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: When did the Tevatron Collider Run II start and when is it expected to end?

Read the following text and answer briefly.

Context 1: \section{INTRODUCTION}
The Tevatron Collider Run II started in March 2002 and is expected
to continue until the end of this decade  The Tevatron and the 
two detectors, CDF and D\O, have been performing  well in 2004,
each experiment is collecting data at the rate 
of $\approx$10 pb$^{-1}$ per week 
The total  luminosity accumulated by August 2004 is $\approx$500 pb$^{-1}$
per detector 
The rich physics program includes the
production and precision measurement of properties of  standard model (SM)
objects, as well as searches for phenomena beyond standard model 
In this brief review we focus on areas of most interest 
to the lattice community  We present
new results on the top quark mass
and their implication for the mass of the SM Higgs boson, 
on searches for the SM Higgs boson, on evidence for the $X(3872)$ state, 
on searches for pentaquarks, and on $b$ hadron properties 
All Run II results presented here are preliminary  

\section{TOP QUARK MASS}

The experiments CDF and D\O\ published several direct  measurements of
the top quark pole mass, $\ensuremath{M_{\mathrm{top}}}$, 
based on Run I data (1992-1996) 
The ``lepton $+$ jets'' channel yields the most precise determination of
$\ensuremath{M_{\mathrm{top}}}$  Recently, the
D\O\ collaboration published a new measurement~\cite{Mtop1-D0-l+j-new},
based on a powerful analysis technique yielding  greatly improved precision 
The differential probability 
that the measured variables in any event correspond to the signal
is calculated as a function of $\ensuremath{M_{\mathrm{top}}}$  
The maximum in the product of the individual event probabilities 
provides the best estimate of $\ensuremath{M_{\mathrm{top}}}$ 
The critical differences from previous analyses 
in the lepton $+$ jets decay channel lie in 
the assignment of more 
weight to events that are well measured or more likely to correspond to  
$t \bar t$ signal, 
and  the handling of the combinations of final-state objects
(lepton, jets, and imbalance in transverse momentum) 
and their identification with
top-quark decay products in an event  
The new combined value for the top-quark mass from Run I is 
$\ensuremath{M_{\mathrm{top}}}  =  178 0\pm4 3~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 

In Run II, both collaborations  have been exploring several different techniques 
for $\ensuremath{M_{\mathrm{top}}}$
measurements  The best single CDF result comes from a dynamic likelihood method
(DLM)  The method is similar to
the technique used in Ref ~\cite{Mtop1-D0-l+j-new} 
The result is $\ensuremath{M_{\mathrm{top}}} = 177 8^{+4 5}_{-5 0} (stat) \pm  6 2 (syst) ~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
The joint likelihood of the selected events is shown in Fig  ~\ref{fig:cdf_tml}  
The Run II goal is a 1\% uncertainty on $\ensuremath{M_{\mathrm{top}}}$  




\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=5 8cm,width=8 1cm]  {data_22ev_likelihood eps}
\vspace*{-1 2cm}
\caption{The joint likelihood of top candidates(CDF) }
\label{fig:cdf_tml}
\end{figure}




\section{SEARCH FOR SM HIGGS BOSON}


The constraints on the SM Higgs ($H$)  boson  mass from
published  measurements, updated to include the new D\O\ top mass
measurement~\cite{Mtop1-D0-l+j-new}, are
$M_H = 117 ^{+67}_{-45}~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$, $M_H < 251~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ at 95\% C L 
The  new most likely  value of $M_H$
is above the experimentally excluded range,
and sufficiently low for $H$ to be observed at the Tevatron 


\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=7 5cm,width=7 8cm]  {d0_wbb_fig_3_err eps}
\vspace*{-1 1cm}
\caption{Distribution of the dijet
invariant mass for $W+2 b$-tagged jets  events,
compared to the expectation (D\O)  
}
\label{fig:d0_wbb_2tag}
\end{figure}



D\O\  has conducted a search for $H$ at $M_H < 140~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
in the production channel  
$p \bar{p} \rightarrow WH \rightarrow  e \nu b \bar{b}$  
The experimental signature of  $WH \rightarrow e \nu b \bar{b}$
is a final state with 
one high $p_T$ electron, two  $b$ jets, and
large missing transverse energy  resulting from
the undetected neutrino 
The dominant backgrounds to $WH$ production
are  $W b \bar{b}$, $t \bar{t}$ and single-top production 
The distribution 
of the dijet mass for events with two $b$-tagged jets is shown in
Fig ~\ref{fig:d0_wbb_2tag}  
Also shown is the  expected contribution ($0 06$ events)  
from the $b \bar{b}$ decay of a
SM Higgs boson with $M_H =$ 115 $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
No events are observed in the  dijet mass window of 85--135  $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
D\O\ sets a limit on the cross section
for $\sigma( p\bar{p} \rightarrow WH) \times B(H \rightarrow b \bar{b}) $
of 9 0 pb at the 95\% C L ,  for a 115  $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ Higgs boson 
The results for mass points 105, 125, and 135 $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$
 are 11 0, 9 1 and 12 2 pb,  respectively 



\begin{figure}[htb]
\vspace*{-1 2cm}
\includegraphics[height=0 33\textheight,width=8 0cm]{whww_aps04_bw eps}

\vspace*{-1 2cm}
\caption{95\% limits on the $H$ production (CDF) }
\label{fig:cdf_whww}
\end{figure}


CDF  has done  a similar search, allowing either an  electron or a muon  
in the final state   Both groups have also searched for $H$ produced in
gluon-gluon fusion, with subsequent decay to a pair of $W$ bosons 
The CDF results for both channels  are shown in Fig ~\ref{fig:cdf_whww}  



\section{THE STATE X(3872)}


\begin{figure}[htb]

\includegraphics[height=8 0cm,width=7 5cm]  {X3872cdfPRL1FullM eps}
\vspace*{-1cm}
\caption{The $X(3872)$ signal (CDF) }
\label{fig:cdf_x}
\end{figure}




 The existence of the $X(3872)$ state discovered by 
the Belle Collaboration~\cite{Belle-X}
 has been confirmed 
 in $p \bar{p}$ collisions by  CDF~\cite{cdf-X} (see Fig ~\ref{fig:cdf_x})
and D\O~\cite{d0-X} 
 It is still unclear whether this particle is a $c\bar{c}$ state,
 or a more complex object   When the data are separated according to
production and decay variables, D\O\  finds no significant
differences between the $X(3872)$ and
the $c \bar{c}$ state $\psi(2S)$ 
CDF has analysed the ``lifetime'' distribution of the $X(3872)$ events in order to
quantify what fraction of this state arises from decay of $B$ hadrons, as opposed to
those produced promptly  The authors find that for the selected samples
28 3$\pm$1 0$(stat)\pm$0 7$(syst)$\% of $\psi(2S)$ candidates are from $b$ decays,
whereas 16 1$\pm$4 9$(stat)\pm$2 0$(syst)$\% of $X$ mesons arise from such decays 





\section{SEARCH FOR PENTAQUARKS}



\begin{figure}[htb]

\includegraphics[height=0 27\textheight,width=7 6cm]  {mpks_1stminbias eps}
\vspace*{-1 2cm}

\caption{Invariant mass distribution of an identified proton and a $K^0_s$ candidate  (CDF)
}
\label{fig:pqtheta}
\end{figure}



\begin{figure}[htb]

\vspace*{-0 9cm}
\includegraphics[height=0 25\textheight,width=8 0cm]  {CM_xicst_cc_1 eps}
\vspace*{-1 2cm}
\caption{Invariant mass distribution of the $(\Xi^-,\pi^+)$ system  (CDF) 
}
\label{fig:pqxi}
\end{figure}


\begin{figure}[htb]
\vspace*{-0 9cm}

\includegraphics[height=0 25\textheight,width=7 6cm]  {theta_note_dstp_dedx_pt eps}
\vspace*{-1 2cm}
\caption{Mass of the ($D^{*+}\bar p$) system  The arrow indicates the position of 
the $\Theta_c$ state (CDF) }
\label{fig:pqthetac}
\end{figure}



Following reports of evidence for exotic
baryons containing five quarks (pentaquarks), CDF has analysed 
its data for evidence of the following pentaquarks:
$\Theta^+$ ($uud\bar d \bar s$), doubly strange states 
$\Xi_{3/2}$, charmed states $\Theta_c$, and, most recently, 
a state $(udus\bar b)$, dubbed $R^+_s$, through its weak decay to $(J/\psi, p)$  
With its excellent particle indentification and mass resolution,
CDF has a unique capability to search for  pentaquark states 
The signals of known states: $\phi$, $\Lambda$,
$\Lambda(1520)$, $K^*$, $\Xi$, 
compare favorably with those provided
by the authors of  the pentaquark evidence 
The group finds no evidence for pentaquark states, see Figs 
~\ref{fig:pqtheta},{\ref{fig:pqxi},\ref{fig:pqthetac} 
This can be interpreted as an indication that the pentaquark production 
in $p \bar p$ collisions is heavily suppressed compared to the conventional
hadron production, or as an evidence against the existence of pentaquarks 

\clearpage

\section{RECENT B PHYSICS RESULTS}


\subsection{Spectroscopy}

CDF has measured the mass of $b$ hadrons in exclusive $J/\psi$ channels 
The measurements of the $B_s$ and $\Lambda_b$ (Fig  \ref{fig:masslb})
masses are the current world's best \\

$m(B^+)$ = 5279 10$\pm$0 41$(stat)\pm$0 36$(syst)$,

$m(B^0)$ = 5279 63$\pm$0 53$(stat)\pm$0 33$(syst)$,

$m(B_s)$ = 5366 01$\pm$0 73$(stat)\pm$0 33$(syst)$,

$m(\Lambda_b)$ = 5619 7$\pm$1 2$(stat)\pm$1 2$(syst)$ MeV/$c^2$ \\


\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0 30\textheight,width=7 5cm]  {lambdav1c eps}
\vspace*{-1cm}

\caption{The mass spectrum of $\Lambda_b$ candidates (CDF) }
\label{fig:masslb}
\end{figure}


D\O\ reports the first observation of the excited $B$ mesons 
$B_1$ and $B^*_2$ as two separate states in fully reconstructed
decays to $B^{(*)}\pi$  The mass of $B_1$ is measured to be
5724$\pm$4$\pm$7 MeV/c$^2$, and the mass difference $\Delta M$ between
$B^*_2$ and $B_1$ is 23 6$\pm$7 7$\pm$3 9 MeV/c$^2$
(Fig   \ref{fig:d0_bexc}) 

D\O\ observes semileptonic $B$ decays to narrow $D^{**}$ states,
the orbitally excited states  of the $D$ meson
seen as resonances in the $D^{*+}\pi^-$ invariant mass spectrum 
The $D^*$ mesons are reconstructed through the decay sequence 
$D^{*+} \rightarrow D^0\pi^+$, $D^0\rightarrow K^-\pi^+$ 
The invariant mass  of oppositely charged $(D^*,\pi)$ pairs
is shown in Fig   \ref{fig:d0_dstst} 
The mass peak between 2 4 and 2 5 GeV/$c^2$ can be interpreted as two merged 
narrow $D^{**}$ states, $D^0_1(2420)$ and $D^0_2(2460)$ 
The combined branching fraction is 
$ {\cal B}(B\rightarrow D^0_1,D^0_2)\cdot {\cal B}(D^0_1,D^0_2\rightarrow D^{*+}\pi^-)=(0 280\pm0 021(stat)\pm0 088(syst)$\%  The systematic error includes the unknown phase between the
two resonances  Work is in progress on extracting the two Breit-Wigner
amplitudes 


\begin{figure}[htb]
\vspace*{-2mm}
\hspace*{-3mm}
\includegraphics[height=0 28\textheight,width=8 3cm]  {B08F02 eps}

\vspace*{-1cm}
\caption{Mass difference $\Delta M = M(B\pi)-M(B)$ for exclusive $B$ decays 
The background-subtracted signal is a sum of 
$B^*_1 \rightarrow B^* \pi$, $B^* \rightarrow B \gamma $ (open area)
and $B^*_2 \rightarrow B^*\pi$ $B^*\rightarrow B \gamma$ (lower peak in the shaded area)
and $B^*_2 \rightarrow B \pi$ (upper peak in the shaded area)  
(D\O) }
\label{fig:d0_bexc}
\end{figure}


\begin{figure}[htb]
\includegraphics[height=0 25\textheight,width=7 5cm]  {B05F03 eps}

\vspace*{-1cm}
\caption{The invariant mass distribution of
$(D^*,\pi)$ pairs, opposite sign (points) and same-sign (solid histogram) }
\label{fig:d0_dstst}
\end{figure}






\subsection{Lifetimes}


CDF and D\O\ have measured  lifetimes of $b$ hadrons through the exclusively
reconstructed decays $B^+ \rightarrow J/\psi K^+$, $B^0 \rightarrow J/\psi K^{*0}$,
$B_s \rightarrow J/\psi \phi$, 
and $\Lambda_b \rightarrow J/\psi \Lambda$
(Fig  \ref{fig:d0_lbctau}) 
The latest results are:  \\



 $\tau(B^+)$=1 65 $\pm$ 0 08 $^{+0 096}_{-0 123}$  ps ~(D\O\ 2003),

 $\tau(B^+)$=1 662 $\pm$ 0 033  $\pm$ 0 008  ps ~(CDF),

 $\tau(B^0_d)$=1 473  $^{+0 052}_{-0 050}$ $\pm$ 0 023    ps ~(D\O) 

 $\tau(B^0_d)$=1 539 $\pm$ 0 051  $\pm$ 0 008  ps ~(CDF),

 $\tau(B^0_s)$=1 444   $^{+0 098}_{-0 090}$ $\pm$ 0 020   ps ~(D\O),

 $\tau(B^0_s)$=1 369 $\pm$ 0 100 $\pm$ $^{+0 008}_{0 010}$  ps ~(CDF),


 $\tau(\Lambda_b)$=1 221 $^{+0 217}_{-0 179}$ $\pm$ 0 043  ps ~(D\O),


 $\tau(\Lambda_b)$=1 25 $\pm$ 0 26 $\pm$ 0 10  ps ~(CDF 2003) \\



The measured lifetimes correspond to the following lifetime ratios:\\

$\tau(B^+)/\tau(B^0_d)$   =  1 080$\pm$0 042     ~(CDF),
 
$\tau(B^0_s)/\tau(B^0_d)$ =  0 890$\pm$0 072    ~(CDF),

$\tau(B^0_s)/\tau(B^0_d)$ = 0 980$ ^{+0 075}_{-0 070}   \pm$0 003    ~(D\O),

$\tau(\Lambda_b)/\tau(B^0_d)$ =  0 874$ ^{+0 169}_{-0 142}   \pm$0 028    ~(D\O) \\



\begin{figure}[htb]
\includegraphics[height=0 3\textheight,width=8 2cm]  {d0_lbctau_B11F02 eps}
\vspace*{-1cm}

\caption{ Fit projection on  $c\tau$ for the $\Lambda_b$ candidates   (D\O)}
\label{fig:d0_lbctau}
\end{figure}


The $B_s$ lifetime measurements listed above are results of
a single-lifetime fit to data, integrated over the decay angles 
Because  of the presence of  final
states  common to ${B_s^0}$\ and its charge conjugate ${\overline{B}_s^0}$,
the two meson states   are expected
to mix in such a way that the two CP  eigenstates may have a relatively
large lifetime difference 
It is possible to
separate the two CP components of ${B_s^0 \rightarrow J/\psi \phi}$\ and thus to measure the
lifetime difference by studying the time evolution of the
polarization states of the vector mesons in the final state 
CDF has carried out a combined analysis of $B_s$ lifetimes 
and polarization amplitudes  The results for the lifetimes of the
low mass (CP even) and high mass (CP odd) eigenstates, and the relative 
width difference are:\\

 $\tau_L = 1 05 ^{+0 16}_{-0 13} \pm 0 02$ ~ps,
 
 $\tau_H = 2 07 ^{+0 58}_{-0 46} \pm 0 03$ ~ps,

 $\Delta \Gamma /\overline \Gamma   = 0 65 ^{+0 25}_{-0 33} \pm 0 01$ \\

Figure \ref{fig:cdf_dg} shows  the scan of the likelihood function 
for $\Delta \Gamma /\overline \Gamma$ 
Pseudoexperiments tossed with $\Delta \Gamma /\overline \Gamma =0$
yield the betting odds for observing the above results at
1/315  For $\Delta \Gamma /\overline \Gamma = 0 12$ (SM prediction,
which has recently been updated to 0 14$\pm$0 05~\cite{dg_un}) the betting odds are
1/84 

\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0 3\textheight,width=8 2cm]  {cdf_scan-dg-un eps}

\vspace*{-1cm}
\caption{Scan of the likelihood function 
for $\Delta \Gamma /\overline \Gamma$ (CDF) 
}
\label{fig:cdf_dg}
\end{figure}




D\O\ has used a novel technique to  measure the lifetime ratio
of the charged and neutral $B$ mesons, exploiting the large
semileptonic sample  $B$ hadrons were reconstructed in the channels
$B\rightarrow \mu^+ \nu D^*(2010)^-X$, which are dominated by $B^0$ decays, 
and  $B\rightarrow \mu^+ \nu D^0X$, which are dominated by $B^+$ decays 
The lifetime ratio was
obtained from the variation of the ratio of the number of events in these two
processes at different decay lengths 
The result is \\


$\tau(B^+)/\tau(B^0_d)$   =  1 093$\pm$0 021$\pm$0 022       ~(D\O)




\subsection{Towards $B_s$ mixing}

Measurement of the $B_s$ oscillation frequency via ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing
will provide an important constraint on the CKM matrix  The oscillation
frequency is proportional to the mass difference between the mass eigenstates,
$\Delta m_s$, and is related to the CKM matrix through 
$\Delta m_s \propto |V_{tb}V_{ts}|$  When combined with the
$B_d$ mass difference, $\Delta m_d$ it helps in extraction of $|V_{td}|$,
and thereby the CP violating phase  

As a benchmark for future $B_s$ oscillation measurement, both groups
study  $B_d$ mixing, gaining an understanding of the different components
of a $B$ mixing analysis (sample composition, flavor tagging, vertexing,
asymmetry fitting)  For a sample of partially reconstructed decays
$B\rightarrow D^*(2010)^+\mu^-X$, D\O\ obtains 
$\Delta m_d = 0 506 \pm 0 055 (stat) \pm  0 049 (syst))$ ps$^{-1}$ and
$\Delta m_d = 0 488 \pm 0 066 (stat) \pm  0 044 (syst))$ ps$^{-1}$
when employing  opposite side muon tagging and the same side tagging,
respectively 

The CDF result for semileptonic channels is
$\Delta m_d = 0 536 \pm 0 037 (stat) \pm  0 009 (s c ) \pm 0 015 (syst)$ ps$^{-1}$ 
CDF also reports a result on $B$ oscillations using fully reconstructed
decays:
$\Delta m_d = 0 526 \pm 0 056 (stat) \pm  0 005 (syst))$ ps$^{-1}$ 

Reconstructing $B_s$ decays into different final states is another
important
 step in the ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing analysis 
Thanks to the  large muon and tracking coverage,   D\O\ is accumulating
a  high statistics sample of semileptonic $B_s$ decays 
D\O\ reconstructs the $B_s \rightarrow D^+_s \mu^- X$ decays, with
$D^+_s \rightarrow \phi \pi^+ $ and
$D^+_s \rightarrow K^* K^+ $,
at a rate of $\approx$ 40(25) events per pb$^{-1}$,  respectively 
Figure \ref{fig:d0_bsdsphipi} shows the mass distribution of the
$D^+_s \rightarrow \phi \pi$ candidates 


\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=0 3\textheight,width=8 0cm]  {blds-250 eps}
\vspace*{-1 2cm}
\caption{  $D^+_s \rightarrow \phi \pi^+$  signal  (D\O)}
\label{fig:d0_bsdsphipi}
\end{figure}


\begin{figure}[htb]
\vspace*{-10mm}
\hspace*{-4mm}
\includegraphics[height=0 35\textheight,width=7 9cm]  {cdf_Bs-DsPi-PhiPi eps}

\vspace*{-1 0cm}
\caption{ $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$  signal   (CDF)}
\label{fig:cdf_bsdsphipi}
\end{figure}


CDF has clean signals for fully hadronic, flavor-specific  $B_s$ decays,
providing the best sensitivity to $B_s$ oscillations at high
$\Delta m_s$  Figure \ref{fig:cdf_bsdsphipi} shows the signal for
the best channel, $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$ 

\clearpage


\subsection{Rare decays}

The purely leptonic decays $B_{d,s}^0 \rightarrow \mu^+
\mu^-$ are flavor-changing neutral current (FCNC) processes 
In the standard model, these decays are forbidden at the tree level and
proceed at a very low rate through higher-order diagrams 
The latest SM prediction~\cite{sm_ref3}
is ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-)=(3 42\pm 0 54)\times
10^{-9}$, where the error is dominated by non-perturbative uncertainties  The
leptonic branching fraction of the $B_d^0$ decay is suppressed by CKM matrix elements $|V_{td}/V_{ts}|^2$
leading to a predicted SM branching fraction of $(1 00\pm0 14)\times 10^{-10}$ 
The best published experimental bound (Fig ~\ref{fig:cdf_bsmumu})
 for the branching fraction
of $B^0_s$ $(B^0_d)$ is presently
${\cal B}(B^0_s \, (B^0_d) \rightarrow \mu^+\mu^-)<7 5\times 10^{-7}\, 
(1 9\times 10^{-7})$ at the 95\% C L ~\cite{cdfII} 
The decay amplitude of $B^0_{d,s} \rightarrow \mu^+ \mu^-$ can be
significantly enhanced in some extensions of the SM  

\begin{figure}[htb]
\includegraphics[height=8 3cm,width=7 9cm]  {cdfbsmumu_results_prl eps}

\vspace*{-1cm}
\caption{Invariant mass for the events passing all requirements  (CDF)}
\label{fig:cdf_bsmumu}
\end{figure}


Assuming no contributions 
from the decay $B^0_d\rightarrow \mu^+\mu^-$ in the signal region,
D\O\  finds the conservative upper limit on the branching fraction 
to be ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-) \leq 4 6\times 10^{-7}$ 
at the 95\% C L  (Fig ~\ref{fig:d0_bsmumu}) 






\begin{figure}[htb]
\includegraphics[height=5 0cm,width=8 0cm]  {B06F03 eps}
\vspace*{-1cm}
\caption{Invariant mass for the events  passing all requirements  (D\O)}
\label{fig:d0_bsmumu}
\end{figure}


Context 3: 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,   



ground_truth: The Tevatron Collider Run II started in March 2002 and is expected to continue until the end of this decade.

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 position did Simon English hold in the 2008 general election?

Read the following text and answer briefly.

Context 2: Sir Simon William English  (born 30 December 1961) is a New Zealand former National Party politician who served as the 39th prime minister of New Zealand from 2016 to 2017  He had previously served as the 17th deputy prime minister of New Zealand and minister of finance from 2008 to 2016 under John Key and the Fifth National Government 

A farmer and public servant before entering politics, English was elected to the New Zealand Parliament in  as the National Party's candidate in the Wallace electorate  He was elevated to Cabinet in 1996 and in 1999 was made minister of finance, although he served for less than a year due to his party's loss at the 1999 general election  In October 2001, English replaced Jenny Shipley as the leader of the National Party (and consequently as Leader of the Opposition)  He led the party to its worst defeat at the 2002 general election, and as a consequence, in October 2003 he was replaced as leader by Don Brash 

In November 2006, after Brash's resignation, English became deputy leader under John Key  After National's victory at the 2008 general election, he became deputy prime minister and was also made minister of finance for the second time  Under English's direction New Zealand's economy maintained steady growth during National's three terms of government  He became a list-only MP after stepping down as an electorate MP at the 2014 general election 

John Key resigned as leader of the National Party and prime minister in December 2016  English won the resulting leadership election unopposed and was sworn in as prime minister on 12 December 2016  His tenure was only ten months, and included a three-month election campaign  In the 2017 general election, National won the largest number of seats but fell short of a majority  The parties holding the balance of power declined to support the existing government, and English was subsequently replaced as prime minister by Jacinda Ardern, leader of the Labour Party  English initially continued on as Leader of the Opposition, but resigned as leader of the National Party on 27 February 2018 and left parliament two weeks later 

Early life
English was born on 30 December 1961 at Lumsden Maternity Centre in Lumsden  He is the eleventh of twelve children of Mervyn English and Norah (née O'Brien) English  His parents purchased Rosedale, a mixed sheep and cropping farm in Dipton, Southland from Mervyn's uncle, Vincent English, a bachelor, in 1944  English was born in the maternity unit at Lumsden 

English attended St Thomas's School in Winton, then boarded at St  Patrick's College in Upper Hutt, where he became head boy  He played in the first XV of the school's rugby team  English went on to study commerce at the University of Otago, where he was a resident at Selwyn College, and then completed an honours degree in English literature at Victoria University of Wellington 

After finishing his studies, English returned to Dipton and farmed for a few years  From 1987 to 1989, he worked in Wellington as a policy analyst for the New Zealand Treasury, at a time when the free market policies favoured by Labour's finance minister Roger Douglas (known collectively as "Rogernomics") were being implemented 

English joined the National Party in 1980, while at Victoria University  He served for a period as chairman of the Southland branch of the Young Nationals, and became a member of the Wallace electorate committee  After moving to Wellington, he served for periods on the Island Bay and Miramar electorate committees, respectively 

Fourth National Government (1990–1999)

At the 1990 general election, English stood as the National candidate in Wallace, replacing the retiring Derek Angus, and was elected with a large majority  He would hold this seat, renamed Clutha-Southland in 1996, until 2014  He and three other newly elected National MPs (Tony Ryall, Nick Smith, and Roger Sowry) were soon identified as rising stars in New Zealand politics, and at various points were dubbed the "brat pack", the "gang of four", and the "young Turks"  In his first term in parliament, English chaired a select committee into social services  He was made a parliamentary under-secretary in 1993, serving under the Minister of Health 

First period in cabinet (1996–1999)
In early 1996, English was elevated to cabinet by Prime Minister Jim Bolger, becoming the Minister for Crown Health Enterprises and Associate Minister of Education (to Wyatt Creech)  He was 34 at the time, becoming the cabinet's youngest member  After the 1996 general election, the National Party was forced into a coalition with New Zealand First to retain government  In the resulting cabinet reshuffle, English emerged as Minister of Health  However, as a condition of the coalition agreement, NZ First's Neil Kirton (a first-term MP) was made Associate Minister of Health, effectively becoming English's deputy  This arrangement was described in the press as a "shotgun marriage", and there were frequent differences of opinion between the two ministers  After their relationship became unworkable, Kirton was sacked from the role in August 1997, with the agreement of NZ First leader Winston Peters 

As Minister of Health, English was responsible for continuing the reforms to the public health system that National had begun after the 1990 general election  The reforms were unpopular, and health was perceived as one of the government's weaknesses, with the health portfolio consequently being viewed as a challenge  English believed that the unpopularity of the reforms was in part due to a failure in messaging, and encouraged his National colleagues to avoid bureaucratic and money-focused language (such as references to "balance sheets" and "user charges") and instead talk about the improvements to services the government's reforms would bring  He also rejected the idea that public hospitals could be run as commercial enterprises, a view which some of his colleagues had previously promoted 

By early 1997, as dissatisfaction with Bolger's leadership began to grow, English was being touted as a potential successor, along with Jenny Shipley and Doug Graham  His age (35) was viewed as the main impediment to a successful leadership run  National's leadership troubles were resolved in December 1997, when Bolger resigned and Shipley was elected to the leadership unopposed  English had been a supporter of Bolger as leader, but Shipley reappointed him Minister of Health in her new cabinet 

English was promoted to Minister of Finance in a reshuffle in January 1999, a position which was at the time subordinate to the Treasurer, Bill Birch  After a few months, the pair switched positions as part of Birch's transition to retirement, with English assuming the senior portfolio  In early interviews, he emphasised his wish to be seen as a pragmatist rather than an ideologue, and said that the initiatives of some of his predecessors (Roger Douglas's "Rogernomics" and Ruth Richardson's "Ruthanasia") had focused on "fruitless, theoretical debates" when "people just want to see problems solved" 

Opposition (1999–2008)

After the National Party lost the 1999 election to Helen Clark's Labour Party, English continued on in the shadow cabinet as National's spokesperson for finance  He was elected deputy leader of the party in February 2001, following the resignation of Wyatt Creech, with Gerry Brownlee being his unsuccessful opponent 

Leader of the Opposition
In October 2001, after months of speculation, Jenny Shipley resigned as leader of the National Party after being told she no longer had the support of the party caucus  English was elected as her replacement unopposed (with Roger Sowry as his deputy), and consequently became Leader of the Opposition  However, he did not openly organise against Shipley, and according to The Southland Times "there was almost an element of 'aw, shucks, I'll do it then' about Mr English's ascension" 

Aged 39 when he was elected, English became the second-youngest leader in the National Party's history, after Jim McLay (who was 38 when elected in 1984)  He also became only the third Southlander to lead a major New Zealand political party, after Joseph Ward and Adam Hamilton  However, English failed to improve the party's performance  In the 2002 election, National suffered its worst electoral defeat ever, gaining barely more than twenty percent of the vote  English described it as "the worst day of my political life"  Both party insiders and the general public were split as to how much to blame him for the loss, but most of the party believed that English would be able to rebuild National's support 

By late 2003, however, National's performance in opinion polls remained poor  The party had briefly increased its popularity in the year following the election, but by October its support had fallen to levels only slightly better than what it achieved in the last ballot  English also appeared in a boxing match for a charity against entertainer Ted Clarke  This did not boost his polling or that of the National party either, with suggestions that it devalued his image as a serious politician  Don Brash, former governor of the Reserve Bank and a relative newcomer to politics, began to build up support to replace English  On 28 October, Brash gained sufficient backing in Caucus to defeat English in a leadership contest 

Shadow cabinet roles and deputy leader
On 2 November 2003, when Brash changed responsibilities for certain MPs, English became National's spokesman for education, ranked at fifth place in the party's parliamentary hierarchy  He remained in parliament after the 2005 election  In his new shadow education portfolio, English performed strongly, and remained a party favourite despite his election defeat as leader in 2002, eventually being returned to the finance portfolio in August 2004 as deputy spokesman (while still retaining responsibility for education) 

In November 2006, Brash resigned as leader  English was considered as a potential replacement leader (running against John Key) or deputy leader (against incumbent Gerry Brownlee) in the ensuing leadership election  However, a contest was avoided when the MPs agreed a Key/English ticket would run unopposed in a display of party unity  English took over the deputy leadership and the finance portfolio in the Key shadow cabinet 

Fifth National Government (2008–2017)

Deputy Prime Minister and Minister of Finance (2008–2016)

At the 2008 election, English was re-elected by his electorate, winning by a margin of about 15,500 votes  He became Deputy Prime Minister of New Zealand and Minister of Finance in the fifth National Government, being sworn into office on 19 November 2008 and continued to serve in those roles until becoming Prime Minister on 12 December 2014  He was also made Minister of Infrastructure in National's first term of government and Minister responsible for Housing New Zealand Corporation and minister responsible for the New Zealand flag consideration process in its third 

He was comfortably re-elected in Clutha-Southland in the 2011 election but opted to run as a party-list candidate in 2014  

The pairing of John Key as leader of the National Party and English as his deputy has been compared to that of Bob Hawke and Paul Keating (in Australia) and Tony Blair and Gordon Brown (in the UK) 

English acceded to the role of Finance Minister in the continuing wake of the financial crisis  In response to New Zealand's rising debt, English made budget deficit-reduction his main priority  His first budget outlined three focuses in New Zealand's financial recovery: "improving the business environment and removing roadblocks to growth; investment in productive infrastructure; and improving the way government works"  One of his first acts was creating the National Infrastructure Unit, charged with formulating a plan for infrastructure projects and investments  He commissioned a government-wide spending review, with an aim to reducing government expenditure—with the exceptions of a two-year stimulus package and long-term increases on infrastructure spending 

In April 2011, the Opposition criticised English for suggesting that New Zealand businesses could use New Zealand's low wages to help it compete with Australia  The National Government campaigned for re-election in 2011 on its economic record  The Government boasted growth for five consecutive quarters up to mid-2010, totalling 1 6% of real GDP 

Strong growth resulted in a surplus of $473 million for the 2015/16 financial year, projected to rise to $8 5 billion by 2020/21  In his 2016 Economic and Fiscal Update address, English stated that reducing debt and tackling the costs of the 2016 Kaikōura earthquake were higher priorities than reducing rates of tax 

Allowances issue
In 2009, the media, including TVNZ and TV3 revealed that English was receiving about NZ$900 a week as part of a living allowance for ministers, to live in his own NZ$1 2 million Wellington home  At the time, English also received $276,200 in his annual salary as Deputy Prime Minister  It was also revealed other ministers with homes in the capital city were also claiming accommodation allowances  On 3 August 2009, Prime Minister John Key started a review of the housing allowances claimed by cabinet ministers  English subsequently paid back $12,000 and only claimed about $24,000 a year in living allowances  The Auditor-General's office said in September 2009 that they were making "preliminary enquiries" into parliamentary housing expenses in response to a letter of complaint from Progressive party leader Jim Anderton  Two days later English stated that he would no longer take up any housing allowance and had paid back all the allowance he had received since the November 2008 election 

Prime Minister (2016–2017)

John Key resigned on 12 December, and endorsed English as his successor in the resulting leadership election  Following the drop-out of both Judith Collins and Jonathan Coleman from the leadership election, English was sworn in as the 39th Prime Minister of New Zealand on 12 December 2016 

English appointed his first cabinet on 18 December  In a reshuffle, he appointed Steven Joyce to succeed him as Finance Minister, while most ministerial portfolios remained the same 

In February 2017, English did not attend Waitangi Day commemorations at the historic treaty grounds, reportedly in response to the Ngāpuhi iwi's decision to stop the Prime Minister from speaking at the marae  Ngāpuhi have protested the Government's negotiation of the Trans Pacific Partnership Agreement (TPPA), which the iwi believe infringes upon Māori sovereignty, and thus does not adhere to the Treaty of Waitangi  English had been invited to attend in an official capacity; his non-attendance was criticised by a Ngāpuhi elder and Opposition leader Andrew Little 

In his first overseas trip as Prime Minister, English travelled to Europe to discuss trade ties, including a prospective New Zealand–European Union free trade agreement  He first travelled to London on 13 January 2017 to meet British Prime Minister Theresa May  Discussing trade relations, English said the two nations were "natural partners" and would "continue to forge ties" after the UK's withdrawal from the EU  He also arranged to meet with London Mayor Sadiq Khan, Belgian Prime Minister Charles Michel and German Chancellor Angela Merkel   In a meeting with Merkel, English received crucial backing from Germany for a trade deal with the EU  On 16 January, English stated that his government would continue to promote TPPA, despite the United States' decision to withdraw from the agreement  He explained that Southeast Asian countries would now be treated as a priority in negotiations—he also asserted that the United States was ceding influence to China by its rejection of the trade pact 

At a press conference at the Beehive on 1 February 2017, English announced that the 2017 general election would be held on 23 September  The Prime Minister later confirmed that his party would approach ACT, United Future and the Māori Party if confidence and supply agreements were required to form a government following the election  In his second cabinet reshuffle on 24 April, English appointed Gerry Brownlee as his new Foreign Affairs Minister; he also promoted Nikki Kaye to the portfolio of Education Minister, and moved Mark Mitchell into the cabinet to become Defence Minister  The reshuffle was perceived as an election preparation 

On 13 February 2017, English welcomed Australian Prime Minister Malcolm Turnbull to Wellington  The two leaders reaffirmed their shared trade agenda, and discussed changes to the Australian citizenship pathway which will affect permanent residents originating from New Zealand 

On 19 June, it was reported that Todd Barclay, who succeeded English as MP for Clutha-Southland, had clandestinely recorded one of his employee's conversations the previous year, and that John Key's leaders' budget was used to pay a confidential settlement after the employee resigned  English admitted that he had been aware of the illegal recording and the settlement, and thus implicated in the scandal 

During the 2017 National campaign launch, English introduced a $379 million social investment package including digital learning academies for high school students, more resources for mathematics, and boosting support for teaching second languages in schools, and maintaining National Standards in the school curriculum  Prime Minister English also sought to defend National's financial management and economic track record and claimed that the opposition Labour Party would raise taxes  Early opinion polling had forecast a poor showing in the election for the Labour Party, but in early August 37-year-old Jacinda Ardern took over as Labour leader and seemingly energised younger voters 

At the 2017 general election, National won the largest share of the party vote (44 4%) and the largest number of seats (56) in the House Representatives  However, National lacked enough seats to govern alone due to two of the party's support partners, the Māori Party and United Future, losing their parliamentary seats  In response, English stated that the party would be entering into talks to form a coalition with New Zealand First  Following talks with the two largest parties, New Zealand First entered a coalition arrangement with the Labour Party  English was succeeded as prime minister by Jacinda Ardern on 26 October 

Opposition (2017–2018)

Leader of the Opposition
English was re-elected as National Party leader on 24 October 2017  At the time of his re-election, English announced his intention to stay on as leader until the next general election  On 13 February 2018, however, he stood down as National Party leader due to personal reasons, and instructed the party to put into motion the processes to elect a new leader  He also retired from Parliament  English's resignation followed weeks of speculation that he would step aside for a new leader  On 27 February, he was succeeded as party leader by Simon Bridges as the result of the leadership election held that day 

Post-premiership 
In 2018, English joined the board of Australian conglomerate, Wesfarmers  English serves in Chairmanships of Mount Cook Alpine Salmon, Impact Lab Ltd and Manawanui Support Ltd  He is also a director of The Instillery, Centre for Independent Studies and The Todd Corporation Limited, and is a member of the Impact Advisory Group of Macquarie Infrastructure and Real Assets 

Political and social views

English is regarded as more socially conservative than his predecessor, John Key  He has stated his opposition to voluntary euthanasia and physician-assisted suicide, same-sex civil unions, and the decriminalisation of prostitution  As Prime Minister he opposed any "liberalisation" of abortion law 

In 2004, English voted against a bill to establish civil unions for both same-sex and opposite-sex couples  In 2005, he voted for the Marriage (Gender Clarification) Amendment Bill, which would have amended the Marriage Act to define marriage as only between a man and a woman  English voted against the Marriage (Definition of Marriage) Amendment Bill, a bill that legalised same-sex marriage in New Zealand  However, in December 2016 he stated, "I'd probably vote differently now on the gay marriage issue  I don't think that gay marriage is a threat to anyone else's marriage" 

In 2009, English voted against the Misuse of Drugs (Medicinal Cannabis) Amendment Bill, a bill aimed at amending the Misuse of Drugs Act so that cannabis could be used for medical purposes 

Personal life 
English met his future wife, Mary Scanlon, at university  She was studying medicine at the time, and became a general practitioner  Both her parents were immigrants, her father being Samoan and her mother Italian, born on the island of Stromboli  They have six children: a daughter and five sons 

English is a practising Roman Catholic, but has stated that he considers his religious beliefs personal and thus separate from politics 

In June 2002, English took part in TV3's Fight For Life, a celebrity boxing fundraiser to raise money for the Yellow Ribbon anti-youth-suicide campaign, influenced by the death of a teenage nephew in 1997  He lost a split decision to former university colleague Ted Clarke 

Honours
In the 2018 Queen's Birthday Honours, English was appointed a Knight Companion of the New Zealand Order of Merit, for services of over 27 years to the State 

See also

List of New Zealand governments
Politics of New Zealand

References

External links

Profile at National Party 
Profile on Parliament nz
Releases and speeches at Beehive govt nz

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1961 births
21st-century New Zealand politicians
Candidates in the 2017 New Zealand general election
Deputy Prime Ministers of New Zealand
Leaders of the Opposition (New Zealand)
Living people
Members of the Cabinet of New Zealand
Members of the New Zealand House of Representatives
New Zealand farmers
New Zealand finance ministers
New Zealand list MPs
New Zealand MPs for South Island electorates
New Zealand National Party MPs
New Zealand National Party leaders
New Zealand Roman Catholics
New Zealand people of Irish descent
People educated at St  Patrick's College, Silverstream
People from Dipton, New Zealand
People from Lumsden, New Zealand
Prime Ministers of New Zealand
University of Otago alumni
Victoria University of Wellington alumni
Knights Companion of the New Zealand Order of Merit
New Zealand politicians awarded knighthoods
Context 3: , 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



ground_truth: He became deputy prime minister and minister of finance.

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.

Context 3:  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: By using a propagator in the latent space.