the sake of simplicity. Young’s modulus $Y=200$
GPa and Poisson’s ratio $\nu=0.31$ (e.g., for a typical nickel alloy) are
chosen as a representative example [29]. XFEM is used to model FCG under
uniaxial tension and additional shear components. In XFEM, special functions
are added to the continuous displacement fields to model discontinuity
problems such as cracks and interface, which solves difficulties in meshing
high-stress areas near the crack tip in FEA [30]. The step of crack
advancement is set to $0.3\ {\rm mm}$, which is minute enough in comparison to
the size of the structural components ($10\ {\rm mm}$) to ensure convergence.
The angle of crack deflection ($\theta$) is determined by the maximum
tangential stress criterion [31],

$\theta=\arccos{\frac{3K_{\rm II}^{2}+\sqrt{K_{\rm I}^{4}+8K_{\rm I}^{2}K_{\rm
II}^{2}}}{K_{\rm I}^{2}+9K_{\rm II}^{2}}},$ (2)

where $K_{\rm I}$ and $K_{\rm II}$ are mode-$\rm I$ and the $\rm II$ stress
intensity factors (SIFs), respectively. The remaining life of structures with
an existing crack is evaluated by integration using the Paris-Erdogan equation
[32],

${\rm d}a/{\rm d}N=C(\Delta K)^{m},$ (3)

where $\Delta K$ is the difference between the maximum and minimum SIFs in a
load cycle, $C=9.7\times 10^{-12}$ and $m=3.0$ are material coefficients [33].

A path-slicing technique is proposed for the uncertainties in service
conditions, where the structure is discretized into $N_{\rm s}$ segments. The
loading conditions change across their boundaries, which follow the
distribution of statistical noises (assumed to be Gaussian, Figure 3A). Tails
of the Gaussian distribution with a relative probability below $0.05$ are
considered as rare events. The produced digital libraries contain $1,000$
voxel datasets of the crack patterns and their corresponding life to failure.

### Model reduction

Fatigue cracks can be represented as curves or surfaces for 2D or 3D models.
In this work, we use VAE for nonlinear dimensionality reduction into the
latent representations, where the fatigue crack patterns in the digital
libraries follow Gaussian distributions [22]. The encoder and decoder in VAE
approximate the posterior and likelihood distributions, respectively,

$p(\mathbf{z}\mid\mathbf{x})=\frac{p(\mathbf{x}\mid\mathbf{z})p(\mathbf{z})}{p(\mathbf{x})},$
(4)

where $p(\mathbf{z}\mid\mathbf{x})$ is the posterior,
$p(\mathbf{x}\mid\mathbf{z})$ is the likelihood, and $p(\mathbf{z})$ is the
prior distribution, $\mathbf{x}$ is the fatigue crack and $\mathbf{z}$ is the
latent representation in reduced dimension. The high-dimensional data is fed
into the encoder to reduce the dimensionality. The decoder then recovers the
high-dimensional data from the latent reduced representation. The loss
function of the training process is defined by the reconstruction errors and
the KL divergence [22],

${\rm Loss}=\|\mathbf{x}-\mathbf{\widehat{x}}\|+{\rm
KL}(\mathcal{N}(\mathbf{\mu},\bf{\Sigma})\|\mathcal{N}(\bf{0},\bf{I})),$ (5)
${\rm
KL}(\mathcal{N}(\mathbf{\mu},\bf{\Sigma})\|\mathcal{N}(\bf{0},\bf{I}))=\int\mathcal{N}(\mathbf{\mu},\bf{\Sigma})\ln\frac{\mathcal{N}(\mathbf{\mu},\bf{\Sigma})}{\mathcal{N}(\bf{0},\bf{I})}{\rm
d}\mathbf{z},$ (6)

where $\mathbf{\widehat{x}}$ is the reconstruction of the fatigue crack,
$\mathbf{\mu}$ and $\mathbf{\Sigma}$ are the mean and covariance matrix of
latent representations learned by neural networks. The latent representations
in reduced dimension are used to learn the history or path dependence and the
nonlinear mapping in downstream tasks.

### Statistical learning

FCG depends on the loading conditions and the instantaneous crack
configuration with loading-history or crack-path dependence. LSTM and FNN are
integrated to achieve efficient prediction of the path and life of FCG at the
structural level. LSTM is used to train and predict FCG from the observed
fatigue crack patterns. The encoder and decoder both have two LSTM layers
followed by a fully connected layer. There are $100$ neural units in each
layer unless otherwise noted. The Adam optimizer is adopted to update the
model parameters in training with hyperparameters, $\eta=10^{-3}$ (the
learning rate), $\beta_{1}=0.9$, $\beta_{1}=0.99$, and $\epsilon=10^{-7}$
[34]. A re-weighting technique is proposed here for the rare events, where the
weight of fatigue cracks with low probability is boosted by $500$ times. The
predicted growing crack patterns are fed to FNN to forecast the remaining
life.

### Verification and validation

The root mean square error (RMSE) between the predicted crack paths
($\mathbf{r}^{\rm pred}$) and the ground truth ($\mathbf{r}^{\rm truth}$)
obtained from FEA is defined as

${\rm RMSE}=\sqrt{\sum_{i=k}^{n}\frac{(\mathbf{r}^{\rm
pred}_{i}-\mathbf{r}^{\rm truth}_{i})^{2}}{n-k+1}},$ (7)

where the full path of a crack is discretized into $n$ points and points $1-k$
are known from observation. The structural similarity (SSIM) is calculated as

${\rm SSIM}=\frac{(2\mu_{\rm p}\mu_{\rm t}+c_{1})(2\sigma_{\rm
pt}+c_{2})}{(\mu_{\rm p}^{2}+\mu_{\rm t}^{2}+c_{1})(\sigma_{\rm
p}^{2}+\sigma_{\rm t}^{2}+c_{2})},$ (8)

where $\mu_{\rm p}$ and $\mu_{\rm t}$ are the mean values of the prediction
and ground-truth voxel data, respectively. $\sigma_{\rm p}$ and $\sigma_{\rm
t}$ are their variances, and $\sigma_{\rm pt}$ is their covariance.
$c_{1}=(0.01R)^{2}$ and $c_{2}=(0.03R)^{2}$ are parameters defined by the
range of the voxel values, $R$ [35, 36]. A low RMSE value or a high SSIM score
indicates a high accuracy of prediction.

### The complexity of loading profiles

The complexity of loading profiles with various levels of statistical noises
and rare events is measured by employing the symbolic aggregate approximation
(SAX) [24]. SAX transforms time series data such as the loading profiles into
symbolic representations (e.g., words) using piecewise aggregate approximation
(PAA) and discretization [24]. The loading profile within a time span of $m$
($t_{1},t_{2},\cdots,t_{m}$) can be represented by a vector $\mathbf{A}$ in a
reduced $w$-dimensional space according to PAA, that is

$a_{i}=\frac{w}{m}\sum_{j=\frac{m}{w}(i-1)+1}^{\frac{m}{w}i}t_{j},$ (9)
$\mathbf{A}=[a_{1},a_{2},\cdots,a_{w}].$ (10)

$a_{i}$ ($i=1,2,\cdots,w$) are then discretized to letters ($\widehat{a}_{i}$)
into $l$ ($=10$) equally sized regions according to the value of $a_{i}$, that
is

$\widehat{a}_{i}={\rm letter}_{j},\ {\rm if}\
\beta_{j-1}\leq{a}_{i}<\beta_{j},$ (11)

where $\beta_{j-1}$ ($j=1,2,\cdots,l$). The loading profile is then
represented as a word $\widehat{A}$ with $w$ letters.

$\widehat{A}=\widehat{a}_{1},\widehat{a}_{2},\cdots,\widehat{a}_{w},$ (12)

and data complexity of the loading profile can be estimated by the number of
possible words as

${\rm data\ complexity}=l^{w},$ (13)

and $l$ is the size of the alphabet. To ensure the accuracy of PAA, we choose
large values of $w$ as the complexity of the time series increases, which are
$1,\ 4,\ 8,\ 9$ for constant loads, weak statistical noises, strong
statistical noises, and rare events, respectively.

## Acknowledgments

This study was supported by the National Natural Science Foundation of China
through grants 11825203, 11832010, 11921002, 52090032, 12122204, and 11872150.
The computation was performed on the Explorer 100 cluster system of the
Tsinghua National Laboratory for Information Science and Technology.

## Author contributions

Z.X. conceived and supervised the research. Y.Z. and Y.L. performed the finite
element simulations and analysis. Y.Z. developed the statistical learning
codes. All authors wrote the manuscript.

## Declaration of interests

The authors declare that they have no competing financial interests.

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## Figure legends

Figure 1. The landscape of fatigue life prediction (FLP) in engineering

Empirical models can be constructed by fitting to experimental data. Finite
element analysis (FEA) can solve fatigue crack growth (FCG) problems with
complex geometries. Statistical learning models equipped with path slicing and
re-weighting can handle statistical noises and rare events based on physical
simulation data. Digital twins allow real-time structural health monitoring
(SHM), FLP, and maintenance management decision-making.

Figure 2. The statistical learning framework for FLP

The framework contains three modules. FEA is utilized to model the complexity
of FCG by considering the effect of statistical noises and rare events in the
construction of digital libraries. FEA data are fed to dimensionality
reduction and neural network architectures in model training. FLP is achieved
by utilizing the well-trained model in engineering applications.

Figure 3. Statistical noises and rare events in the loading conditions

(A) The procedures of path slicing and re-weighting used to handle the
uncertainties in loading profiles. (B) The distribution of fatigue crack
patterns in the space of major features extracted by variational autoencoder
(VAE). (C) Unsupervised clustering of fatigue crack patterns based on the
probability density of fatigue crack patterns.

Figure 4. Performance of path slicing and re-weighting

(A) Fatigue crack path predictions for a flat plate without path slicing and
re-weighting. The predicted results are not corrected along with the
observation time, $t_{\rm obs}$. (B) Predictions with path slicing, which is
corrected by observation data as $t_{\rm obs}$ increases (shown by gradual
shades). (C) Predictions with path slicing and re-weighting, which are
corrected by observation data and offer accurate forecasting by considering
the effects of statistical noises and rare events. (D) The comparison of
prediction errors by different models.

Figure 5. Performance assessment of FLP

(A) Schematic diagram for calculating the structural similarity (SSIM) of
fatigue cracks in a flat plate. (B) The average accuracy of FLP measured by
SSIM for all samples in the test set, which increases with $t_{\rm obs}$. (C)
The predicted life, the ground truth, and the accuracy of a representative
sample in the test set. The accuracy is defined as
$1-|\tau_{i}-\widehat{\tau}_{i}|/\sum_{i}|\tau_{i}-\widehat{\tau}_{i}|$, where
$\tau_{i}$ and $\widehat{\tau}_{i}$ are the ground truth and predictions of
fatigue life with $t_{\rm obs}$. (D) The average error in FLP measured by root
mean square error (RMSE) for all samples in the test set. The RMSE decreases
with $t_{\rm obs}$. The standard deviations are reported in the error bars.

Figure 6. Reduction of the loading-profile complexity

(A) FLP with path-slicing and re-weighting under loading conditions with
continuous profiles for a flat plate. (B) The convergence analysis of the size
of training sets and $N_{\rm s}$, which is measured by SSIM. (C) The
complexity of loading profiles with various strengths of uncertainties, which
is significantly reduced by introducing path slicing and re-weighting. (D) FCG
in turbine blades with a complex 3D geometry.