prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What molecule was the focus of the study?

Read the following text and answer briefly.

Context 1: \section{Introduction}

Spectral line surveys have revealed that high-mass star-forming
regions are rich reservoirs of molecules from simple diatomic species
to complex and larger molecules (e g ,
\citealt{schilke1997b,hatchell1998b,comito2005,bisschop2007}) 
However, there have been rarely studies undertaken to investigate the
chemical evolution during massive star formation from the earliest
evolutionary stages, i e , from High-Mass Starless Cores (HMSCs) and
High-Mass Cores with embedded low- to intermediate-mass protostars
destined to become massive stars, via High-Mass Protostellar Objects
(HMPOs) to the final stars that are able to produce Ultracompact H{\sc
  ii} regions (UCH{\sc ii}s, see \citealt{beuther2006b} for a recent
description of the evolutionary sequence)  The first two evolutionary
stages are found within so-called Infrared Dark Clouds (IRDCs)   While
for low-mass stars the chemical evolution from early molecular
freeze-out to more evolved protostellar cores is well studied (e g ,
\citealt{bergin1997,dutrey1997,pavlyuchenkov2006,joergensen2007}),
it is far from clear whether similar evolutionary patterns are present
during massive star formation 

To better understand the chemical evolution of high-mass star-forming
regions we initiated a program to investigate the chemical properties
from IRDCs to UCH{\sc ii}s from an observational and theoretical
perspective  We start with single-dish line surveys toward a large
sample obtaining their basic characteristics, and then perform
detailed studies of selected sources using interferometers on smaller
scales  These observations are accompanied by theoretical modeling of
the chemical processes   Long-term goals are the chemical
characterization of the evolutionary sequence in massive star
formation, the development of chemical clocks, and the identification
of molecules as astrophysical tools to study the physical processes
during different evolutionary stages  Here, we present an initial
study of the reactive radical ethynyl (C$_2$H) combining single-dish
and interferometer observations with chemical modeling   Although
C$_2$H was previously observed in low-mass cores and Photon Dominated
Regions (e g , \citealt{millar1984,jansen1995}), so far it was not
systematically investigated in the framework of high-mass star
formation 

\section{Observations}
\label{obs}

The 21 massive star-forming regions were observed with the Atacama
Pathfinder Experiment (APEX) in the 875\,$\mu$m window in fall 2006 
We observed 1\,GHz from 338 to 339\,GHz and 1\,GHz in the image
sideband from 349 to 350\,GHz   The spectral resolution was
0 1\,km\,s$^{-1}$, but we smoothed the data to
$\sim$0 9\,km\,s$^{-1}$  The average system temperatures were around
200\,K, each source had on-source integration times between 5 and 16
min  The data were converted to main-beam temperatures with forward
and beam efficiencies of 0 97 and 0 73, respectively
\citep{belloche2006}  The average $1\sigma$ rms was 0 4\,K   The main
spectral features of interest are the C$_2$H lines around 349 4\,GHz
with upper level excitation energies $E_u/k$ of 42\,K (line blends of
C$_2$H$(4_{5,5}-3_{4,4})$ \& C$_2$H$(4_{5,4}-3_{4,3})$ at
349 338\,GHz, and C$_2$H$(4_{4,4}-3_{3,3})$ \&
C$_2$H$(4_{4,3}-3_{3,2})$ at 349 399\,GHz)  The beam size was $\sim
18''$ 

The original Submillimeter Array (SMA) C$_2$H data toward the
HMPO\,18089-1732 were first presented in \citet{beuther2005c}  There
we used the compact and extended configurations resulting in good
images for all spectral lines except of C$_2$H  For this project, we
re-worked on these data only using the compact configuration  Because
the C$_2$H emission is distributed on larger scales (see
\S\ref{results}), we were now able to derive a C$_2$H image  The
integration range was from 32 to 35\,km\,s$^{-1}$, and the achieved
$1\sigma$ rms of the C$_2$H image was 450\,mJy\,beam$^{-1}$   For more
details on these observations see \citet{beuther2005c} 

\section{Results}
\label{results}

The sources were selected to cover all evolutionary stages from IRDCs
via HMPOs to UCH{\sc ii}s  We derived our target list from the samples
of \citet{klein2005,fontani2005,hill2005,beltran2006}   Table
\ref{sample} lists the observed sources, their coordinates, distances,
luminosities and a first order classification into the evolutionary
sub-groups IRDCs, HMPOs and UCH{\sc ii}s based on the previously
available data  Although this classification is only based on a
limited set of data, here we are just interested in general
evolutionary trends  Hence, the division into the three main classes
is sufficient 

Figure \ref{spectra} presents sample spectra toward one source of each
evolutionary group  While we see several CH$_3$OH lines as well as
SO$_2$ and H$_2$CS toward some of the HMPOs and UCH{\sc ii}s but not
toward the IRDCs, the surprising result of this comparison is the
presence of the C$_2$H lines around 349 4\,GHz toward all source types
from young IRDCs via the HMPOs to evolved UCH{\sc ii}s   Table
\ref{sample} lists the peak brightness temperatures, the integrated
intensities and the FWHM line-widths of the C$_2$H line blend at
349 399\,GHz  The separation of the two lines of 1 375\,MHz already
corresponds to a line-width of 1 2\,km\,s$^{-1}$  We have three C$_2$H
non-detections (2 IRDCs and 1 HMPO), however, with no clear trend with
respect to the distances or the luminosities (the latter comparison is
only possible for the HMPOs)  While IRDCs are on average colder than
more evolved sources, and have lower brightness temperatures, the
non-detections are more probable due to the relatively low sensitivity
of the short observations (\S\ref{obs})  Hence, the data indicate
that the C$_2$H lines are detected independent of the evolutionary
stage of the sources in contrast to the situation with other
molecules   When comparing the line-widths between the different
sub-groups, one finds only a marginal difference between the IRDCs and
the HMPOs (the average $\Delta v$ of the two groups are 2 8 and
3 1\,km\,s$^{-1}$)   However, the UCH{\sc ii}s exhibit significantly
broader line-widths with an average value of 5 5\,km\,s$^{-1}$ 

Intrigued by this finding, we wanted to understand the C$_2$H spatial
structure during the different evolutionary stages   Therefore, we
went back to a dataset obtained with the Submillimeter Array toward
the hypercompact H{\sc ii} region IRAS\,18089-1732 with a much higher
spatial resolution of $\sim 1''$ \citep{beuther2005c}   Albeit this
hypercompact H{\sc ii} region belongs to the class of HMPOs, it is
already in a relatively evolved stage and has formed a hot core with a
rich molecular spectrum   \citet{beuther2005c} showed the spectral
detection of the C$_2$H lines toward this source, but they did not
present any spatially resolved images  To recover large-scale
structure, we restricted the data to those from the compact SMA
configuration (\S\ref{obs})  With this refinement, we were able to
produce a spatially resolved C$_2$H map of the line blend at
349 338\,GHz with an angular resolution of $2 9''\times 1 4''$
(corresponding to an average linear resolution of 7700\,AU at the
given distance of 3 6\,kpc)  Figure \ref{18089} presents the
integrated C$_2$H emission with a contour overlay of the 860\,$\mu$m
continuum source outlining the position of the massive protostar  In
contrast to almost all other molecular lines that peak along with the
dust continuum \citep{beuther2005c}, the C$_2$H emission surrounds the
continuum peak in a shell-like fashion 

\section{Discussion and Conclusions}

To understand the observations, we conducted a simple chemical
modeling of massive star-forming regions  A 1D cloud model with a mass
of 1200\,M$_\sun$, an outer radius of 0 36\,pc and a power-law density
profile ($\rho\propto r^p$ with $p=-1 5$) is the initially assumed
configuration  Three cases are studied: (1) a cold isothermal cloud
with $T=10$\,K, (2) $T=50$\,K, and (3) a warm model with a temperature
profile $T\propto r^q$ with $q=-0 4$ and a temperature at the outer
radius of 44\,K  The cloud is illuminated by the interstellar UV
radiation field (IRSF, \citealt{draine1978}) and by cosmic ray
particles (CRP)  The ISRF attenuation by single-sized $0 1\mu$m
silicate grains at a given radius is calculated in a plane-parallel
geometry following \citet{vandishoeck1988}  The CRP ionization rate is
assumed to be $1 3\times 10^{-17}$~s$^{-1}$ \citep{spitzer1968}  The
gas-grain chemical model by \citet{vasyunin2008} with the desorption
energies and surface reactions from \citet{garrod2006} is used 
Gas-phase reaction rates are taken from RATE\,06 \citep{woodall2007},
initial abundances, were adopted from the ``low metal'' set of
\citet{lee1998} 

Figure \ref{model} presents the C$_2$H abundances for the three models
at two different time steps: (a) 100\,yr, and (b) in a more evolved
stage after $5\times10^4$\,yr  The C$_2$H abundance is high toward the
core center right from the beginning of the evolution, similar to
previous models (e g , \citealt{millar1985,herbst1986,turner1999}) 
During the evolution, the C$_2$H abundance stays approximately
constant at the outer core edges, whereas it decreases by more than
three orders of magnitude in the center, except for the cold $T=10$~K
model   The C$_2$H abundance profiles for all three models show
similar behavior 

The chemical evolution of ethynyl is determined by relative removal
rates of carbon and oxygen atoms or ions into molecules like CO, OH,
H$_2$O  Light ionized hydrocarbons CH$^+_{\rm n}$ (n=2  5) are quickly
formed by radiative association of C$^+$ with H$_2$ and hydrogen
addition reactions: C$^+$ $\rightarrow$ CH$_2^+$ $\rightarrow$
CH$_3^+$ $\rightarrow$ CH$_5^+$   The protonated methane reacts with
electrons, CO, C, OH, and more complex species at later stage and
forms methane   The CH$_4$ molecules undergo reactive collisions with
C$^+$, producing C$_2$H$_2^+$ and C$_2$H$_3^+$  An alternative way to
produce C$_2$H$_2^+$ is the dissociative recombination of CH$_5^+$
into CH$_3$ followed by reactions with C$^+$   Finally, C$_2$H$_2^+$
and C$_2$H$_3^+$ dissociatively recombine into CH, C$_2$H, and
C$_2$H$_2$  The major removal for C$_2$H is either the direct
neutral-neutral reaction with O that forms CO, or the same reaction
but with heavier carbon chain ions that are formed from C$_2$H by
subsequent insertion of carbon  At later times, depletion and
gas-phase reactions with more complex species may enter into this
cycle   At the cloud edge the interstellar UV radiation
instantaneously dissociates CO despite its self-shielding,
re-enriching the gas with elemental carbon 

The transformation of C$_2$H into CO and other species proceeds
efficiently in dense regions, in particular in the ``warm'' model
where endothermic reactions result in rich molecular complexity of the
gas (see Fig ~\ref{model})   In contrast, in the ``cold'' 10\,K model
gas-grain interactions and surface reactions become important   As a
result, a large fraction of oxygen is locked in water ice that is hard
to desorb ($E_{\rm des} \sim 5500$~K), while half of the elemental
carbon goes to volatile methane ice ($E_{\rm des} \sim 1300$~K)  Upon
CRP heating of dust grains, this leads to much higher gas-phase
abundance of C$_2$H in the cloud core for the cold model compared to
the warm model  The effect is not that strong for less dense regions
at larger radii from the center 

Since the C$_2$H emission is anti-correlated with the dust continuum
emission in the case of IRAS\,18089-1732 (Fig \,\ref{18089}), we do
not have the H$_2$ column densities to quantitatively compare the
abundance profiles of IRAS\,18089-1732 with our model  However, data
and model allow a qualitative comparison of the spatial structures 
Estimating an exact evolutionary time for IRAS\,18089-1732 is hardly
possible, but based on the strong molecular line emission, its high
central gas temperatures and the observed outflow-disk system
\citep{beuther2004a,beuther2004b,beuther2005c}, an approximate age of
$5\times10^4$\,yr appears reasonable   Although dynamical and chemical
times are not necessarily exactly the same, in high-mass star
formation they should not differ to much: Following the models by
\citet{mckee2003} or \citet{krumholz2006b}, the luminosity rises
strongly right from the onset of collapse which can be considered as a
starting point for the chemical evolution  At the same time disks and
outflows evolve, which should hence have similar time-scales   The
diameter of the shell-like C$_2$H structure in IRAS\,18089-1732 is
$\sim 5''$ (Fig \,\ref{18089}), or $\sim$9000\,AU in radius at the
given distance of 3 6\,kpc   This value is well matched by the modeled
region with decreased C$_2$H abundance (Fig \,\ref{model})   Although
in principle optical depths and/or excitation effects could mimic the
C$_2$H morphology, we consider this as unlikely because the other
observed molecules with many different transitions all peak toward the
central submm continuum emission in IRAS\,18089-1732
\citep{beuther2005c}  Since C$_2$H is the only exception in that rich
dataset, chemical effects appear the more plausible explanation 

The fact that we see C$_2$H at the earliest and the later evolutionary
stages can be explained by the reactive nature of C$_2$H: it is
produced quickly early on and gets replenished at the core edges by
the UV photodissociation of CO  The inner ``chemical'' hole observed
toward IRAS\,18089-1732 can be explained by C$_2$H being consumed in
the chemical network forming CO and more complex molecules like larger
carbon-hydrogen complexes and/or depletion 

The data show that C$_2$H is not suited to investigate the central gas
cores in more evolved sources, however, our analysis indicates that
C$_2$H may be a suitable tracer of the earliest stages of (massive)
star formation, like N$_2$H$^+$ or NH$_3$ (e g ,
\citealt{bergin2002,tafalla2004,beuther2005a,pillai2006})  While a
spatial analysis of the line emission will give insights into the
kinematics of the gas and also the evolutionary stage from chemical
models, multiple C$_2$H lines will even allow a temperature
characterization  With its lowest $J=1-0$ transitions around 87\,GHz,
C$_2$H has easily accessible spectral lines in several bands between
the 3\,mm and 850\,$\mu$m   Furthermore, even the 349\,GHz lines
presented here have still relatively low upper level excitation
energies ($E_u/k\sim42$\,K), hence allowing to study cold cores even
at sub-millimeter wavelengths   This prediction can further be proved
via high spectral and spatial resolution observations of different
C$_2$H lines toward young IRDCs 

\acknowledgments{H B  acknowledges financial support
  by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft
  (DFG, grant BE2578)  }



Context 2: Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG  5  Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG  7  Comparison between predictions and reference solutions for a new initial condition fort = 1 25, 3 75, 7 5, 12 5, 20, 30  (from left to right and top to down) We note that with longer prediction time the uncertainty bounds increases Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12 5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering  However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial  We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data 
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space  It readily allows for the inclusion of sparse and irregularly sampled training data  The learned, latent dynamics are interpretable and reveal the different temporal scales involved 
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix  Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation 
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework 

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making  They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations   In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution 
In most cases, data-driven reduced-order models are not interpretable  In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics  A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure 
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data  Other frameworks are tailored to specific problems such as molecular dynamics  
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics  The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model  
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics  Therefore we do not have to enforce any parametrizations on the Koopman matrix  
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase  By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed  Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function 
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately   Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed   We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis  
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons  The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II 
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics  In section III we present three numerical illustrations, i e  a system of linear ODEs, a hidden Markov Model and the discretized KS-equation  We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation 
We conclude with a summary and a short discussion about possible next steps  We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process  We consider data from high-dimensional time series x n ∈ R f with n = 1,    , T   We remark that the intervals between the different states do not need to be uniformly spaced 

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f   We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder  The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued  The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ  As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them 

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system  In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables  In addition the latent variables are treated independently  The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters 
This is in contrast to temporal propagators such as LSTMs   For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states 
The symbol is used to indicate a component-wise multiplication  We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue  The real part is associated with growth and decay whereas the imaginary part is representing the periodic component 
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition   In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space 

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data 
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator  At a designated time step we are using the decoder to reconstruct the high-dimensional solution  We applied our algorithm to three systems  First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues 
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified  As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation  Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z  The loss function was optimized using the Adam algorithm   As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm 
We observe in Figure that the algorithm was able to recover the correct values, i e  the eigenvalues 7 and 3 of the given linear ODE  The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value  Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution 
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components 

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process  In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W  
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component  dp 2 dt = (−0 9 + 1 5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points 
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0 0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder  The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3  
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure   We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process  With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i e  we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time 
Afterwards, we employ the decoder to reconstruct the full high-dimensional system  The results can be found in Figure and show very good agreement between predictions and reference data  This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different 
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged 

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]  For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data 
;   The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs  This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers 
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space  We obtained the λ's in Figure   Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying 
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data  We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data  The results for the phase space can be seen in Figure  
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution  All phase-spaces were obtained by using a finite-difference operator on the data or predictions  These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions 
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging  We believe that a probabilistic version of our algorithm could be advantageous here  This section contains a fully probabilistic formulation for the deterministic model discussed before 
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE  The loss function which we optimize is the Evidence-Lower Bound (ELBO) 

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent  Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems 
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i )  The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section)  These parameters along with the state variables z t have to be inferred from the data x t  
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1  As a consequence, a priori, the latent dynamics are stationary  A derivation and reasoning for this choice can be found in Appendix A  Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics 

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n   In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section 

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ  We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed 
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters  In the context of variational inference, we use the following factorization of the approximate posterior i e  we infer only the mean µ and variance σ for each state variable based on the given data points 
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before  It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B  We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm  

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation  We used the same settings as for the deterministic approach but considered up to 10 complex latent variables  The obtained λ's are in Figure   The probabilistic model allows us to quantify the uncertainty in predictions  In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition 
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon  Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds 
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution  The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model  As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar 



ground_truth: The focus of the study was on the reactive radical ethynyl (C$_2$H).

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the framework capture the reduced-order dynamics?

Read the following text and answer briefly.

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Title: Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
Publish Date: March 7, 2023
Author List: Qianying Cao (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology), Anteng Chang (from College of Engineering, Ocean University of China), Junfeng Du (from College of Engineering, Ocean University of China), Lin Lu (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)

Figure

Fig  1: Procedure to compute the response by a combination of Volterra series and Laguerre polynomials
Fig  2: Linear frequency response function: (a) Modulus of H 1 (ω), (b) phase angle of H 1 (ω)
Fig  6: Comparison of h 1 (t) based on the analytical and reconstructed by Laguerre polynomials
Fig  11: Response for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  18: Comparison of original excitations and reconstructed results: (a) Case 1, (b) Case 2 (c) Case 3
Fig  19: Response to irregular excitation for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  23: Input-output dataset used to identify Volterra series: (a) input excitation, (b) output response
Fig  26: Comparison of responses between the predicted and numerical results: (a) response to regular excitation, (b) response to irregular excitation
Parameter values of the irregular excitation

abstract

Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering  To solve these kinds of problems, time-consuming stepby-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used 
In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients  Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods 
Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained  In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated 
For each system, regular excitations and complex irregular excitations with different parameters are studied  Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge-Kutta method 

Introduction

Most real dynamic systems, as encountered in mechanical and civil engineering, are inherently nonlinear and include geometric nonlinearities, nonlinear constitutive relations in material or nonlinear resistances, etc    Nonlinear problems are attracting increasing attention from engineers and scientists 
This work focuses on solving nonlinear system vibration problems, i e , computing transient responses of nonlinear oscillators under arbitrary irregular excitations based on a combination of a pole-residue operation and Volterra series  Because Volterra series are single-valued, the scope of the present study is restricted to nonlinear behaviours without bifurcations  
To analyse nonlinear vibration problems, researchers have performed extensive studies and developed various mathematical methods  Popular methods include step-by-step numerical integration methods in the time domain, such as the Runge-Kutta method  This kind of method not only requires a small time-step resolution for obtaining high-precision solutions but also is prone to numerical instability  
For a long response with small time steps, the time domain methods are very costly in computational time  Volterra series is another widely used method, which is the extension of the Duhamel integral for linear systems   Volterra series can reproduce many nonlinear phenomena, but they are very complex due to higher-dimensional convolution integrals  
Since 1980's, significant progress has been made in the general area of the Volterra series  The reader is referred to Ref  for a quite thorough literature review on the relevant topics  After 2017, most papers focus on Volterra series identification  De Paula and Marques proposed a method for the identification of Volterra kernels, which was based on time-delay neural networks 
Son and Kim presented a method for a direct estimation of the Volterra kernel coefficients  Dalla Libera et al  introduced two new kernels for Volterra series identification  Peng et al  used the measured response to identify the kernel function and performed the nonlinear structural damage detection  Only a few papers concentrated on simplifying the computation of convolution integrals 
Traditional methods for computing convolution integrals involved in the Volterra series have been performed in three distinct domains: time, frequency and Laplace  The time domain method based on Volterra series refers to discrete time convolution methods, which also suffer computational cost problems  
Both the frequency domain method and the Laplace domain method based on the Volterra series consist of three steps: (1) Volterra series are transformed into an algebraic equation in the frequency domain or Laplace domain; the algebraic equation is solved by purely algebraic manipulations; and (3) the solution in Step ( ) is transformed back to the time domain 
Many researchers have used the frequency domain method to compute the responses of nonlinear systems  Billings et al  developed a new method for identifying the generalized frequency response function (GFRF) of nonlinear systems and then predicted the nonlinear response based on these GFRFs  Carassale et al  introduced a frequency domain approach for nonlinear bridge aerodynamics and aeroelasticity 
Ho et al  computed an output frequency domain function of a nonlinear damped duffing system modelled by a Volterra series under a sinusoidal input  Kim et al  identified the higher order frequency response functions by using the nonlinear autoregressive with exogenous input technique and the harmonic probing method 
This type of frequency domain method is much more efficient than the time domain method due to the fast Fourier transform algorithm  However, the frequency domain method not only is limited by frequency resolutions but also suffers from leakage problems due to the use of discrete Fourier transforms  In addition, the frequency domain method calculates only a steady-state response 
A natural response generated by initial conditions and a cross response caused by interactions between a system and an excitation are ignored  In contrast, the Laplace domain method can calculate all response components because initial conditions are considered in the computational procedure  However, it has been restricted to analytical operations for simple excitations, such as sinusoidal excitations and exponential excitations  
The proposed method falls into the category of the Volterra series method computed in the Laplace domain  Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the proposed method follows a similar path as a pole-residue method for linear systems , the proposed method to solve nonlinear system vibration problems is called the generalized pole-residue method 
The main concept of the pole-residue method developed by Hu et al  was that the poles and residues of the response could be easily obtained from those of the input and system transfer function to obtain the closed-form response solution of linear systems  This method included three steps: (1) writing the system transfer function into pole-residue form; (2) writing the excitation into pole-residue form by the Prony-SS method; (3) computing the poles and residues of the response by an algebraic operation based on those from system and excitation 
Compared to Hu et al  , which was regarded as an efficient tool to compute responses of linear systems, the generalized pole-residue method in this paper is introduced to compute responses of nonlinear systems  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the generalized pole-residue method is how to deal with the higher-order poles and their corresponding coefficients  Similar to the Taylor series, the Volterra series representation is an infinite series, and convergence conditions are needed to assure that the representation is meaningful 
Because the proposed method is based on the Volterra series, only the system with convergent Volterra series representation can be treated by the proposed method  The paper is organized as follows  In Section 2, the nonlinear response is modelled by a Volterra series, and Volterra kernel functions are decoupled by Laguerre polynomials 
Then, the pole-residue method for computing explicit responses is developed in Section 3  Numerical studies and discussions are given in Section 4  Finally, the conclusions are drawn in Section 5 

Response calculation based on Volterra series

A nonlinear oscillator, whose governing equation of motion is given by where z(t, y, ẏ) represents an arbitrary nonlinear term; m, c, and k are the mass, damping and linear stiffness, respectively; y(t), ẏ(t) and ÿ(t) are the displacement, velocity and acceleration, respectively; and f (t) is the time-dependent excitation 
If the energy of excitation f (t) is limited, the nonlinear response under zero initial conditions (i e , zero displacement and zero velocity) can be represented by the Volterra series : where N is the order of Volterra series and In Eq  3, h 1 (τ ) is called the first-order Volterra kernel function, which represents the linear behaviour of the system; h n (τ 1 ,      
, τ n ) for n > 1 are the higher-order Volterra kernel functions, which describe the nonlinear behaviour of the system  The complete formulation of y(t) includes infinite series where the labour of calculating the n th term increases quickly with the growth of n  Fortunately, the response accuracy may be ensured by the first several order Volterra series 
This is proved here in numerical studies  The commonly known Laguerre polynomials are represented as : where p i is the order of the Laguerre polynomials and a i is the damping rate  The Laguerre polynomials satisfy the orthogonal relationship expressed as: By using Laguerre polynomials, the Volterra kernel function h n (t 1 ,      
, t n ) in Eq  3 can be decoupled as follows : where the coefficient is computed resorting to the orthogonal relationship in Eq  5: Substituting Eq  6 into Eq  3 yields       The above operation that uses the Laguerre polynomials to decouple Volterra higher order kernel functions has been well-developed 
The reader is referred to Refs  for details about the adopted technique  After decoupling Volterra higher order kernel functions in time, one can regroup Eq  8 into:       By denoting Eq  9 becomes The above procedure to compute the nonlinear response by a combination of Volterra series and Laguerre polynomials is schematically shown in Fig   
Volterra kernel functions h n (t 1 ,       , t n ) can be obtained by either an equation of motion or measured input-output signals  To derive a closedform solution of the response, we must obtain a closed-form solution of x i (t) first  In the following presentation, a closed-form solution of the aforementioned x i (t) and y n (t) is derived by using the pole-residue method 
3  Pole-residue method for calculating x i (t) and y n (t) Performing the Laplace transform of x i (t) in Eq  10 yields where in which Eq  13 includes a single pole and several higher-order poles  For k = 0, −a i is a single pole, and b p i (0) is a corresponding coefficient, namely, the residue  For k > 0, −a i are higher-order poles, and b p i (k) are corresponding coefficients 
For an irregular excitation signal f (t) of a finite duration of T , it can always be approximated into a pole-residue form by using the complex exponential signal decomposition method-Prony-SS : where N ℓ is the number of components; α ℓ and λ ℓ are constant coefficients, which either are real numbers or occur in complex conjugate pairs 
We define λ ℓ = −δ ℓ + iΩ ℓ , where Ω ℓ is the excitation frequency and δ ℓ is the damping factor of the ℓ th component  We denote α ℓ = A ℓ e iθ ℓ , where A ℓ is the amplitude and θ ℓ is the sinusoidal initial phase in radians  Taking the Laplace transform of Eq  15 yields Note that the concept of the Prony-SS method is similar to that of a principal component method 
A smooth excitation usually requires just several terms to achieve a good approximation  For high irregular loadings, including more terms would achieve a better approximation  Substituting Eqs  13 and 16 into Eq  12 yields Expressing xi (s) in its pole-residue form yields where λ ℓ are simple poles, and the corresponding residues are easily obtained by
and −a i are higher-order poles, and the corresponding coefficients are firstly derived as: By taking the inverse Laplace transform of Eq  18, a closed-form solution is obtained: Substituting Eqs  11 and 21 into Eq  2 yields Theoretically speaking, the proposed method for deriving the closed-form solution of the nonlinear response is applicable to any order of the Volterra series 
For practical engineering, usually only the first several order responses dominate  By setting up N = 2, Eq  22 can be simplified into three components: where the natural response, which is only related to system poles, is given by and the cross response, which is related to both system poles and excitation poles, is given by
and the forced response, which is related only to excitation poles, is given by The first term in Eq  26 is the first-order forced response governed by the excitation frequency, i e , the imaginary part of the pole λ ℓ   The second term corresponds to the second-order nonlinear forced response, which includes the sum frequency and difference frequency responses governed by λ ℓ + λ j  
Eq  26 straightforwardly offers visible information about the possible nonlinear vibrations by the cooperation of excitation frequencies  Particularly, consider a sinusoidal excitation f (t) = sin ω r t, which can be expressed as f (t) = γe λt + γ * e λ * t , where γ = −0 5i and λ = iω r   Substituting these values into Eq 
26, the second term of Eq  26 is simplified as where the first term is the difference frequency response, and the second term is the sum frequency response 

Numerical studies

In practical engineering, some systems have an accurate equation of motion  Additionally, some systems have difficulty constructing their equations of motion because of complex nonlinear dynamic behaviours and uncertain system parameters  In this article, a system with a known equation of motion is called a known system, and a system with an unknown equation of motion is called an unknown system for simplicity 
In this section, two numerical studies are presented  The first study verifies the proposed method using a known nonlinear oscillator, and the second study demonstrates the applicability of the proposed method to an unknown system  Throughout the numerical studies, the unit system is the metre-kilogramme-second (MKS) system; for conciseness, explicit units for quantities are omitted 

A known nonlinear system

This study chooses a nonlinear oscillator written as: where mass m = 1, damping c = 1, linear stiffness k 1 = 10, quadratic stiffness k 2 = 20 and cubic stiffness k 3 = 20  It is a case that has been studied in a previously published article   The linear natural frequency of the system ω 0 = k 1 /m = 3 16 and the damping ratio ζ = c/(2mω 0 ) = 15 8% 
This kind of oscillator occurs in many engineering problems, such as a model of fluid resonance in a narrow gap between large vessels   In the model, k 1 y represents the linear restoring force of the fluid, and k 2 y 2 and k 3 y 3 are respectively the quadratic and cubic nonlinear restoring forces of the fluid 

Volterra kernel functions

Generally, the first several order responses dominate the total response of a system  Hence, the order of the Volterra series in Eq  22 is chosen to be 3, namely, N = 3  For computing the first three order responses from Eq  22, the first three order Volterra kernel functions need to be known  Since Volterra kernel functions and corresponding frequency response functions are related by a specific Fourier transform pair, we can first write the first three orders of frequency response functions directly from Eq  28 
Then, Volterra kernel functions are obtained by the inverse Fourier transform  Based on the harmonic probing algorithm , the linear frequency response function (LFRF) H 1 (ω), the quadratic frequency response function (QFRF) H 2 (ω 1 , ω 2 ) and the cubic frequency response function (CFRF) H 3 (ω 1 , ω 2 , ω 3 ) are analytically given by:
and Figures show H 1 (ω), H 2 (ω 1 , ω 2 ) and H 3 (ω 1 , ω 2 , ω 3 ), respectively, which agree well with those reported in Ref    As expected, the modulus of H 1 (ω) in Fig  peaks near the linear natural frequency ω 0 , and the phase angle decreases monotonically from 0 to -π with increasing frequency 
Figure shows the sum frequency QFRF, where the energy converges along the line of ω 1 +ω 2 ≈ ω 0   Therefore, when the sum frequency of a two-tone excitation equals the linear resonant frequency, the second-order response may reach its maximum  Additionally, those pairs of excitations in line ω 1 + ω 2 ≈ ω 0 may produce non-negligible vibration magnitudes due to second-order nonlinear effects 
For the difference frequency QFRF in Fig  (b), the energy converges along two main lines, i e , ω 1 ≈ ω 0 and ω 2 ≈ ω 0   Figures show moduli of H 3 (ω, ω, ω) and H 3 (ω, ω, −ω), which are diagonal terms of the sum frequency CFRF and the difference frequency CFRF, respectively  While the modulus of H 3 (ω, ω, ω) peaks near ω ≈ ω 0 /3 and ω 0 , that of H 3 (ω, ω, −ω) peaks near ω ≈ ω 0 with a small hump around ω ≈ ω 0 /2 
Values at ω ≈ ω 0 /3 and ω 0 /2 may be magnified by higher-order stiffness terms in Eq  28  By performing the inverse fast Fourier transform to Eqs  29-31, the corresponding linear impulse response function h 1 (t), quadratic impulse response function h 2 (t 1 , t 2 ) and cubic impulse response function h 3 (t 1 , t 2 , t 3 ) are obtained 
Here, h 1 (t) and h 2 (t 1 , t 2 ) are plotted in Figs  , respectively, and h 3 (t, t, t) is shown in Fig    In the numerical implementation, Eqs  29-31 have been utilized with the frequency interval ∆ω = 0 1, number of frequency components N n = 1025, and cut-off frequencies 102 4 and −102 4  For decoupling Volterra kernel functions by using Laguerre polynomials, the damping rate and number of Laguerre polynomials for each order Volterra kernel function need to be determined (see Eqs  4 and 6) 
In this example, we set a 1 = a 2 = a 3 = 2 and R 1 = R 2 = R 3 = 24 because coefficients c p 1    pn become very small when R n > 24, n = 1, 2, 3  According to Eq  7, the coefficients of the first three order Volterra kernel functions are calculated, which are shown in Figs  9 and 10  For convenience, Fig  plots only c p 1 p 2 p 3 for p 3 = 0 
With the increase of the order of Laguerre polynomials, coefficients in Figs  9 and 10 gradually decrease, which illustrates how the first several orders of Laguerre polynomials dominate all orders of the Volterra kernel function  With the known Laguerre polynomials and corresponding coefficients, Volterra kernel functions are reconstructed by Eq  6 
For comparison, reconstructed Volterra kernel functions are also plotted in Figs    The reconstructed results agree well with the analytical values, which verifies the accuracy of the decomposition 

Sinusoidal excitation

From Eq  28, we consider a sinusoidal excitation where A and Ω are the amplitude and the frequency, respectively  Five cases of A and Ω are shown in Table   Excitation frequencies in Cases 1 and 2 are larger than the linear natural frequency (ω 0 ≈ 3 16), those in Case 3 are very close to ω 0 , and those in Cases 4 and 5 are smaller than ω 0  
All cases have same amplitudes  The poles of a sinusoidal excitation are λ 1,2 = ±iΩ, and the residues are α 1,2 = ∓iA/2  Numerical values of excitation poles and residues for different cases are listed in Table   Table : Parameter values, poles and residues of the sinusoidal excitation Substituting poles and residues of the excitation, as well as those of the system into Eqs 
20 and 19, response coefficients β p i ,k corresponding to system poles −a i and response coefficients γ p i ,ℓ corresponding to excitation poles λ ℓ are calculated, respectively  According to Eq  22, the first three orders of responses for each case in Table are calculated  Figures )-15(a) show the comparison of responses obtained by the proposed method and the fourth-order Runge-Kutta method with ∆t = 10 −4  
For Cases 1 and 2, the first-order responses agree well with the total responses obtained by the Runge-Kutta method, and the higher-order responses only slightly improve the transient parts  For Cases 3-5, the sum of the first three orders of responses is in good agreement with the Runge-Kutta solution 
When the response nonlinearity increases, higher-order responses need to be considered  In other words, the proposed method can accurately compute the nonlinear responses by choosing a small number N of Volterra series terms  Figures )-15(b) show the contributions of the three response components for the five cases 
In each case, the first-order response is the most dominant component, and the contributions of secondand third-order responses are much less than those of the first-order response  Especially for Cases 1 and 2, whose excitation frequencies are far from the linear natural frequency, second-and thirdorder responses are close to zero 
This may be because the QFRF and CFRF approach zero when the frequency is larger than 4 rad/s (see Figs  )  Furthermore, the mean values of the first-order responses are approximately zero, and those of the second-order responses are always smaller than zero, which are the difference frequency components in Eq  27 
Moreover, it is clearly observed that second-order responses for Cases 3-5 exhibit a periodic oscillation with a period near half of that for the first-order response, which is excited by the sum frequency component of the excitation (see second part of Eq  27)  Compared with steady-state solutions of first-and second-order responses, those of third-order responses in Cases 3-5 are no longer single regular motions 
By performing the FFT, frequency spectra of these three third-order responses are shown in Fig    We find that these three third-order responses are all dominated by their own fundamental harmonic component and the third harmonic (triple frequency) component  Figure shows the computational time to calculate the response of the oscillator for Case 1 by the proposed method, the fourth-order Runge-Kutta method and the convolution method 
The proposed method, which has an explicit solution, is much more efficient in computational time than the latter two methods, which need small time steps to obtain high-precision solutions  In particular, the efficiency of the proposed method increases with the length of the response time  Computation time (sec )
t=0 02s Convolution, t=0 02s Convolution, t=0 001s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the proposed method, the fourth-fifth order Runge-Kutta method and the convolution method regular loading in Case 1

Irregular excitation

In Eq  28, considering an irregular excitation consisting of several cosine functions where N f is the number of cosine components; A n , Ω n and θ n are the amplitude, frequency and phase angle of the n th component, respectively  Table lists three cases of these parameters  In each case, the amplitudes of all components are the same, and phase angles θ n uniformly distributed between 0 and 2π are randomly generated 
To decompose the excitation into a pole-residue form, the Prony-SS method is used, whose concept is similar to that of a principal component method  The readers are referred to Ref  for details  The chosen rank of each case is also shown in Table   Figure shows the comparison of original excitations and reconstructed results of these three cases, which all have excellent agreement 
show the results computed by the fourth-order Runge-Kutta method  In all cases, the sums of the first three orders of responses agree well with those obtained by the Runge-Kutta method  The contributions of the first three orders of responses for each case are plotted in Figs  )-21(b)  Similarly, the system vibration is dominated by the first-order response 
However, the contributions of second-and third-order significantly grow with increasing excitation magnitude and frequency number  Furthermore, when the magnitude of the nonlinear response becomes large, sharp troughs are present  This phenomenon may be induced by the nonlinear stiffness  While the first-order response fails to capture these troughs, the higher-order responses successfully capture these troughs 
Figure plots the computational time to calculate the response of the oscillator for the irregular loading in Case 1 by the proposed method and the fourth-fifth order Runge-Kutta method, respectively  While the fourth-fifth order Runge-Kutta method is more efficient under a small response length, the proposed method becomes much more efficient when the response length is larger than about 130 s 
In addition, the proposed method obtains the explicit response solution, so one can directly obtain the response value at a specific time t p instead of integrating from 0 to t p for traditional numerical methods  Computation time (sec ) Proposed, t=0 02s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the method and the fourth-fifth order Runge-Kutta method for irregular loading in Case 1

An unknown nonlinear

To check the applicability of the proposed method to an unknown nonlinear system, a known input excitation and its corresponding response are used to identify its Volterra kernel functions  When the Volterra kernel functions are known, we can follow the procedure in Section 4 1 to predict system responses 
In this study, the input excitation is white noise with a constant power spectrum S 0 = 0 001, and the corresponding response is obtained by solving Eq  28 by the fourth-order Runge-Kutta method, which is shown in Fig    From Section 4 1, we determine that the sum of the first two orders of responses agrees well with the total response 
In this study, the order of Volterra series N is chosen to be 2, damping rates of Laguerre polynomials are a 1 = a 2 = 2, and numbers of Laguerre polynomials are R 1 = R 2 = 24  To estimate the first two orders of Volterra kernel functions, a matrix equation is constructed using excitation data and response data 
By using the least square method to solve this matrix equation, coefficients c p 1 and c p 1 p 2 in Eq  8 are identified  Figure plots c p 1 and c p 1 p 2 , respectively, which have good agreement with the exact results in Fig    Then, the first two order Volterra kernel functions are constructed by Eq  6 
Compared with the exact results in Figs  , the identified Volterra kernel functions in Fig  completely agree well with the exact solutions  Note that the white noise excitation, which can excite more frequency components of the response, is chosen to obtain good Volterra kernel functions  A regular excitation f (t) = sin(πt) and an irregular excitation f (t) = N f n=1 A n cos(Ω n t + θ n ) with A n = 0 3 and Ω n varying from 0 to 40 with equal interval 1 are chosen as input excitations 
The predicted responses, along with results obtained by the fourth-order Runge-Kutta method, are shown in Fig    In both cases, the proposed method accurately predicts system responses  As presented in Eq  23, a nonlinear response is the sum of three terms: natural response y s (t), forced response y f (t) and cross response y c (t) 
These individual terms, as well as their sum to two excitations, are shown in Figs  27 and 28, respectively  As shown in Figs  and 28, both first-and second-order responses include the natural response y s (t) and the forced response y f (t), but the cross response y c (t) only exists in second-order responses 
When t becomes larger, both y s (t) and y c (t) diminish due to the presence of system damping, and the total response is entirely governed by y f (t)  Moreover, we notice some features at t = 0 for these components, including y s (0) = −y f (0) for the first-order response and y s (0) + y f (0) = −y c (0) for the second-order response, which are due to imposed zero initial conditions 

Conclusions

Considering arbitrary irregular excitations, an efficient generalized pole-residue method to compute the nonlinear dynamic response modelled by the Volterra series was developed  A core of the proposed method was obtaining poles and corresponding coefficients of Volterra kernel functions, then those of each order response modelled by each order Volterra series 
Once the poles and corresponding coefficients of Volterra kernel functions and excitations were both available, the remaining derivation could follow a similar pole-residue method that had been developed for ordinary linear oscillators  To obtain the poles and corresponding coefficients of Volterra kernel functions, two steps were included: (1) using Laguerre polynomials to decouple higher-order Volterra kernel functions with respect to time and (2) obtaining poles and corresponding coefficients of Laguerre polynomials in the Laplace domain 
Because the proposed method gave an explicit, continuous response function of time, it was much more efficient than traditional numerical methods  Moreover, many meaningful physical and mathematical insights were gained because not only each order response but also the natural response, the forced response and the cross response of each order were obtained in the solution procedure 
To demonstrate that the proposed method was not only suitable for a system with a known equation of motion but also applicable to a system with an unknown equation of motion, two numerical studies were conducted  For each study, regular excitations and complex irregular excitations with different parameters were investigated 
The efficiency of the proposed method was verified by the fourth-order Runge-Kutta method  This paper only computes the response under zero initial conditions  The response under non-zero initial conditions will be investigated in our future work 



ground_truth: By using a propagator in the latent space.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What does the paper aim to solve?

Read the following text and answer briefly.

Context 4: 
All men are my brothers  I would have liked to have said it then, and I would like to say it now: all men are my brothers  But all men are not my brothers  Why? Because all women are my sisters  And the brother who denies the rights of his sister: that brother is not my brother  At the very best, he is my half-brother - by definition  Osama is not my brother 
Religion is sensitive ground, as well it might be  Here we walk on eggshells  Because religion is itself an eggshell  Today, in the West, there are no good excuses for religious belief - unless we think that ignorance, reaction and sentimentality are good excuses  This is of course not so in the East, where, we acknowledge, almost every living citizen in many huge and populous countries is intimately defined by religious belief  The excuses, here, are very persuasive; and we duly accept that 'faith' - recently and almost endearingly defined as 'the desire for the approval of supernatural beings' - is a world-historical force and a world-historical actor  All religions, unsurprisingly, have their terrorists, Christian, Jewish, Hindu, even Buddhist  But we are not hearing from those religions  We are hearing from Islam 
Let us make the position clear  We can begin by saying, not only that we respect Muhammad, but that no serious person could fail to respect Muhammad - a unique and luminous historical being  Judged by the continuities he was able to set in motion, he remains a titanic figure, and, for Muslims, all-answering: a revolutionary, a warrior, and a sovereign, a Christ and a Caesar, 'with a Koran in one hand', as Bagehot imagined him, 'and a sword in the other'  Muhammad has strong claims to being the most extraordinary man who ever lived  And always a man, as he always maintained, and not a god  Naturally we respect Muhammad  But we do not respect Muhammad Atta 
Until recently it was being said that what we are confronted with, here, is 'a civil war' within Islam  That's what all this was supposed to be: not a clash of civilisations or anything like that, but a civil war within Islam  Well, the civil war appears to be over  And Islamism won it  The loser, moderate Islam, is always deceptively well-represented on the level of the op-ed page and the public debate; elsewhere, it is supine and inaudible  We are not hearing from moderate Islam  Whereas Islamism, as a mover and shaper of world events, is pretty well all there is 
So, to repeat, we respect Islam - the donor of countless benefits to mankind, and the possessor of a thrilling history  But Islamism? No, we can hardly be asked to respect a creedal wave that calls for our own elimination  More, we regard the Great Leap Backwards as a tragic development in Islam's story, and now in ours  Naturally we respect Islam  But we do not respect Islamism, just as we respect Muhammad and do not respect Muhammad Atta 
المؤامرات على الأديان وجميع الانقلابات المخربة والثورات على القيم والمبادئ خرجت من هذا التراث    وان كل معول هدم كان وراءه توجيه يهودي 
•تذكروا أن الشعب الذي لا يهلك غيره يهلك نفسه 
•يجب ان نخلق الجيل الذي لا يخجل من كشف عورته (ألا تفسر لنا هذه الجملة موجة العرى في الافلام والموضات التى تسود العالم الآن) 
 علينا ان نشعل حربا بين الشعوب ونضرب الدول بعضها ببعض فبهذا يصبح جميع المتحاربين في حاجة الى أموالنا فنفرض عليهم شروطنا 
•الجماهير عمياء فاشتروها بالمال وسوقوها كالبهائم الى أهدافكم 
•سيطروا على الانتخابات ووسائل الاعلام والصحافة (وهم قد سيطروا عليها بالمال والجنس والمرأة في الغرب الرأسمالي وبالحزب والسلطة في العالم الاشتراكي) 
•ادفعوا الجماهير العمياء الى الثورة وسلموهم مقاليد الحكم ليحكموا في غوغائية وغباء (وقد فعلوا هذه في الثورة الفرنسية) وحينئذ نأتي نحن ونعدمهم فنكون منقذين للعالم (وقد اعدموهم جميعاً من روبسبير الى ميرابوا) 
•ارفعوا شعار الحرية واهدموا بها الاخلاق والاسرة والقومية والوطنية 
 ارفعوا شعار العلم واهدموا به الدين    وهذا ما فعله كمال أتاتورك (حفيد مزاراحي) حينما اقام الدولة العلمانية في تركيا ووقف يخطب في البرلمان التركي عام 1923 ساخراً من القرآن 
نحن الآن في القرن العشرين لا نستطيع ان نسير وراء كتاب تشريع يبحث عن التين والزيتون 
•الذي يعرقل مؤامراتكم اوقعوه في فضائح ثم هددوه بكشفها (وقد فعلوها في ووترجيت) او في مآزق مالية ثم تقدموا لانقاذه (وقد فعلها دزرائيلي مع الخديو واستولى على القنال)   وإذ تعذر الامر سارعوا الى اغتياله (وقد فعلوها بكنيدي) ثم اقتلوا قاتله لتدفنوا اسرارنا معه الى الأبد (وقد فعلوها بقاتل كنيدي) 
•اقتلوا القوميات والوطنيات بالدعوة الى الاممية والمواطنة العالمية وقد فعلها ماركس في الشيوعية 
•كل ما عدا اليهود حيوانات ناطقة سخرها الله في خدمة اليهود 
واليهودية ترى ان الله واحد ولكنها تحتكره لنفسها فلا عمل لله الا الحفاظ على اسرائيل وتسخير جميع الشعوب لخدمتها 
واللاهوت اليهودي لا يؤمن بآخرة، وقد شطبوا كل ما جاء عن الآخرة في التوراة    والقيامة عندهم هي قيامة دولتهم في فلسطين والبعث بعثها والنشر نشرها    ويوم الحساب هو اليوم الذي يحاسبون فيه كل الأمم يوم يعود المسيح ويباركهم ويختارهم نواباً له في حكم العالم وإقامة ملكوت الله على الأرض    والعجيب انهم كفروا بالمسيح حينما جاء ثم أعلنوا إيمانهم بعودته وشرطوا هذه العودة بانها رجعة من المسيح ليختارهم رؤساء وحكاماً للعالم الى الأبد 
والفكر اليهودي يلقى غلالة من الأسرار والطلاسم والكتمان والغموض على كل شئ    والكبالا والسحر وعلم الأعداد والحروف وتسخير الشياطين من علومهم التى شغفوا بها وروجوها ونشروها 
وكانت وسيلتهم الى هدم الكتب السماوية هى تفسيرها بالتأويل وذلك برفض المعاني الظاهرة واختراع معان باطنة تهدم الغرض الديني وتفسد هدفه 
ونستطيع ان نرى اثر التوجيه الهودي في الفلسفات العبثية والدمية والمادية والفوضية والإباحية    واحيانا نلمح اسماء يهودية خلفها مثل : سارتر – فرويد – ماركس – ماركوز 
واذا فتحنا ملف الديانة البهائية فإننا نجد اثر التوجيه اليهودي واضحا في كتبها 
عبد الهاء تأليف سليم قبعين القاهرة مطبعة العمران 1922 
مفواضات عبد البهاء الطبعة الاولى 1928م  موعود كل الامم 
جورج تاوزنه مطبوع بإذن من المحفل الروحاني لمصر والسودان 
•اكثر فلاسفة اليونان تعلموا الحكمة من بنى اسرائيل 
•رسالة عبد البهاء هي توحيد المسلمين والنصارى واليهود وجمعهم على أصل نواميس موسى 
•عمل موسى لا يس



ground_truth: The paper aims to solve nonlinear system vibration problems efficiently.