#  CP VIOLATION IN THE $\bm{B^{0}_{(s)}}$ SYSTEM AT LHCb

F. DORDEI
on behalf of the LHCb Collaboration

The study of $C\\!P$ violation in decays of neutral B mesons provides an
important test of the Standard Model (SM) predictions and it is a sensitive
probe to search for non-SM physics. In these proceedings I present
measurements of $C\\!P$ violation in the neutral B meson system. The discussed
analyses are based on $pp$-collision data corresponding to an integrated
luminosity of 1 $\mathrm{fb}^{-1}$ or 3 $\mathrm{fb}^{-1}$ recorded by the
LHCb experiment.

##  1 Introduction

The Standard Model description of $C\\!P$ violation has been very successful
in describing existing data. $\\!{}^{{\bf?}}$ However, the source of $C\\!P$
violation in the SM is not sufficient $\\!{}^{{\bf?}}$ to explain the matter-
antimatter asymmetry that results in the matter dominated universe we observe
today. Therefore, precision measurements of $C\\!P$ violation are mandatory in
order to confirm SM predictions and to check whether non-SM contributions are
also present.
LHCb $\\!{}^{{\bf?}}$ is one of the four large particle experiments located at
the LHC. It is specialised for precision measurements of $b$\- and $c$\-
hadron decays, searching for indirect effects of new virtual particles in
quantum loops, predicted in New Physics models. In particular decays of
neutral $B$-mesons provide an ideal laboratory to study $C\\!P$ violation
originating from a non-trivial complex phase in the CKM quark mixing matrix
$\\!{}^{{\bf?}}$. Since the weak interaction eigenstates are not the same as
the mass eigenstates, the flavour content of neutral mesons changes as a
function of time, giving rise to the phenomenon known as neutral meson mixing.
Thanks to this effect, neutral B mesons allow to test all the three different
kinds of $C\\!P$ violation: $C\\!P$ violation in the decay, $C\\!P$ violation
in the mixing and $C\\!P$ violation in the interference between mixing and
decay. The former is possible in both neutral and charged meson decays and it
consists in a difference between absolute value of the amplitude of the decay
of a meson to a certain final state and the amplitude of the $C\\!P$
conjugated decay. The latter two are instead only possible for neutral mesons.
The phenomenological aspects linked to neutral B meson mixing are described in
many articles $\\!{}^{{\bf?}}$; only the main parameters involved are briefly
introduced here. $B^{0}_{(s)}$ and $\overline{B}^{0}_{(s)}$ are flavour
eigenstates with the quark content $\overline{b}d(s)$ and
$b\overline{d}(\overline{s})$, respectively. Any arbitrary combination of
flavour eigenstates, $B^{0}_{(s)}$ and $\overline{B}^{0}_{(s)}$, has a time
evolution described by an effective Schrödinger equation (we adopt units such
that $\hbar=\textit{c}=1$)

$\textit{i}\,\,\frac{d}{dt}\begin{pmatrix}B^{0}_{(s)}\\\
\overline{B}^{0}_{(s)}\\\
\end{pmatrix}=\begin{pmatrix}M_{11}-i\,\Gamma_{11}/2&M_{12}-i\,\Gamma_{12}/2\\\
M_{21}-i\,\Gamma_{21}/2&M_{22}-i\,\Gamma_{22}/2\\\
\end{pmatrix}\begin{pmatrix}B^{0}_{(s)}\\\ \overline{B}^{0}_{(s)}\\\
\end{pmatrix}\,,$ (1)

where M and $\bm{\Gamma}$ are $2\times 2$ hermitian matrices. The heavy (H)
and light (L) mass eigenstates of the Schrödinger equation are obtained
diagonalizing the matrix in Eq.1

$|B_{H/L}\rangle=p|{B}^{0}_{(s)}\rangle\pm q|\overline{B}^{0}_{(s)}\rangle\,,$
(2)

where the complex coefficients p and q obey the normalization condition
$|p|^{2}+|q|^{2}=1$. The system can be defined in terms of five different
experimental observables. They are the mass difference $\Delta m_{(s)}$ and
the decay width difference $\Delta\Gamma_{(s)}$ between the mass eigenstates,
defined by

$\Delta m_{(s)}\equiv m_{H}-m_{L}\simeq
2|M_{12}|,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Delta\Gamma_{(s)}\equiv\Gamma_{L}-\Gamma_{H}\simeq
2|\Gamma_{12}|\cos(\phi_{12}),$ (3)

the average mass and width, defined by

$m_{(s)}=\frac{m_{H}+m_{L}}{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma_{(s)}=\frac{\Gamma_{L}+\Gamma_{H}}{2},$
(4)

and the so-called flavour-specific or semileptonic asymmetry, which is a
dimensionless quantity that parametrizes $C\\!P$ violation in the mixing and
is approximately given by

$a_{sl}\simeq\frac{\Delta\Gamma}{\Delta M}\tan\phi_{12}\,.$ (5)

In the SM model the phase $\phi_{12}$ is very small, and in particular for
$B^{0}_{s}$ mixing, $a^{s}_{sl}$ is predicted $\\!{}^{{\bf?}}$ to be $(+1.9\pm
0.3)\cdot 10^{-5}$.
The interference between $B^{0}_{s}$-mesons decaying to a $J/\psi hh$ ($h$=
$K$ or $\pi$) final state either directly or via
$B^{0}_{s}-\overline{B}^{0}_{s}$ mixing gives rise to a $C\\!P$-violating
phase called $\phi_{s}$. In the SM, neglecting sub-leading penguin
contributions, this phase is predicted $\\!{}^{{\bf?}}$ to be
$-2\,\,\mathrm{arg}(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*})=(3.40^{+1.32}_{-0.77})\times
10^{-3}$, where $V_{ij}$ are elements of the CKM matrix.
In the following a selection of recent LHCb measurements of $C\\!P$ violation
in the neutral B meson system are presented. They are based on 2011 only or
2011+2012 data sets, corresponding to 1 $\mathrm{fb}^{-1}$ or 3
$\mathrm{fb}^{-1}$ of integrated luminosity of $pp$ collisions, respectively.

##  2 Measurement of CP asymmetries, and direct CP violation in $B^{0}\to\phi
K^{*}(892)^{0}$

An angular analysis of the decay $B^{0}\to\phi K^{*}(892)^{0}$ has been
performed $\\!{}^{{\bf?}}$ using a data set which corresponds to
$1.0\,\,\mathrm{fb}^{-1}$ of integrated luminosity. It is a pseudoscalar to
vector-vector transition. For a P-wave state angular momentum conservation
allows three possible helicity configurations of the vector meson pair,
parametrised by a longitudinal polarisation $A_{0}$, and two tansverse
polarisations, $A_{\perp}$ and $A_{\parallel}$. Moreover for the first time
also the contributions where either the $KK$ or $K\pi$ are produced in an
S-wave state are taken into account, with the amplitudes $A_{s}^{KK}$ and
$A_{s}^{K\pi}$, respectively.
The polarisation amplitudes are extracted by means of an untagged time-
independent angular analysis. An almost equal mixture of longitudinal and
transverse polarisations is found with greater precision than by previous
measurements. The fraction of the longitudinal polarisation is measured to be
$f_{L}=0.497\pm 0.019\,\,\mathrm{(stat.)}\pm 0.015\,\,\mathrm{(syst.)}$.
Moreover, a significant $S$-wave contributions is found in both the $K\pi$ and
the $KK$ system. Using the information that the B meson flavour at decay can
be identified by the charge of the kaon from the $K^{*}$, the sample is
splitted in two sub-samples and the difference in polarisation amplitudes and
phases between the two samples is calculated. All differences are found to be
consistent with zero within the uncertainties.
The polarisation amplitudes and phases are also used to calculate triple
product asymmetries. Non-zero triple product asymmetries arise either due to a
$T$-violating phase or a $C\\!P$-violating phase and final-state interactions
(so-called fake asymmetry). The former case (so-called true asymmetry) implies
that $C\\!P$ is violated, assuming $C\\!PT$ is conserved. The true asymmetries
are found consistent with zero, showing no evidence for $C\\!P$ violation. In
contrast, all but one of the fake asymmetries are significantly different from
zero, indicating the presence of final-state interactions introducing strong
phases.
After splitting the sample according to the flavour of the B meson at decay,
the following raw asymmetry is measured

$A=\frac{N(\overline{B}^{0}\to\phi\overline{K}^{*}(892)^{0})-N(B^{0}\to\phi
K^{*}(892)^{0})}{N(\overline{B}^{0}\to\phi\overline{K}^{*}(892)^{0})+N(B^{0}\to\phi
K^{*}(892)^{0})}.$ (6)

Correcting for particle-antiparticle production and detection asymmetries,
calculated using the control channel $B^{0}\to J/\psi K^{*}(892)^{0}$, the
direct $C\\!P$ asymmetry is determined to be

$A_{C\\!P}(\phi K^{*}(892)^{0})=(+1.5\pm 3.2\,\mathrm{(stat.)}\pm
0.5\,\mathrm{(syst.)})\%.$ (7)

This is consistent with zero and a factor of two more precise than the values
reported by Babar and Belle $\\!{}^{{\bf?}}$.

##  3 Measurement of time-dependent CP violation in $B^{0}_{s}\to K^{+}K^{-}$
and $B^{0}\to\pi^{+}\pi^{-}$

The study of $C\\!P$ violation in charmless charged two-body decays of neutral
B mesons is a good probe of the SM predictions. Moreover, precise measurements
in this sector are important to constrain hadronic factors that cannot be
accurately calculated from quantum chromodynamics at present. In these
proceedings the first measurement of time-dependent $C\\!P$-violating
asymmetries in $B^{0}_{s}\to K^{+}K^{-}$ decays $\\!{}^{{\bf?}}$ is discussed.
Furthermore, a measurement of the corresponding quantities for
$B^{0}\to\pi^{+}\pi^{-}$ has been performed. The analysis is based on a data
sample corresponding to $1\,\,\mathrm{fb}^{-1}$ of integrated luminosity.
Assuming $C\\!PT$ invariance, the $C\\!P$ asymmetry as a function of time for
neutral B mesons decaying to a $C\\!P$ eigenstate $f$ is given by

$A^{\mathrm{C\\!P}}(t)=\frac{\Gamma_{\overline{B}^{0}_{(s)}\to
f}(t)-\Gamma_{B^{0}_{(s)}\to f}(t)}{\Gamma_{\overline{B}^{0}_{(s)}\to
f}(t)+\Gamma_{B^{0}_{(s)}\to f}(t)}=\frac{-C_{f}\cos(\Delta
m_{(s)}t)+S_{f}\sin(\Delta
m_{(s)}t)}{\cosh\left(\frac{\Delta\Gamma_{(s)}}{2}t\right)-\mathcal{A}^{\Delta\Gamma_{(s)}}_{f}\sinh\left(\frac{\Delta\Gamma_{(s)}}{2}t\right)}\,.$
(8)

The quantities $C_{f}$, $S_{f}$, and $\mathcal{A}^{\Delta\Gamma_{(s)}}_{f}$
are

$C_{f}=\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}},\,\,\,\,\,\,\,\,\,\,\,\,S_{f}=\frac{2\mathrm{Im}\lambda_{f}}{1+|\lambda_{f}|^{2}}\,,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,\,\,\,\,\,\,\,\mathcal{A}^{\Delta\Gamma_{(s)}}_{f}=-\frac{2\mathrm{Re}\lambda_{f}}{1+|\lambda_{f}|^{2}}\,,$
(9)

where the parameter $\lambda_{f}=(q\,\overline{A}_{f})/(p\,A_{f})$ is related
to the $B^{0}_{(s)}$ meson mixing (via $q/p$) and to the decay amplitudes of
the $B^{0}_{(s)}\to f$ decay ($A_{f}$) and of the $\overline{B}^{0}_{(s)}\to
f$ decay ($\overline{A}_{f}$).
The parameter $\mathcal{A}^{\Delta\Gamma_{(s)}}_{f}$ can be expressed as
$\mathcal{A}^{\Delta\Gamma}_{f}=\pm\sqrt{1-C_{f}^{2}-S_{f}^{2}}$. For
$B^{0}_{s}$ decays the positive solution is taken, which is consistent with
measurement of the $B^{0}_{s}\to K^{+}K^{-}$ effective lifetime, while for
$B^{0}$ decays, due to the fact that the width difference of the $B^{0}$ meson
is negligible, the ambiguity is not relevant. By means of a time-dependent
analysis, where the initial $B^{0}_{(s)}$ flavour is identified through a
flavour-tagging algorithm calibrated using flavour-specific $B^{0}\to
K^{+}\pi^{-}$ events, the parameters for $B^{0}_{s}\to K^{+}K^{-}$ decay are
found to be

$\displaystyle C_{KK}=0.14\pm 0.11\,\mathrm{(stat.)}\pm
0.03\,\mathrm{(syst.)},$ (10) $\displaystyle S_{KK}=0.30\pm
0.12\,\mathrm{(stat.)}\pm 0.04\,\mathrm{(syst.)},$

with a statistical correlation coefficient of 0.02. The results for the
$B^{0}\to\pi^{+}\pi^{-}$ decay are

$\displaystyle C_{\pi\pi}=-0.38\pm 0.15\,\mathrm{(stat.)}\pm
0.02\,\mathrm{(syst.)},$ (11) $\displaystyle S_{\pi\pi}=-0.71\pm
0.13\,\mathrm{(stat.)}\pm 0.02\,\mathrm{(syst.)},$

with a statistical correlation coefficient of 0.38. The significances for
$(C_{KK},\,S_{KK})$ to differ from $(0,0)$ are determined to be 2.7$\,\sigma$
and 5.6$\,\sigma$, respectively. These results are compatible with SM
predictions and are important inputs in the determination of the unitarity
triangle angle $\gamma$ using decays affected by penguin processes.

##  4 Measurement of the semileptonic asymmetry $a^{s}_{sl}$

The LHCb collaboration has performed a measurement $\\!{}^{{\bf?}}$ of the
semileptonic asymmetry $a^{s}_{sl}$ in decays of $B^{0}_{s}\to
D^{+}_{s}\mu^{-}X$, using a data sample corresponding to an integrated
luminosity of $1\,\,\mathrm{fb}^{-1}$. The time integrated asymmetry in the
untagged yields of $D_{s}^{-}\mu^{+}$ and $D^{+}_{s}\mu^{-}$ events is related
to $a_{sl}^{s}$ by

$A^{\mathrm{C\\!P}}_{\mathrm{measured}}=\frac{\Gamma[D^{-}_{s}\mu^{+}]-\Gamma[D^{+}_{s}\mu^{-}]}{\Gamma[D^{-}_{s}\mu^{+}]+\Gamma[D^{+}_{s}\mu^{-}]}=\frac{a^{s}_{sl}}{2}+a_{D}+\left[a_{p}-\frac{a^{s}_{sl}}{2}\right]\cdot\frac{\int
e^{-\Gamma_{s}t}\cos(\Delta m_{s}t)\varepsilon(t)dt}{\int
e^{-\Gamma_{s}t}\cosh(\Delta\Gamma_{s}/2t)\varepsilon(t)dt},$ (12)

where $\varepsilon(t)$ is the decay time acceptance function and $a_{p}$ is
the particle-antiparticle production asymmetry. The latter is expected to be
at most a few percent at the LHC. However, thanks to the large value of the
oscillation frequency $\\!{}^{{\bf?}}$, $\Delta m_{s}=17.768\pm
0.024\,\mathrm{ps}^{-1}$, the integral on the right-hand side is small and the
third term on the right-hand side of Eq.12 can be neglected.
Using data-driven methods to measure efficiency ratios, the measured $C\\!P$
asymmetry is corrected for tracking efficiency asymmetries and background
asymmetries ($a_{D}$) and the semileptonic asymmetry $a^{s}_{sl}=(-0.06\pm
0.50\pm 0.36)\%$ is derived. This result is the most precise measurement to
date and is consistent with the SM.

##  5 Measurement of the CP-violating phase $\phi_{s}$

The decays $B^{0}_{s}\rightarrow J/\psi\phi$ and $B^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ are used to measure the $C\\!P$ violating phase,
$\phi_{s}$. The decay $B^{0}_{s}\rightarrow J/\psi\phi$ is a pseudo-scalar to
vector-vector decay, so angular momentum conservation implies that the final
state is an admixture of $C\\!P$-even and $C\\!P$-odd components. By
performing a tagged time-dependent angular analysis, it is possible to
statistically disentangle the different $C\\!P$ eigenstates by the
differential decay rate for $B^{0}_{s}$ and $\overline{B}^{0}_{s}$ mesons,
produced as flavour eigenstates at $t=0$. Using $1$ $\mathrm{fb}^{-1}$ of data
collected in $pp$ collisions at $\sqrt{s}=7$ TeV with the LHCb detector, the
$C\\!P$ violating phase $\phi_{s}$ as well as $\Delta\Gamma_{s}$ and
$\Gamma_{s}$ are extracted $\\!{}^{{\bf?}}$ by performing an unbinned maximum
log-likelihood fit to the $B^{0}_{s}$ mass, decay time $t$, angular
distributions and initial flavour tag of the selected $B^{0}_{s}\rightarrow
J/\psi\phi$ events. The $B^{0}_{s}\rightarrow J/\psi\pi^{+}\pi^{-}$ final
state is predominantly $C\\!P$-odd. Thus there is no need for an angular
analysis. The results of a simultaneous fit to both $B^{0}_{s}\rightarrow
J/\psi\phi$ and $B^{0}_{s}\rightarrow J/\psi\pi^{+}\pi^{-}$ are:
$\phi_{s}=0.01\pm 0.07\,\mathrm{(stat.)}\pm 0.01\,\mathrm{(syst.)}$,
$\Gamma_{s}=(0.661\pm 0.004\,\mathrm{(stat.)}\pm
0.006\,\mathrm{(syst.)})\,\mathrm{ps}^{-1}$ and $\Delta\Gamma_{s}=(0.106\pm
0.011\,\mathrm{(stat.)}\pm 0.007\,\mathrm{(syst.)})\,\mathrm{ps}^{-1}$.
These results are in good agreement with the SM prediction. The systematic
uncertainty on $\phi_{s}$ is dominated by the size of the Monte Carlo sample
used to determine the angular acceptance, the one on $\Delta\Gamma_{s}$ is
dominated by the background subtraction method and the decay time acceptance,
the one on $\Gamma_{s}$ by the decay time acceptance determination. All of
them are expected to decrease in the near future.

##  6 Measurement of resonant and CP components in $\overline{B}^{0}_{s}\to
J/\psi\,\,\pi^{+}\pi^{-}$ decays

Using the dataset corresponding to $3$ $\mathrm{fb}^{-1}$ of integrated
luminosity, the resonant structure of the decay $\overline{B}^{0}_{s}\to
J/\psi\,\,\pi^{+}\pi^{-}$ has been studied $\\!{}^{{\bf?}}$. The
$\pi^{+}\pi^{-}$ invariant mass and all three decay angular distributions are
used to determine the resonant and non-resonant components. The decay
$\overline{B}^{0}_{s}\to J/\psi\,\,\pi^{+}\pi^{-}$ can be described by the
interfering sum of five resonant components: $f_{0}(980)$, $f_{0}(1500)$,
$f_{0}(1790)$, $f_{2}(1270)$ and $f^{\prime}_{2}(1525)$. An alternative model
including these states and a non-resonant $J/\psi\pi^{+}\pi^{-}$ component has
been checked and it also provides a good description of data. In both models
the largest component of the decay is $f_{0}(980)$, while the $f_{0}(500)$ and
the $\rho(770)$ components are found to be not signficant. An upper limit for
the fit fractions of these two component is derived, corresponding to $3.4\%$
and $1.7\%$ at 90$\%$ confidence level (from the solution where also the non-
resonant component is included), respectively.
The final state is found to be compatible with being entirely $C\\!P$-odd; the
$C\\!P$-even part is measured to be less than 2.3$\%$ at 95$\%$ CL. This upper
limit is the same as of a previous measurement $\\!{}^{{\bf?}}$ performed with
$1$ $\mathrm{fb}^{-1}$ of integrated luminosity, while the current measurement
also adds a possible $f^{\prime}_{2}(1525)$ contribution.
Also of importance is the limit on the absolute value of the mixing angle,
$\phi_{m}$, between the $f_{0}(500)$ and the $f_{0}(980)$ resonances. The
limit is found to be

$|\phi_{m}|<7.7^{\circ}\,\,\,\,\,\mathrm{at}\,\,90\%\,\,\mathrm{CL},$ (13)

and it is the most constraining ever placed. It is consistent with the
tetraquark model, which predicts zero within a few degrees.

##  7 Measurement of the $\overline{B}^{0}_{s}\to D^{+}_{s}D^{-}_{s}$
effective lifetime

Measurements of the $B_{s}^{0}$ effective lifetime in decays to $C\\!P$-odd
and $C\\!P$-even flavour specific final states, $f$, allow to probe the decay
width difference, $\Delta\Gamma_{s}$, and the $C\\!P$-violating phase,
$\phi_{s}$, of $B^{0}_{s}-\overline{B}^{0}_{s}$ mixing box-diagram. We define
the effective lifetime of the decay $B^{0}_{s}\rightarrow f$ as the time
expectation value of the untagged rate

$\tau_{f}\equiv\frac{\int_{0}^{\infty}t\,\,\langle\Gamma(B^{0}_{s}(\overline{B}^{0}_{s})(t)\rightarrow
f)\rangle
dt}{\int_{0}^{\infty}\langle\Gamma(B^{0}_{s}(\overline{B}^{0}_{s})(t)\rightarrow
f)\rangle dt}\,,$ (14)

which is equivalent to the lifetime that results from fitting the untagged
decay time distribution with a single exponential. By making use of the usual
definition $y_{s}=\Delta\Gamma_{s}/2\Gamma_{s}$ and using
$\tau_{B^{0}_{s}}=\Gamma_{s}^{-1}$, one can express the effective lifetime as

$\tau_{f}=\tau_{B^{0}_{s}}+\tau_{B^{0}_{s}}\,\mathcal{A}^{\Delta\Gamma_{s}}_{f}\,y_{s}+\tau_{B^{0}_{s}}\,[2-(\mathcal{A}^{\Delta\Gamma_{s}}_{f})^{2}]\,y_{s}^{2}+\mathcal{O}(y_{s}^{3})\,.$
(15)

The parameter $\mathcal{A}^{\Delta\Gamma_{s}}_{f}$ is defined in Eq. 9, and is
a function of $\phi_{s}$ $\\!{}^{{\bf?}}$. So the corresponding lifetime
measurements can be used to constrain $y_{s}$ (or $\Delta\Gamma_{s}$) with
respect to $\phi_{s}$, with the advantage that only an untagged analysis is
needed.
Here the first measurement of the effective lifetime of the
$\overline{B}^{0}_{s}$ meson in the decay $\overline{B}^{0}_{s}\to
D^{+}_{s}D^{-}_{s}$ is reported $\\!{}^{{\bf?}}$, using a dataset
corresponding to $3\,\,\mathrm{fb}^{-1}$ of integrated luminosity. The
lifetime of $\overline{B}^{0}_{s}\to D^{+}_{s}D^{-}_{s}$ meson is measured
relative to the well known $B^{-}$ lifetime, using the normalisation decay
channel $B^{-}\to D^{0}D^{-}_{s}$, which has a similar topology and kinematic
properties. As a result, many of the systematic uncertainties cancel in the
ratio. The measured value of the $\overline{B}^{0}_{s}\to D^{+}_{s}D^{-}_{s}$
effective lifetime is $\tau=(1.379\pm 0.026\,\mathrm{(stat.)}\pm
0.017\,\mathrm{(syst.))}$ ps. The $D^{+}_{s}D^{-}_{s}$ final state is
$C\\!P$-even hence the effective lifetime is approximately equal to
$\Gamma^{-1}_{L}$. Using this fact a value of $\Gamma_{L}=(1.52\pm
0.15\,\mathrm{(stat.)}\pm 0.001\,\mathrm{(syst.)})\,\,\mathrm{ps}^{-1}$ is
derived.

##  8 Conclusions

Recent LHCb results of $C\\!P$ violation in the neutral B meson system have
been presented. All measurements are in good agreement with SM predictions.
The selection of $C\\!P$ violation presented here is not a complete list of
the analyses performed by the LHCb collaration. Updated results based on the
full $3\,\,\mathrm{fb}^{-1}$ of data are expected soon and will allow to
constrain even stronger contributions of beyond the SM physics.

## Acknowledgments

The author acknowledges the support received from the ERC under FP7 and the
support by the International Max Planck Research School for Precision Tests of
Fundamental Symmetries.

## References

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# Automorphisms of Veronese subalgebras of polynomial algebras and free
Poisson algebras

###### Abstract.

The Veronese subalgebra $A_{0}$ of degree $d\geq 2$ of the polynomial algebra
$A=K[x_{1},x_{2},\ldots,x_{n}]$ over a field $K$ in the variables
$x_{1},x_{2},\ldots,x_{n}$ is the subalgebra of $A$ generated by all monomials
of degree $d$ and the Veronese subalgebra $P_{0}$ of degree $d\geq 2$ of the
free Poisson algebra $P=P\langle x_{1},x_{2},\ldots,x_{n}\rangle$ is the
subalgebra spanned by all homogeneous elements of degree $kd$, where $k\geq
0$.

If $n\geq 2$ then every derivation and every locally nilpotent derivation of
$A_{0}$ and $P_{0}$ over a field $K$ of characteristic zero is induced by a
derivation and a locally nilpotent derivation of $A$ and $P$, respectively.
Moreover, we prove that every automorphism of $A_{0}$ and $P_{0}$ over a field
$K$ closed with respect to taking all $d$-roots of elements is induced by an
automorphism of $A$ and $P$, respectively.

Bakhyt Aitzhanova111Department of Mathematics, Wayne State University,
Detroit, MI 48202, USA, e-mail<EMAIL_ADDRESS>Leonid Makar-
Limanov222Department of Mathematics, Wayne State University, Detroit, MI
48202, USA and Faculty of Mathematics and Computer Science, Weizmann Institute
of Science, Rehovot, 7610001, Israel, e-mail<EMAIL_ADDRESS>and Ualbai
Umirbaev333Department of Mathematics, Wayne State University, Detroit, MI
48202, USA and Institute of Mathematics and Mathematical Modeling, Almaty,
050010, Kazakhstan, e-mail<EMAIL_ADDRESS>
Mathematics Subject Classification (2020): 14R10, 14J50, 13F20.

Key words: Automorphism, derivation, free Poisson algebra, polynomial algebra.

##  1\. Introduction

Let $K$ be an arbitrary field and let $\mathbb{A}^{n}$ and $\mathbb{P}^{n}$ be
the affine and the projective $n$-spaces over $K$, respectively. The Veronese
map of degree $d$ is the map

$\displaystyle\nu_{n,d}:\mathbb{P}^{n}\to\mathbb{P}^{m}$

that sends $[x_{0}:\ldots:x_{n}]$ to all $m+1$ possible monomials of total
degree $d$, where

$\displaystyle m={n+d\choose d}-1.$

It is well known that the image $V_{n,d}$ of the Veronese map $\nu_{n,d}$ is a
projective variety and is called the Veronese variety [7].

If $Y_{i_{0}\ldots i_{n}}=x_{0}^{i_{0}}\ldots x_{n}^{i_{n}}$,
$i_{0}+\ldots+i_{n}=d$, then the Veronese variety is determined by the set of
quadratic relations

(1) $\displaystyle Y_{i_{0}\ldots i_{n}}Y_{j_{0}\ldots j_{n}}=Y_{k_{0}\ldots
k_{n}}Y_{r_{0}\ldots r_{n}},$

where $i_{0}+j_{0}=k_{0}+r_{0},\ldots,i_{n}+j_{n}=k_{n}+r_{n}$ [19].

The affine cone of $V_{n,d}$ or the affine Veronese variety is the affine
subvariety $\mathbb{A}^{m+1}$ determined by the set of relations (1). The
algebra of polynomial functions on the affine Veronese cone $V_{n,d}$ is
isomorphic to the subalgebra of $K[x_{0},\ldots,x_{n}]$ generated by all
monomials of degree $d$ [21].

Veronese surfaces play an important role in the description of
quasihomogeneous affine surfaces given by M.H. Gizatullin [4] and V.L. Popov
[18]. They form one of the main examples of the so-called Gizatullin surfaces
[10]. The structure of the automorphism groups of Veronese surfaces are
studied in [1, 2, 5, 6, 11, 12]. The derivations and locally nilpotent
derivations of affine Veronese surfaces are described in [1].

Recently J. Kollar devoted two interesting papers [8, 9] to the study of
automorphism groups of some more general affine varieties. In particular, he
described the group of automorphisms of all affine Veronese varieties.

Let $A=K[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra over a field $K$
in the variables $x_{1},x_{2},\ldots,x_{n}$. Consider the grading

$\displaystyle A=A_{0}\oplus A_{1}\oplus\ldots\oplus A_{d-1},$

where $d\geq 2$ and $A_{i}$ is the subspace of $A$ generated by all monomials
of degree $kd+i$ for all $k\geq 0$. This is a $\mathbb{Z}_{d}$-grading of $A$,
i.e., $A_{i}A_{j}\subseteq A_{i+j}$ for all $i,j\in\mathbb{Z}_{d}$. The
subalgebra $A_{0}$ is called the Veronese subalgera of $A$ of degree $d$.

Similarly, let $P=P\langle x_{1},x_{2},\ldots,x_{n}\rangle$ be the free
Poisson algebra over $K$ in the variables $x_{1},x_{2},\ldots,x_{n}$. The
grading

$\displaystyle P=P_{0}\oplus P_{1}\oplus\ldots\oplus P_{d-1},$

where $P_{i}$ is the subspace of $P$ generated by all homogeneous elements of
degree $kd+i$ is a $\mathbb{Z}_{d}$-grading of $P$, i.e., $P_{i}P_{j}\subseteq
P_{i+j}$ and $\\{P_{i},P_{j}\\}\subseteq P_{i+j}$ for all
$i,j\in\mathbb{Z}_{d}$. The subalgebra $P_{0}$ is called the Veronese
subalgera of $P$ of degree $d$.

We prove that every derivation and every locally nilpotent derivation of
$P_{0}$ (of $A_{0}$) over a field $K$ of characteristic zero is induced by a
derivation and a locally nilpotent derivation of $P$ (of $A$), respectively.
Moreover, we prove that every automorphism of $P_{0}$ (of $A_{0}$) over a
field $K$, closed with respect to taking all $d$-roots of elements, is induced
by an automorphism of $P$ (of $A$).

This paper is organized as follows. In Section 2 we describe a basis of free
Poisson algebras and give some elementary definitions that are necessary in
the future. Derivations of the Veronese subalgebras are studied in Section 3
and automorphisms are studied in Section 4. All results are proven for Poisson
algebras. Analogues of these results for polynomial algebras are formulated in
Section 5. In the same section we give some counter-examples that show the
analogous results do not hold for polynomial algebras in one variable and for
free associative algebras.

##  2\. Free Poisson algebra $P\langle x_{1},\ldots,x_{n}\rangle$

A vector space $P$ over a field $K$ endowed with two bilinear operations
$x\cdot y$ (a multiplication) and $\\{x,y\\}$ (a Poisson bracket) is called a
_Poisson algebra_ if $P$ is a commutative associative algebra under $x\cdot
y$, $P$ is a Lie algebra under $\\{x,y\\}$, and $P$ satisfies the following
identity (the Leibniz identity):

$\\{x,y\cdot z\\}=y\cdot\\{x,z\\}+\\{x,y\\}\cdot z.$

Symplectic Poisson algebras $P_{n}$ appear in many areas of algebra. For each
natural $n\geq 1$ the symplectic Poisson algebra $P_{n}$ of index $n$ is a
polynomial algebra $K[x_{1},y_{1},\ldots,x_{n},y_{n}]$ endowed with the
Poisson bracket defined by

$\displaystyle\\{x_{i},y_{j}\\}=\delta_{ij},\ \ \\{x_{i},x_{j}\\}=0,\ \
\\{y_{i},y_{j}\\}=0,$

where $\delta_{ij}$ is the Kronecker symbol and $1\leq i,j\leq n$.

Let $P$ be a Poisson algebra. A linear map $D:P\to P$ is called a derivation
of the Poisson algebra $P$ if

(2) $\displaystyle D(xy)=D(x)y+xD(y)$

and

(3) $\displaystyle D(\\{x,y\\})=\\{D(x),y\\}+\\{x,D(y)\\}$

for all $x,y\in P$.

We refer to a linear map $D:P\to P$ satisfying (2) as an associative
derivation of $P$ and to a linear map $D:P\to P$ satisfying (3) as a Lie
derivation of $P$. Thus, a derivation of a Poisson algebra is simultaneously
an associative and a Lie derivation. We often refer to them as Poisson
derivations.

The Leibniz identity implies that for any $x\in P$ the map

$\displaystyle\mathrm{ad}_{x}:P\to P\,\,\,(y\mapsto\\{x,y\\})$

is an associative derivation of $P$. The Jacobi identity implies that the map
$\mathrm{ad}_{x}$ is also a Lie derivation of $P$. So for any $x\in P$ the map
$\mathrm{ad}_{x}$ is a Poisson derivation of $P$.

Let $L$ be a Lie algebra with Lie bracket $[\,,]$ over a field $K$ and let
$e_{1},e_{2}\ldots,e_{k},\ldots$ be a linear basis of $L$. Then there exists a
unique bracket $\\{\,,\\}$ on the polynomial algebra
$K[e_{1},e_{2},\ldots,e_{k},\ldots]$ defined by

$\displaystyle\\{e_{i},e_{j}\\}=[e_{i},e_{j}]$

for all $i,j$ and satisfying the Leibniz identity. With this bracket

$\displaystyle P(L)=\langle K[e_{1},e_{2},\ldots],\cdot,\\{\,,\\}\rangle$

becomes a Poisson algebra. This Poisson algebra $P(L)$ is called the Poisson
enveloping algebra [23] (or Poisson symmetric algebra [15]) of $L$. Note that
the bracket $\\{,\\}$ of the algebra $P(L)$ depends on the structure of $L$
but does not depend on a chosen basis.

Let $L=\mathrm{Lie}\langle x_{1},\ldots,x_{n}\rangle$ be the free Lie algebra
with free generators $x_{1},\ldots,x_{n}$. It is well-known (see, for example
[20]) that $P(L)$ is the free Poisson algebra over $K$ in the variables
$x_{1},\ldots,x_{n}$. We denote it by $P\langle x_{1},\ldots,x_{n}\rangle$.

Let us choose a multihomogeneous linear basis

$\displaystyle
x_{1},\ldots,x_{n},[x_{1},x_{2}],\ldots,[x_{1},x_{n}],\ldots,[x_{n-1},x_{n}],[[x_{1},x_{2}],x_{3}],\ldots$

of a free Lie algebra $L$ and denote the elements of this basis by

(4) $\displaystyle e_{1},e_{2},\ldots,e_{s},\ldots.$

The algebra $P=P\langle x_{1},\ldots,x_{n}\rangle$ coincides with the
polynomial algebra on the elements (4). Consequently, the monomials

(5) $\displaystyle u=e_{i_{1}}e_{i_{2}}\ldots e_{i_{s}},$

where $i_{1}\leq i_{2}\leq\ldots\leq i_{s}$ form a linear basis of $P$.

Denote by $\deg$ the standard degree function on $P$, i.e., $\deg(x_{i})=1$
for all $1\leq i\leq n$. If $u$ is an element of the form (5) then

$\displaystyle\deg u=\deg e_{i_{1}}+\deg e_{i_{2}}+\ldots+\deg e_{i_{s}}.$

Set also $d(u)=s$ and call it the polynomial length of $u$. Note that

$\displaystyle\deg\\{f,g\\}=\deg f+\deg g$

if $f$ and $g$ are homogeneous and $\\{f,g\\}\neq 0$.

Denote by $Q(P)=P(x_{1},\ldots,x_{n})$ the field of fractions of the
polynomial algebra $K[e_{1},e_{2},\ldots]$ in the variables $(\ref{h4})$. The
Poisson bracket $\\{\cdot,\cdot\\}$ on $K[e_{1},e_{2},\ldots]=P$ can be
uniquely extended to a Poisson bracket on the field of its fractions $Q(P)$
and

(6)
$\displaystyle\Bigl{\\{}\frac{a}{b},\frac{c}{d}\Bigr{\\}}=\frac{\\{a,c\\}bd-\\{a,d\\}bc-\\{b,c\\}ad+\\{b,d\\}ac}{b^{2}d^{2}}$

for all $a,b,c,d\in P$ with $bd\neq 0$.

The field $Q(P)=P(x_{1},x_{2},\ldots,x_{n})$ with this Poisson bracket is
called the free Poisson field over $K$ in variables $x_{1},\ldots,x_{n}$ [16].

Several combinatorial results on the structure of free Poisson algebras and
free Poisson fields are proven in [13, 14, 15, 16, 17, 22].

We fix a grading

(7) $\displaystyle P=P_{0}\oplus P_{1}\oplus\ldots\oplus P_{d-1}$

of the free Poisson algebra $P=P\langle x_{1},\ldots,x_{n}\rangle$, where
$P_{i}$ is the linear span of all elements of degree $i+ds$,
$i=0,1,\ldots,d-1$, and $s$ is an arbitrary nonnegative integer. This is a
$\mathbb{Z}_{d}$-grading of $P$, i.e.,

$\displaystyle P_{i}P_{j}\subseteq P_{i+j},\ \ \\{P_{i},P_{j}\\}\subseteq
P_{i+j},$

where $i,j\in\mathbb{Z}_{d}=\mathbb{Z}/d\mathbb{Z}$. For shortness we will
refer to this grading as the $d$-grading.

An automorphism $\phi\in\mathrm{Aut}\,P$ is called a graded automorphism with
respect to grading (7) if $\phi(x_{1}),\phi(x_{2}),\ldots,\phi(x_{n})\in
P_{1}$. A graded automorphism is called graded tame if it is a product of
graded elementary automorphisms.

We will call a graded automorphism of $P$ with respect to grading (7) a
d-graded automorphism for shortness. Obviously, every $d$-graded automorphism
induces an automorphism of the algebra $P_{0}$. A derivation $D$ of $P$ will
be called a d-graded derivation if $D(x_{1}),D(x_{2}),\ldots,D(x_{n})\in
P_{1}$.

##  3\. Derivations of $P_{0}$

In this section we assume that $K$ is an arbitrary field of characteristic
zero.

######  Lemma 1.

Every derivation of the Poisson algebra $P=P\langle
x_{1},x_{2},\ldots,x_{n}\rangle$ over $K$ can be uniquely extended to a
derivation of the Poisson field $Q(P)=P(x_{1},x_{2},\ldots,x_{n})$.

Proof. Let $D$ be an arbitrary derivation of the free Poisson algebra $P$. In
particular, $D$ is a derivation of the polynomial algebra
$P=K[e_{1},\ldots,e_{s},\ldots]$. It is well known [25, p. 120] that $D$ can
be uniquely extended to an associative derivation $S$ of the quotient field
$Q(P)=K(e_{1},\ldots,e_{s},\ldots)$. We will show that $S$ is a Lie derivation
of $Q(P)$, i.e., that $S$ satisfies (3). We have to check that

(8) $\displaystyle
S\left(\Bigl{\\{}\frac{a}{b},\frac{c}{d}\Bigr{\\}}\right)=\Bigl{\\{}S\left(\frac{a}{b}\right),\frac{c}{d}\Bigr{\\}}+\Bigl{\\{}\frac{a}{b},S\left(\frac{c}{d}\right)\Bigr{\\}}$

for all $a,b,c,d\in P\langle x_{1},\dots,x_{n}\rangle$ with $bd\neq 0$. Using
(6) we get

$\displaystyle
S\left(\Bigl{\\{}\frac{a}{b},\frac{c}{d}\Bigr{\\}}\right)=S\left(\frac{\\{a,c\\}bd-\\{a,d\\}bc-\\{b,c\\}ad+\\{b,d\\}ac}{b^{2}d^{2}}\right)$
$\displaystyle=S\left(\frac{\\{a,c\\}}{bd}\right)-S\left(\frac{\\{a,d\\}c}{bd^{2}}\right)-S\left(\frac{\\{b,c\\}a}{b^{2}d}\right)+S\left(\frac{\\{b,d\\}ac}{b^{2}d^{2}}\right)$
$\displaystyle=\frac{S(\\{a,c\\})bd-\\{a,c\\}S(bd)}{b^{2}d^{2}}-\frac{S(\\{a,d\\}c)bd^{2}-\\{a,d\\}cS(bd^{2})}{b^{2}d^{4}}$
$\displaystyle-\frac{S(\\{b,c\\}a)b^{2}d-\\{b,c\\}aS(b^{2}d)}{b^{4}d^{2}}+\frac{S(\\{b,d\\}ac)b^{2}d^{2}-\\{b,d\\}acS(b^{2}d^{2})}{b^{4}d^{4}}$
$\displaystyle=\frac{S(\\{a,c\\})}{bd}-\frac{\\{a,c\\}S(b)}{b^{2}d}-\frac{\\{a,c\\}S(d)}{bd^{2}}-\frac{S(\\{a,d\\})c}{bd^{2}}-\frac{\\{a,d\\}S(c)}{bd^{2}}+\frac{\\{a,d\\}cS(b)}{b^{2}d^{2}}$
$\displaystyle+\frac{2\\{a,d\\}cS(d)}{bd^{3}}-\frac{S(\\{b,c\\})a}{b^{2}d}-\frac{\\{b,c\\}S(a)}{b^{2}d}+\frac{2\\{b,c\\}aS(b)}{b^{3}d}+\frac{\\{b,c\\}aS(d)}{b^{2}d^{2}}$
$\displaystyle+\frac{S(\\{b,d\\})ac}{b^{2}d^{2}}+\frac{\\{b,d\\}S(a)c}{b^{2}d^{2}}+\frac{\\{b,d\\}aS(c)}{b^{2}d^{2}}-\frac{2\\{b,d\\}acS(b)}{b^{3}d^{2}}-\frac{2\\{b,d\\}acS(d)}{b^{2}d^{3}}$
$\displaystyle=\frac{\\{S(a),c\\}}{bd}+\frac{\\{a,S(c)\\}}{bd}-\frac{\\{a,c\\}S(b)}{b^{2}d}-\frac{\\{a,c\\}S(d)}{bd^{2}}-\frac{\\{S(a),d\\}c}{bd^{2}}$
$\displaystyle-\frac{\\{a,S(d)\\}c}{bd^{2}}-\frac{\\{a,d\\}S(c)}{bd^{2}}+\frac{\\{a,d\\}cS(b)}{b^{2}d^{2}}+\frac{2\\{a,d\\}cS(d)}{bd^{3}}-\frac{\\{S(b),c\\}a}{b^{2}d}$
$\displaystyle-\frac{\\{b,S(c)\\}a}{b^{2}d}-\frac{\\{b,c\\}S(a)}{b^{2}d}+\frac{2\\{b,c\\}aS(b)}{b^{3}d}+\frac{\\{b,c\\}aS(d)}{b^{2}d^{2}}+\frac{\\{S(b),d\\}ac}{b^{2}d^{2}}$
$\displaystyle+\frac{\\{b,S(d)\\}ac}{b^{2}d^{2}}+\frac{\\{b,d\\}S(a)c}{b^{2}d^{2}}+\frac{\\{b,d\\}aS(c)}{b^{2}d^{2}}-\frac{2\\{b,d\\}acS(b)}{b^{3}d^{2}}-\frac{2\\{b,d\\}acS(d)}{b^{2}d^{3}}.$

Direct calculations give that

$\displaystyle\Bigl{\\{}S\left(\frac{a}{b}\right),\frac{c}{d}\Bigr{\\}}+\Bigl{\\{}\frac{a}{b},S\left(\frac{c}{d}\right)\Bigr{\\}}=\Bigl{\\{}\frac{S(a)b-aS(b)}{b^{2}},\frac{c}{d}\Bigr{\\}}+\Bigl{\\{}\frac{a}{b},\frac{S(c)d-cS(d)}{d^{2}}\Bigr{\\}}$
$\displaystyle=\Bigl{\\{}\frac{S(a)}{b},\frac{c}{d}\Bigr{\\}}-\Bigl{\\{}\frac{aS(b)}{b^{2}},\frac{c}{d}\Bigr{\\}}+\Bigl{\\{}\frac{a}{b},\frac{S(c)}{d}\Bigr{\\}}-\Bigl{\\{}\frac{a}{b},\frac{cS(d)}{d^{2}}\Bigr{\\}}$
$\displaystyle=\frac{\\{S(a),c\\}}{bd}-\frac{\\{S(a),d\\}c}{bd^{2}}-\frac{\\{b,c\\}S(a)}{b^{2}d}+\frac{\\{b,d\\}S(a)c}{b^{2}d^{2}}-\frac{\\{a,c\\}S(b)}{b^{2}d}$
$\displaystyle-\frac{\\{S(b),c\\}a}{b^{2}d}+\frac{\\{a,d\\}S(b)c}{b^{2}d^{2}}+\frac{\\{S(b),d\\}ac}{b^{2}d^{2}}+\frac{2\\{b,c\\}aS(b)}{b^{3}d}-\frac{2\\{b,d\\}aS(b)c}{b^{3}d^{2}}$
$\displaystyle+\frac{\\{a,S(c)\\}}{bd}-\frac{\\{a,d\\}S(c)}{bd^{2}}-\frac{\\{b,S(c)\\}a}{b^{2}d}+\frac{\\{b,d\\}aS(c)}{b^{2}d^{2}}-\frac{\\{a,c\\}S(d)}{bd^{2}}$
$\displaystyle-\frac{\\{a,S(d)\\}c}{bd^{2}}+\frac{2\\{a,d\\}cS(d)}{bd^{3}}+\frac{\\{b,c\\}S(d)a}{b^{2}d^{2}}+\frac{\\{b,S(d)\\}ac}{b^{2}d^{2}}-\frac{2\\{b,d\\}acS(d)}{b^{2}d^{3}}.$

These two equalities imply (8). $\Box$

Consider the grading (7) of $P$. A Poisson derivation $D$ of $P$ will be
called a d-graded Poisson derivation if $D(x_{i})\in P_{1}$ for all
$i=1,\ldots,n$. Obviously, every $d$-graded Poisson derivation of $P$ induces
a Poisson derivation of $P_{0}$. The reverse is also true.

######  Lemma 2.

Every Poisson derivation of $P_{0}$ can be uniquely extended to a $d$-graded
Poisson derivation of $P=P\langle x_{1},x_{2},\ldots,x_{n}\rangle$.

Proof. Let $D$ be a Poisson derivation of $P_{0}$. In particular, $D$ is a
derivation of the associative and commutative algebra $P_{0}$. Since $P_{0}$
is a domain, $D$ can be uniquely extended [25, p. 120] to a derivation $T$ of
the field of fractions $Q(P_{0})$ of $P_{0}$. The field extension

$Q(P_{0})\subseteq Q(P)$

is algebraic since every $e_{i}$ is a root of the polynomial
$p(t)=t^{d}-e_{i}^{d}\in Q(P_{0})[t]$ for all $i$. This extension is separable
since $K$ is a field of characteristic zero. By Corollaries $2$ and
$2^{\prime}$ in [25, pages 124–125], the associative derivation $T$ of the
field $Q(P_{0})$ can be uniquely extended to an associative derivation $S$ of
the field $Q(P)$.

Suppose that

$\displaystyle S(e_{j})=\frac{f_{j}}{g_{j}},$

where $f_{j}\in P$, $0\neq g_{j}\in P$, and the pairs $f_{j},g_{j}$ are
relatively prime for all $j$. Notice that $e_{j}^{d}\in P_{0}$ and

$\displaystyle D(e_{j}^{d})=S(e_{j}^{d})=de_{j}^{d-1}\frac{f_{j}}{g_{j}}\in
P_{0}.$

Consequently,

(9) $\displaystyle g_{j}|e_{j}^{d-1}$

since the pair $f_{j},g_{j}$ is relatively prime, that is, $g_{j}$ is a power
of $e_{j}$.

If $d|\deg e_{j}$ then $e_{j}\in P_{0}$ and $S(e_{j})=D(e_{j})\in P_{0}$,
i.e., we may assume that $g_{j}=1$. If $d\not|\deg e_{j}$ then there exist
$1\leq i\leq n$ and $1\leq k<d$ such that $e_{i}=x_{i}\neq e_{j}$ and
$x_{i}^{k}e_{j}\in P_{0}$. Then

$\displaystyle
D(x_{i}^{k}e_{j})=S(x_{i}^{k}e_{j})=kx_{i}^{k-1}\frac{f_{i}}{g_{i}}e_{j}+x_{i}^{k}\frac{f_{j}}{g_{j}}\in
P_{0}.$

Consequently,

$\displaystyle g_{i}g_{j}|kx_{i}^{k-1}f_{i}g_{j}e_{j}+x_{i}^{k}f_{j}g_{i}.$

This implies that $g_{j}|x_{i}^{k}f_{j}g_{i}$. Since $f_{j},g_{j}$ are
relatively prime, we get $g_{j}|x_{i}^{k}g_{i}$. By (9) $g_{i}$ is a power of
$e_{i}=x_{i}$ and $g_{j}$ is a power of $e_{j}$. Since $i\neq j$, this implies
that $g_{j}\in K$ for all $j$. Consequently, $S(e_{j})\in P$ and
$S(P)\subseteq P$.

Let us now show that the restriction of $S$ to $P$ is a Lie derivation, i.e.,

(10) $\displaystyle S(\\{u,v\\})=\\{S(u),v\\}+\\{u,S(v)\\}$

for all $u,v$ of the form (5). We prove (10) by induction on the polynomial
length $d(u)+d(v)$. Suppose that $u=e_{i}$ and $v=e_{j}$. Since
$e_{i}^{d},e_{j}^{d}\in P_{0}$, we get

$\displaystyle
S(\\{e_{i}^{d},e_{j}^{d}\\})=D(\\{e_{i}^{d},e_{j}^{d}\\})=\\{D(e_{i}^{d}),e_{j}^{d}\\}+\\{e_{i}^{d},D(e_{j}^{d})\\}$
$\displaystyle=\\{S(e_{i}^{d}),e_{j}^{d}\\}+\\{e_{i}^{d},S(e_{j}^{d})\\}=\\{de_{i}^{d-1}S(e_{i}),e_{j}\\}de_{j}^{d-1}+de_{i}^{d-1}\\{e_{i},de_{j}^{d-1}S(e_{j})\\}$
$\displaystyle=d^{2}(d-1)e_{i}^{d-2}e_{j}^{d-1}S(e_{i})\\{e_{i},e_{j}\\}+d^{2}e_{i}^{d-1}e_{j}^{d-1}\\{S(e_{i}),e_{j}\\}$
$\displaystyle+d^{2}(d-1)e_{i}^{d-1}e_{j}^{d-2}S(e_{j})\\{e_{i},e_{j}\\}+d^{2}e_{i}^{d-1}e_{j}^{d-1}\\{e_{i},S(e_{j})\\}.$

On the other hand,

$\displaystyle\\{e_{i}^{d},e_{j}^{d}\\}=d^{2}e_{i}^{d-1}e_{j}^{d-1}\\{e_{i},e_{j}\\}$

and

$\displaystyle
S(\\{e_{i}^{d},e_{j}^{d}\\})=S(d^{2}e_{i}^{d-1}e_{j}^{d-1}\\{e_{i},e_{j}\\})$
$\displaystyle=d^{2}S(e_{i}^{d-1})e_{j}^{d-1}\\{e_{i},e_{j}\\}+d^{2}e_{i}^{d-1}S(e_{j}^{d-1})\\{e_{i},e_{j}\\}+d^{2}e_{i}^{d-1}e_{j}^{d-1}S(\\{e_{i},e_{j}\\})$
$\displaystyle=d^{2}(d-1)e_{i}^{d-2}e_{j}^{d-1}S(e_{i})\\{e_{i},e_{j}\\}+d^{2}(d-1)e_{i}^{d-1}e_{j}^{d-2}S(e_{j})\\{e_{i},e_{j}\\}+d^{2}e_{i}^{d-1}e_{j}^{d-1}S(\\{e_{i},e_{j}\\}).$

Comparing two values of $S(\\{e_{i}^{d},e_{j}^{d}\\})$, we get

$\displaystyle
S(\\{e_{i},e_{j}\\})=\\{S(e_{i}),e_{j}\\}+\\{e_{i},S(e_{j})\\}.$

Suppose that $d(v)\geq 2$ and $v=v_{1}v_{2}$. Then

$\displaystyle
S(\\{u,v\\})=S(\\{u,v_{1}v_{2}\\})=S(v_{1}\\{u,v_{2}\\}+\\{u,v_{1}\\}v_{2})$
$\displaystyle=S(v_{1})\\{u,v_{2}\\}+v_{1}S(\\{u,v_{2}\\})+S(\\{u,v_{1}\\})v_{2}+\\{u,v_{1}\\}S(v_{2}).$

By the induction proposition, (10) is true for pairs $u,v_{1}$ and $u,v_{2}$,
i.e.,

$\displaystyle S(\\{u,v_{1}\\})=\\{S(u),v_{1}\\}+\\{u,S(v_{1})\\},\ \ \
S(\\{u,v_{2}\\})=\\{S(u),v_{2}\\}+\\{u,S(v_{2})\\}.$

Then

$\displaystyle
S(\\{u,v\\})=S(v_{1})\\{u,v_{2}\\}+v_{1}\\{S(u),v_{2}\\}+v_{1}\\{u,S(v_{2})\\}$
$\displaystyle+\\{S(u),v_{1}\\}v_{2}+\\{u,S(v_{1})\\}v_{2}+\\{u,v_{1}\\}S(v_{2})$
$\displaystyle=\\{S(u),v_{1}v_{2}\\}+\\{u,S(v_{1})v_{2}+v_{1}S(v_{2})\\}=\\{S(u),v\\}+\\{u,S(v)\\}.$

Consequently, $S$ is a derivation of a Poisson algebra $P$ and induces $D$ on
$P_{0}$. $\Box$

######  Lemma 3.

Every locally nilpotent derivation of the Poisson algebra $P_{0}$ is induced
by a locally nilpotent $d$-derivation of the Poisson algebra $P=P\langle
x_{1},x_{2},\ldots,x_{n}\rangle$.

Proof. Let $D$ be a locally nilpotent derivation of $P_{0}$ and let $S$ be a
unique extension of $D$ to $P$. We have to show that $S$ is a locally
nilpotent derivation of $P$. Notice that

$P_{0}\subset P$

is an integral extension of domains since $e_{i}^{d}\in P_{0}$ for all $i\geq
1$. According to a result of W.V. Vasconcelos [24] (see also Proposition
1.3.37 from [3, p. 41]), $S$ is locally nilpotent. $\Box$

##  4\. Automorphisms of $P_{0}$

As we noticed above, every $d$-graded automorphism of $P\langle
x_{1},x_{2},\ldots,x_{n}\rangle$ induces an automorphism of $P_{0}$. In this
section we prove the reverse of this statement for $n>1$.

######  Theorem 1.

Let $K$ be a field closed with respect to taking all $d$-roots of elements.
Then every automorphism of $P_{0}$ over $K$ is induced by a $d$-graded
automorphism of $P=P\langle x_{1},x_{2},\ldots,x_{n}\rangle$ if $n>1$.

Proof. Let $\alpha$ be an automorphism of $P_{0}$. Denote the extension of
$\alpha$ to the quotient field $Q(P_{0})$ by the same symbol. We have
$\frac{x_{2}}{x_{1}}\in Q(P_{0})$. Suppose that

(11)
$\displaystyle\alpha\left(\frac{x_{2}}{x_{1}}\right)=\frac{f_{2}}{f_{1}},$

where $f_{1},f_{2}$ are relatively prime. Then

$\displaystyle\alpha\left(\frac{x_{2}^{d}}{x_{1}^{d}}\right)=\alpha\left(\frac{x_{2}}{x_{1}}\right)^{d}=\frac{f_{2}^{d}}{f_{1}^{d}}.$

Since $f_{1},f_{2}$ are relatively prime it follows that
$\alpha(x_{1}^{d})=vf_{1}^{d}$ and $\alpha(x_{2}^{d})=vf_{2}^{d}$ for some
$v\in P$. Moreover, $\alpha(x_{1}^{i}x_{2}^{d-i})=vf_{1}^{i}f_{2}^{d-i}$ for
all $0\leq i<d$.

We have $vf_{1}^{d},vf_{2}^{d}\in P_{0}$. If $K$ is a field of characteristic
$p>0$ and $p$ divides $d$, then $f_{1}^{d},f_{2}^{d}\in P_{0}$. Consequently,
$v\in P_{0}$. Assume that $K$ is a field of charateristic $0$ or of
characteristic $p>0$ and $p$ does not divide $d$. Let $\epsilon$ be a
primitive $d$-root of unity. Consider the automorphism $\varepsilon$ of $Q(P)$
such that $\varepsilon(x_{i})=\epsilon x_{i}$ for all $i$. Notice that for any
$f\in Q(P)$ we have $f\in Q(P_{0})$ if and only if $\varepsilon(f)=f$. Then

$\displaystyle\frac{f_{2}}{f_{1}}=\varepsilon(\frac{f_{2}}{f_{1}})=\frac{\varepsilon(f_{2})}{\varepsilon(f_{1})}$

and $\varepsilon(f_{1}),\varepsilon(f_{2})$ are relatively prime. Hence
$f_{1}\varepsilon(f_{2})=\varepsilon(f_{1})f_{2}$ and, say, $f_{1}$ divides
$\varepsilon(f_{1})$ and $\varepsilon(f_{1})$ divides $f_{1}$. Consequently
$f_{1}$ and $\varepsilon(f_{1})$ are proportional which is possible only if
$f_{1}$ is a $d$-homogeneous element. Similarly, $f_{2}$ is a $d$-homogeneous
element. Then $f_{1}^{d},f_{2}^{d}\in P_{0}$ and, consequently, $v\in P_{0}$.

This implies that

$\displaystyle x_{1}^{d}=\alpha^{-1}(v)\alpha^{-1}(f_{1}^{d}),\ \
x_{2}^{d}=\alpha^{-1}(v)\alpha^{-1}(f_{2}^{d}).$

Since $x_{1}^{d}$ is irreducible in $P_{0}$, this is possible only if $v\in
K$. Let $\mu\in K$ be a $d$-root of $v$, i.e., $\mu^{d}=v$. Replacing $f_{1}$
and $f_{2}$ by $\mu f_{1}$ and $\mu f_{2}$, we may assume that

(12) $\displaystyle\alpha(x_{1}^{d})=f_{1}^{d},\ \
\alpha(x_{2}^{d})=f_{2}^{d}.$

By (11) and (12), we get

$\displaystyle\alpha(x_{1}^{i_{1}}x_{2}^{i_{2}})=f_{1}^{i_{1}}f_{2}^{i_{2}},\,\textit{if}\,\,x_{1}^{i_{1}}x_{2}^{i_{2}}\in
P_{0}.$

Consider an arbitrary $e_{i}$ with $i\geq 3$. Suppose that $\deg e_{i}=s$.
Then $y_{i}=\frac{e_{i}}{x_{1}^{s}}\in Q(P_{0})$. Suppose that

$\displaystyle\alpha(y_{i})=\alpha\left(\frac{e_{i}}{x_{1}^{s}}\right)=\frac{f_{i}}{g_{i}},$

where $f_{i},g_{i}$ are relatively prime. Then

$\displaystyle\alpha\left(\frac{e_{i}^{d}}{x_{1}^{sd}}\right)=\alpha\left(\frac{e_{i}}{x_{1}^{s}}\right)^{d}=\frac{f_{i}^{d}}{g_{i}^{d}}.$

Again $\alpha(e_{i}^{d})=vf_{i}^{d}$ and $\alpha(x_{1}^{sd})=vg_{i}^{d}$ for
some $v\in P$. As above, we get that $f_{i}^{d},g_{i}^{d}\in P_{0},v\in K$,
and we can assume that

$\displaystyle\alpha(e_{i}^{d})=f_{i}^{d},\ \ \alpha(x_{1}^{sd})=g_{i}^{d}.$

Then $f_{1}^{sd}=g_{i}^{d}$ and $g_{i}=\lambda f_{1}^{s}$, where $\lambda$ is
a $d$-root of unity. After rescaling, we can assume that $g_{i}=f_{1}^{s}$ and

(13) $\displaystyle\alpha(e_{i}^{d})=f_{i}^{d},\ \
\alpha(y_{i})=\alpha\left(\frac{e_{i}}{x_{1}^{s}}\right)=\frac{f_{i}}{f_{1}^{s}},$

where $s=\deg e_{i}$ and $i\geq 3$. This is true for $i=2$ by (11) and (12).

Let $u=e_{i_{1}}\ldots e_{i_{k}}$ be an arbitrary element of $P$ of the form
(5). We have

(14) $\displaystyle
u=x_{1}^{s}\frac{e_{i_{1}}}{x_{1}^{s_{i_{1}}}}\ldots\frac{e_{i_{k}}}{x_{1}^{s_{i_{k}}}}=x_{1}^{s}y_{i_{1}}\ldots
y_{i_{k}},$

where $s=s_{i_{1}}+\ldots+s_{i_{k}}$. We have $d|s$ since $u\in P_{0}$. Then

$\displaystyle\alpha(u)=f_{1}^{s}\frac{f_{i_{1}}}{f_{1}^{s_{1}}}\ldots\frac{f_{i_{k}}}{f_{1}^{s_{k}}}=f_{i_{1}}\ldots
f_{i_{k}}$

by (11), (12), and (13).

Consequently, the polynomial endomorphism $\beta$ of $P$, determined by
$\beta(e_{i})=f_{i}$ for all $i\geq 1$, induces $\alpha$ on $P_{0}$. First we
show that $\beta$ is a polynomial automorphism of $P$. The elements (4) are
algebraically independent and, consequently, the elements
$e_{1}^{d},\ldots,e_{s}^{d},\ldots$ are algebraically independent. Since
$\alpha$ is an automorphism and $\alpha(e_{i}^{d})=f_{i}^{d}$ for all $i$ by
(12) and (13), the elements $f_{1}^{d},\ldots,f_{s}^{d},\ldots$ are
algebraically independent. Therefore the elements $f_{1},\ldots,f_{s},\ldots$
are algebraically independent and $\beta$ is an injective endomorphism. Then
$\beta$ can be uniquely extended to an endomorphism of the quotient field
$P(x_{1},x_{2},\ldots,x_{n})$ and we denote this extension also by $\beta$.

The restriction of $\beta$ on $Q(P_{0})$ is an automorphism since it coincides
with the $\alpha$. Consider the space

$\displaystyle V=Q(P_{0})P\langle x_{1},x_{2},\ldots,x_{n}\rangle.$

By (14) every element $f\in P$ can be written as

$\displaystyle f=f_{0}+f_{1}x_{1}+\ldots+f_{d-1}x_{1}^{d-1},$

where $f_{0},f_{1},\ldots,f_{d-1}\in K[t,y_{2},\ldots,y_{s},\ldots]$ and
$t=x_{1}^{d}$. Hence $V$ is the $Q(P_{0})$-span of the elements
$1,x_{1},x_{1}^{2},\ldots,x_{1}^{d-1}$. If

$\displaystyle V=b_{1}Q(P_{0})\oplus\ldots\oplus b_{k}Q(P_{0}),$

then

$\displaystyle\beta(V)=\beta(b_{1})Q(P_{0})+\ldots+\beta(b_{k})Q(P_{0})$

since $\beta(Q(P_{0}))=Q(P_{0})$. Notice that $\beta(V)\subseteq V$. If
$\beta(V)\neq V$ then $\dim_{Q(P_{0})}V<k$ and $\mathrm{Ker}\beta\neq 0$. It
is impossible for nonzero field endomorphisms. Consequently, $\beta(V)=V$ and
$e_{i}\in\beta(V)$ for all $i$. Therefore $\beta$ is an automorphism of the
field $P(x_{1},x_{2},\ldots,x_{n})$ and of the polynomial algebra $P\langle
x_{1},x_{2},\ldots,x_{n}\rangle$.

It remains to show that $\beta$ is a Lie automorphism of $P$, i.e.,

(15) $\displaystyle\beta(\\{u,v\\})=\\{\beta(u),\beta(v)\\}$

for all $u,v$ of the form (5). We prove (15) by induction on the polynomial
length $d(u)+d(v)$. Suppose that $u=e_{i}$ and $v=e_{j}$. Since
$e_{i}^{d},e_{j}^{d}\in P_{0}$, we get

$\displaystyle\beta(\\{e_{i}^{d},e_{j}^{d}\\})=\alpha(\\{e_{i}^{d},e_{j}^{d}\\})=\\{\alpha(e_{i}^{d}),\alpha(e_{j}^{d})\\}=\\{\beta(e_{i}^{d}),\beta(e_{j}^{d})\\}$
$\displaystyle=\\{\beta(e_{i})^{d},\beta(e_{j})^{d}\\}=d^{2}\beta(e_{i})^{d-1}\beta(e_{j})^{d-1}\\{\beta(e_{i}),\beta(e_{j})\\}.$

On the other hand,

$\displaystyle\beta(\\{e_{i}^{d},e_{j}^{d}\\})=\beta(d^{2}e_{i}^{d-1}e_{j}^{d-1}\\{e_{i},e_{j}\\})=d^{2}\beta(e_{i})^{d-1}\beta(e_{j})^{d-1}\beta(\\{e_{i},e_{j}\\}).$

Comparing two values of $\beta(\\{e_{i}^{d},e_{j}^{d}\\})$, we get that (15)
holds for $u=e_{i}$ and $v=e_{j}$.

Suppose that $d(v)\geq 2$ and $v=v_{1}v_{2}$. Then

$\displaystyle\beta(\\{u,v\\})=\beta(\\{u,v_{1}v_{2}\\})=\beta(v_{1}\\{u,v_{2}\\}+\\{u,v_{1}\\}v_{2})$
$\displaystyle=\beta(v_{1})\beta(\\{u,v_{2}\\})+\beta(\\{u,v_{1}\\})\beta(v_{2}).$

By the induction proposition, we may assume that (15) is true for pairs
$u,v_{1}$ and $u,v_{2}$. Then

$\displaystyle\beta(\\{u,v\\})=\beta(v_{1})\\{\beta(u),\beta(v_{2})\\}+\\{\beta(u),\beta(v_{1})\\}\beta(v_{2})$
$\displaystyle=\\{\beta(u),\beta(v_{1})\beta(v_{2})\\}=\\{\beta(u),\beta(v)\\}.$

Consequently, $\beta$ is an automorphism of $P$ and induces $\alpha$ on
$P_{0}$. $\Box$

Let $\mathrm{Aut}_{d}P$ be the group of all $d$-graded automorphisms of the
free Poisson algebra $P$.

######  Corollary 1.

Let $K$ be a field closed with respect to taking all $d$-roots of elements and
let $E=\\{\lambda\mathrm{id}|\lambda^{d}=1,\lambda\in K\\}$, where
$\mathrm{id}$ is the identity automorphism of $P$. Then

$\displaystyle\mathrm{Aut}\,P_{0}\cong\mathrm{Aut}_{d}P/E.$

Proof. Consider the homomorphism

(16) $\displaystyle\psi:\mathrm{Aut}_{d}P\to{Aut}\,P_{0}$

defined by $\psi(\alpha)=\overline{\alpha}$, where $\overline{\alpha}$ is the
automorphism of $P_{0}$ induced by the $d$-graded automorphism $\alpha$ of
$P$.

By Theorem 1, $\psi$ is an epimorphism. Let $\alpha\in\mathrm{Ker}\,\psi$.
Then $\alpha(x_{1})^{d}=x_{1}^{d}$. Consequently, $\alpha(x_{1})=\lambda
x_{1}$ for some $d$th root of unity $\lambda\in K$. Extending $\alpha$ to
$Q(P_{0})$, we get $\alpha(x_{i}/x_{1})=x_{i}/x_{1}$. Consequently,
$\alpha(x_{i})=\lambda x_{i}$ for all $i$ and $\alpha=\lambda\mathrm{id}$,
i.e., $\alpha\in E$. Obviously, $E\subseteq\mathrm{Ker}\,\psi$. $\Box$

##  5\. Veronese subalgebras of polynomial algebras

Let $A=K[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra over a field $K$
in the variables $x_{1},x_{2},\ldots,x_{n}$. Consider the grading

$\displaystyle A=A_{0}\oplus A_{1}\oplus\ldots\oplus A_{d-1},$

where $d\geq 2$ and $A_{i}$ is the subspace of $A$ generated by all monomials
of degree $kd+i$ for all $k\geq 0$. This is a $\mathbb{Z}_{d}$-grading of $A$,
i.e., $A_{i}A_{j}\subseteq A_{i+j}$ for all $i,j\in\mathbb{Z}_{d}$. The
subalgebra $A_{0}$ is called the Veronese subalgera of $A$ of degree $d$.

######  Corollary 2.

Let $A=K[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra over a field $K$
of characteristic zero in $n\geq 2$ variables $x_{1},x_{2},\ldots,x_{n}$. Then
every derivation of the Veronese subalgebra $A_{0}$ can be uniquely extended
to a $d$-graded derivation of $K[x_{1},x_{2},\ldots,x_{n}]$.

######  Corollary 3.

Let $A=K[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra over a field $K$
of characteristic zero in $n\geq 2$ variables $x_{1},x_{2},\ldots,x_{n}$. Then
every locally nilpotent derivation of the Veronese subalgebra $A_{0}$ is
induced by a locally nilpotent $d$-derivation of the polynomial algebra
$K[x_{1},x_{2},\ldots,x_{n}]$.

######  Corollary 4.

Let $A=K[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra in $n\geq 2$
variables $x_{1},x_{2},\ldots,x_{n}$ over a field $K$ closed with respect to
taking all $d$-roots of elements. Then every automorphism of the Veronese
subalgebra $A_{0}$ of degree $d$ is induced by a $d$-graded automorphism of
$K[x_{1},x_{2},\ldots,x_{n}]$.

This result is also proven in [8].

Let $\mathrm{Aut}_{d}A$ be the group of all $d$-graded automorphisms of the
polynomial algebra $A$.

######  Corollary 5.

Let $K$ be a field closed with respect to taking all $d$-roots of elements and
let $E=\\{\lambda\mathrm{id}|\lambda^{d}=1,\lambda\in K\\}$, where
$\mathrm{id}$ is the identity automorphism of $A$. Then

$\displaystyle\mathrm{Aut}\,A_{0}\cong\mathrm{Aut}_{d}A/E.$

The proofs of Corollary 2, Corollary 3, Corollary 4, and Corollary 5 repeat
the polynomial parts of the proofs of Lemma 2, Lemma 3, Theorem 1, and
Corollary 1, respectively.

Notice that these statements are not true for the polynomial algebra $A=K[x]$
in one variable $x$. In this case, the Veronese subalgebra $A_{0}$ of degree
$d$ is the polynomial algebra in one variable $x^{d}$. Then the locally
nilpotent derivation of $A_{0}$ determined by

$\displaystyle x^{d}\mapsto 1$

cannot be induced by any derivation of $A$ and the automorphism of $A_{0}$
determined by

$\displaystyle x^{d}\mapsto x^{d}+1$

cannot be induced by any automorphism of $A$.

In addition, analogues of these results are not true for free associative
algebras. In fact, if $B=K\langle x,y\rangle$ is the free associative algebra
in the variables $x,y$ and $d=2$ then the Veronese subalgebra $B_{0}$ of
degree $d$ is the free associative algebra in the variables
$x^{2},xy,yx,y^{2}$. It is easy to check that the locally nilpotent derivation
of $B_{0}$ determined by

$\displaystyle x^{2}\mapsto 1,xy\mapsto 0,yx\mapsto 0,y^{2}\mapsto 0$

cannot be induced by any derivation of $B$ and the automorphism of $B_{0}$
determined by

$\displaystyle x^{2}\mapsto x^{2}+1,xy\mapsto xy,yx\mapsto yx,y^{2}\mapsto
y^{2}$

cannot be induced by any automorphism of $B$.

Acknowledgments

The second and third authors are grateful to Max-Planck Institute für
Mathematik for hospitality and excellent working conditions, where part of
this work has been done.

The third author is supported by the grant of the Ministry of Education and
Science of the Republic of Kazakhstan (project AP14872073).

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# A Data-Driven Hybrid Automaton Framework to Modeling Complex Dynamical
Systems
††thanks: This research was in part supported by the National Science
Foundation, under NSF CAREER Award 2143351, and NSF CNS Award no. 2223035.

Yejiang Yang  School of Electrical Engineering
Southwest Jiaotong University
Chengdu, China
<EMAIL_ADDRESS>Zihao Mo  School of Computer and Cyber Sciences
Augusta University
Augusta, USA
<EMAIL_ADDRESS>Weiming Xiang  School of Computer and Cyber Sciences
Augusta University
Augusta, USA
<EMAIL_ADDRESS>
###### Abstract

In this paper, a computationally efficient data-driven hybrid automaton model
is proposed to capture unknown complex dynamical system behaviors using
multiple neural networks. The sampled data of the system is divided by valid
partitions into groups corresponding to their topologies and based on which,
transition guards are defined. Then, a collection of small-scale neural
networks that are computationally efficient are trained as the local dynamical
description for their corresponding topologies. After modeling the system with
a neural-network-based hybrid automaton, the set-valued reachability analysis
with low computation cost is provided based on interval analysis and a split
and combined process. At last, a numerical example of the limit cycle is
presented to illustrate that the developed models can significantly reduce the
computational cost in reachable set computation without sacrificing any
modeling precision.

###### Index Terms:

data-driven modeling, hybrid automata, neural networks

##  I Introduction

Data-driven methods such as neural networks are widely used in modeling for
their effectiveness without relying on the explicit mathematical model or
prior knowledge of the system in a variety of research activities, e.g.,
modeling nonlinear dynamical systems in the description of Ordinary
Differential Equations (ODEs) [1] such as modeling thermal conductivity of
water-based nanofluid containing magnetic copper nanoparticles in [2],
modeling groundwater-level variation in coastal aquifers in [3], etc. However,
due to the high complexity of large-scale neural network models, some
computationally expensive tasks such as reachability analysis are difficult to
perform on neural-network-based models. Therefore, computationally efficient
modeling methods are in critical need for neural-network-based models.

To improve the performance of the model and ensure that the model is matching
the characteristics of the system, e.g., robustness, stability, etc, modeling
is becoming a challenging task. The training of neural network models has
received particular attention in the machine learning community. For instance,
adding Lyapunov constraints in the training neural networks to enhance
stabilization of learning nonlinear system in [4], studying the adversarial
robustness of neural networks using robust optimization in [5], utilizing the
idea of robust optimization in the training of neural networks in [6],
estimating the Lipschitz constant of neural networks in [7], etc. Besides
training, the verification for neural networks plays a crucial part in the
usability and safety of the neural-network-based model is investigated in
research such as providing reachable set estimation and safety verification
for multi-layer neural networks in [8], verifying the neural network model by
the star set based set-valued reachability analysis in [9], providing bound
propagation-based neural network verifiers in [10, 11], etc. Training and
verification are related to the size of the neural network, i.e., a single
neural network that aims to be trained with all samples may lead to the
training and verification for the neural network model becoming complex and
time-consuming works.

Inspired by [12], a dynamical system can be modeled by a hybrid automaton with
a finite number of local topologies plus transitions among them. Furthermore,
if the dynamical description for each topology of the hybrid automaton is a
neural network, and the neural network only needs to approximate the local
system dynamics within that topology, which means the size of each neural
network can be scaled down compared with using a large-scale neural network
approximating the entire system dynamics. As a result, the computational
complexity in either training or verification will be reduced and moreover,
due to parallel training, the scalability can be further increased. In this
paper, a neural-network-based hybrid automaton is proposed to reduce the
computation cost in training and verification for modeling the dynamical
systems. First, the given region is divided into valid partitions representing
topologies of the proposed model, based on which the guards are defined.
Sample data are selected into different groups with which the neural networks
are trained respectively as the dynamical description of their corresponding
topologies. Then, the Mean Square Error (MSE) and analysis of the set-valued
reachability of our proposed method are provided. Lastly, a numerical example
is given to illustrate the effectiveness of our approach.

The main contributions of this paper lie in the way to model the dynamical
system using our computationally efficient neural-network-based hybrid
automaton and its set-valued analysis. The neural-network-based hybrid
automaton models the system with multiple neural networks with each trained
with the sample group corresponding to its topology, which reduces the
computational complexity while increasing scalability. The set-valued analysis
is given based on interval analysis and a _Split and_ _Combined_ process.

The paper is organized as follows: preliminaries and problem formulation are
given in Section II. The main result, modeling with neural-network-based
hybrid automaton, and the set-valued analysis are given in Section III. In
Section IV, a limit cycle modeling example is provided to evaluate our method.
Conclusions are given in Section V.

##  II Preliminaries and Problem Formulation

The dynamical systems in the paper are in the general discrete-time form of

$\displaystyle x(k+1)=f(x(k),u(k))$ (1)

where $x(k)\in\mathbb{R}^{n_{x}}$ is the system state, and
$u(k)\in\mathbb{R}^{n_{u}}$ is the system input, respectively. The evolution
of system state is governed by
$f:\mathbb{R}^{n_{x}+n_{u}}\to\mathbb{R}^{n_{x}}$, which is an unknown
nonlinear discrete-time process. In this paper, we aim to develop a novel
data-driven, i.e., neural-network-based, modeling method to approximate this
unknown nonlinear mapping $f$.

###### Assumption 1

It is assumed that state $x(k)$ and input $u(k)$ are all measurable from
unknown nonlinear system (1).

Under Assumption 1, the training data for modeling is defined as follows.

###### Definition 1

Sampled data $\mathcal{W}=\\{w_{1},w_{2},\cdots,w_{L}\\}$ of an unknown
dynamical system (1) is a collection of sampled $L$ traces obtained by
measurement, where for each trace $w_{i}$, $i=1,\ldots,L$, is a finite
sequence of time steps and data
$(k_{0,i},d_{0,i}),(k_{1,i},d_{1,i}),\cdots,(k_{M_{i},i},d_{M_{i},i})$ in
which

  * •

$k_{\ell,i}\in(0,\infty)$ and $k_{\ell+1,i}=k_{\ell,i}+1$,
$\forall\ell=0,1,\ldots,M_{i},\ \forall i=1,2,\ldots,L$.

  * •

$d_{\ell,i}=[x^{\top}_{i}(k_{\ell,i}),~{}u^{\top}_{i}(k_{\ell,i})]^{\top}\in\mathbb{R}^{n_{x}+n_{u}}$,
$\forall\ell=0,1,\ldots,M_{i}$, $\forall i=1,2,\ldots,L$, where
$x_{i}(k_{\ell,i}),u_{i}(k_{\ell,i})$ denote the state and input of the system
at $\ell$th step for $i$th trace, respectively.

In this paper, it is assumed that there exist sufficient measurable input and
state traces available in $\mathcal{W}$ for the data-driven modeling of
nonlinear system (1). Instead of using one single large-scale neural network,
e.g., Deep Neural Networks (DNN), as in most of the existing results [8, 13,
9] which commonly suffer overly expensive computation in successive use
especially for safety verification before deployment on safety-critical
systems, we aim to develop a novel neural-network-based hybrid automaton
consisting of a family of small-scale neural networks, i.e., shallow neural
networks, along with inferred transitions to not only accurately approximate
the unknown nonlinear system model $f$, but also hold a great promise in
performing computational-efficient verification.

In this work, we consider feedforward neural networks in the form of
$\Phi:\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{L}}$ defined by the following
recursive equations in the form of

$\displaystyle\begin{cases}\eta_{\ell}=\phi_{\ell}(W_{\ell}\eta_{\ell-1}+b_{\ell}),~{}\ell=1,\ldots,L\\\
\eta_{L}=\Phi(\eta_{0})\end{cases}$ (2)

where $\eta_{\ell}$ denotes the output of the $\ell$-th layer of the neural
network, and in particular $\eta_{0}\in\mathbb{R}^{n_{0}}$ is the input to the
neural network and $\eta_{L}\in\mathbb{R}^{n_{L}}$ is the output produced by
the neural network, respectively. $W_{\ell}\in\mathbb{R}^{n_{\ell}\times
n_{\ell-1}}$ and $b_{\ell}\in\mathbb{R}^{n_{\ell}}$ are weight matrices and
bias vectors for the $\ell$-th layer.
$\phi_{\ell}=[\psi_{\ell},\cdots,\psi_{\ell}]$ is the concatenation of
activation functions of the $\ell$-th layer in which
$\psi_{\ell}:\mathbb{R}\to\mathbb{R}$ is the activation function.

In this paper, we focus on reducing the computational complexity of neural-
network-based models in terms of reachability analysis. The reachable set of
neural network (2) is given as follows.

###### Definition 2

Given neural network (2) with a bounded input set $\mathcal{V}$, the following
set

$\mathcal{Y}\triangleq\\{\eta_{L}\mid\eta_{L}=\Phi(\eta_{0}),~{}\eta_{0}\in\mathcal{V}\subset\mathbb{R}^{n}\\}$
(3)

is called the reachable set of neural network (2).

###### Remark 1

The computation complexity for reachable set computation of $\mathcal{Y}$
heavily relies on the number of layers and neurons. To enable computationally
efficient neural-network-based models in particular for reachability analysis
and successive verification procedures, we have to choose neural networks with
fewer layers and neurons, however, it usually leads to low training accuracy
for complex nonlinear systems. Our goal in this paper is to overcome this
dilemma of computation complexity versus training accuracy.

##  III Main Results

###  III-A Neural-Network-Based Hybrid Automata

The main goal of this paper is to develop an efficiently verifiable data-
driven model for nonlinear system (1). Inspired by the hybridization methods
for the analysis of nonlinear systems in [14, 15] which enable the efficient
verification procedures of hybrid automata with complex, nonlinear dynamics
through an abstraction process of multiple local subsystems of simplified
forms, we propose a hybridization method to model unknown nonlinear dynamics
(1) in data-driven modeling scenarios utilizing a collection of
small/moderate-size neural networks characterizing local system behaviors.

A neural-network-based hybrid automaton consists of variable components
describing both dynamics and the discrete transition logic of a system, which
will be used as the modeling framework in the rest of this paper.

###### Definition 3

A neural-network-based hybrid automaton is defined by a tuple
$\mathcal{H}\triangleq\left\langle\mathcal{Q},\mathcal{X},init,\mathcal{U},\mathcal{E},g,\mathcal{G},inv,\mathcal{F}\right\rangle$
in which the components are defined by:

  * •

_Topologies_ : $\mathcal{Q}\triangleq\\{q_{1},q_{2},\ldots,q_{N}\\}$ is a
finite set of _topologies_.

  * •

_State Variables_ : $\mathcal{X}\subset\mathbb{R}^{n_{x}}$ is the set of state
variables with the _state_ defined by $(q,x)\in\mathcal{Q}\times\mathcal{X}$.

  * •

_Initial conditions_ :
$\emph{init}\subseteq\mathcal{Q}_{0}\times\mathcal{X}_{0}$ is the initial
state set, in which $\mathcal{Q}_{0}\subseteq\mathcal{Q}$ and
$\mathcal{X}_{0}\subseteq\mathcal{X}$.

  * •

_Inputs_ : $\mathcal{U}\triangleq\\{u_{1},u_{2},\ldots,u_{M}\\}$ is the set of
inputs for each topologies.

  * •

_Transitions_ : $\mathcal{E}\subset\mathcal{Q}\times\mathcal{Q}$ is the set of
transitions where a discrete transition from $i$th topology to $j$th topology
is taking place, i.e., $q_{i}\to q_{j}$, $e_{ij}=(q_{i},q_{j})\in\mathcal{E}$.

  * •

_Guard functions_ : $g:\mathcal{E}\to\mathcal{G}$ is the guard function
mapping each transition element $e_{ij}$ to its guard
$g(e_{ij})\in\mathcal{G}$.

  * •

_Guards_ : $\mathcal{G}\subseteq 2^{\mathcal{X}}$ is the guard set which
satisfies $\forall e_{ij}\in\mathcal{E}$, $g(e_{ij})\in\mathcal{G}$. The
_guard_ is satisfied by the state when the hybrid automaton model takes a
transition from the current topology to another given topology, i.e.,
$(q_{k},x_{k})\vDash g(e_{ij})$ if and only if $q_{k}=q_{i}$ and $x_{k}\in
g(e_{ij})$.

  * •

_Invariants_ : $inv:\mathcal{Q}\to 2^{\mathcal{X}}$ is a mapping that assigns
an invariant $inv(q)\subseteq\mathcal{X}$ for each topology
$q\in{\mathcal{Q}}$. An _invariant_ is satisfied by all the states of a hybrid
automata model for a given topology, i.e., $(q,x)\vDash inv(q)$ if and only if
$x\in inv(q)$.

  * •

_Set of Dynamical Description_ : $\mathcal{F}$ is the set of dynamical
description which describes the dynamical process for each given topology
$q\in\mathcal{Q}$. In this paper, neural network
$\Phi_{q}(x(k),u(k))\in\mathcal{F}$ defines the dynamics for each topology
$q\in\mathcal{Q}$ in a given time step $k\in[k_{1},k_{2}]$.

Given sampled data set $\mathcal{W}$ defined by Definition 1, the data-driven
modeling problem for unknown dynamical system (1) is to establish a neural-
network-based hybrid automaton, i.e.,
$\mathcal{H}=\left\langle\mathcal{Q},\mathcal{X},init,\mathcal{U},\mathcal{E},g,\mathcal{G},inv,\mathcal{F}\right\rangle$
with $\mathcal{F}$ represented by a collection of neural networks $\Phi_{q}$,
$q\in\mathcal{Q}$. In addition, the constructed neural-network-based hybrid
automaton model $\mathcal{H}$ is expected to be with low computational
complexity without sacrificing training accuracy. Specifically, the modeling
and verification challenges are described as follows.

###### Problem 1

Given sampled data set $\mathcal{W}$ collected from measurable input and state
traces of dynamical system (1), how does one construct a hybrid automaton
$\mathcal{H}$ embedded with neural network in the form of (2) to accurately
capture the system dynamical behaviors of system (1)?

###### Problem 2

Given neural-network-based hybrid automaton model $\mathcal{H}$ as defined by
Definition 3, how does one perform efficient verification procedures on
$\mathcal{H}$?

###  III-B Data-Driven Modeling Processes

To learn local system behaviors, state space partitioning is required for the
segmentation of training data set $\mathcal{W}$.

###### Definition 4

Given a compact set $\mathcal{X}\subset\mathbb{R}^{n_{x}}$. A finite
collection of sets
$\mathscr{X}\triangleq\\{\mathcal{X}^{(1)},\mathcal{X}^{(2)},\ldots,\mathcal{X}^{(N)}\\}$
is called a partition of $\mathcal{X}$ if (1)
$\mathcal{X}^{(q)}\subseteq\mathcal{X}$, $\forall q=1,\ldots,N$; (2)
$\mathrm{int}(\mathcal{X}^{(q)})\cap\mathrm{int}(\mathcal{X}^{(p)})=\emptyset,\forall
q\neq p$; (3) $\mathcal{X}\subseteq\bigcup_{q=1}^{N}\mathcal{X}^{(i)}$. Each
elements of $\mathcal{X}^{(q)}$ of partition $\mathscr{X}$ is called a cell.

###### Remark 2

In this paper, we use cells defined by hyper-rectangles which are given as
follows: For any bounded state set $\mathcal{X}\subseteq\mathbb{R}^{n_{x}}$,
we define $\mathcal{X}\subseteq\bar{\mathcal{X}}$, where
$\bar{\mathcal{X}}=\\{x\in\mathbb{R}^{n_{x}}\mid\underline{x}\leq
x\leq\bar{x}\\}$, in which $\underline{x}$ and $\bar{x}$ are defined as the
lower and upper bounds of state $x$ in $\mathcal{X}$ as
$\underline{x}=[\mathrm{inf}_{x\in\mathcal{X}}(x_{1}),\ldots,\mathrm{inf}_{x\in\mathcal{X}}(x_{n_{x}})]^{\top}$
and
$\bar{x}=[\mathrm{sup}_{x\in\mathcal{X}}(x_{1}),\ldots,\mathrm{sup}_{x\in\mathcal{X}}(x_{n_{x}})]^{\top}$,
respectively. Then, we are able to partition interval
$\mathcal{I}_{i}=[\mathrm{inf}_{x\in\mathcal{X}}(x_{i}),~{}\mathrm{sup}_{x\in\mathcal{X}}(x_{i})]$,
$i\in\\{1,\ldots,n_{x}\\}$ into $N_{i}$ segments as
$\mathcal{I}_{i,1}=[x_{i,0},x_{i,1}]$, $\mathcal{I}_{i,2}=[x_{i,1},x_{i,2}]$,
$\ldots$, $\mathcal{I}_{i,N_{i}}=[x_{i,N_{i}-1},x_{i,N_{i}}]$, where
$x_{i,0}=\mathrm{inf}_{x\in\mathcal{X}}(x_{i})$,
$x_{i,N_{i}}=\mathrm{sup}_{x\in\mathcal{X}}(x_{i})$ and
$x_{i,n}=x_{i,0}+\frac{m(x_{i,N_{i}}-x_{i,0})}{N_{i}}$,
$m\in\\{0,1,\ldots,N_{i}\\}$. The cells then can be constructed as
$\mathcal{X}_{q}=\mathcal{I}_{1,m_{1}}\times\cdots\times\mathcal{I}_{n_{x},m_{n_{x}}}$,
$q\in\\{1,2,\ldots,\prod\nolimits_{s=1}^{n_{x}}N_{s}\\}$,
$\\{m_{1},\ldots,m_{n_{x}}\\}\in\\{1,\ldots,N_{1}\\}\times\cdots\times\\{1,\ldots,N_{n_{x}}\\}$.
To remove redundant cells, we have to check if the cell has an empty
intersection with $\mathcal{X}$. Cell $\mathcal{X}^{(q)}$ should be removed if
$\mathcal{X}_{q}\cap\mathcal{X}=\emptyset$, and the remaining cells
$\mathcal{X}^{(q)}$ constitute
$\mathscr{X}=\\{\mathcal{X}^{(1)},\mathcal{X}^{(2)},\ldots,\mathcal{X}^{(N)}\\}$.

Based on a collect of sets
$\mathscr{X}=\\{\mathcal{X}^{(1)},\mathcal{X}^{(2)},\ldots,\mathcal{X}^{(N)}\\}$,
we will be able to segment sampled data set $\mathcal{W}$ into a collection of
sets
$\mathscr{W}=\\{\mathcal{W}_{1},\mathcal{W}_{2},\ldots,\mathcal{W}_{N}\\}$
where

$\displaystyle\mathcal{W}_{q}\triangleq\\{x(k)\mid
x(k)\in\mathcal{X}_{q},~{}x(k+1)\in\mathcal{X}^{(q)}\\}$ (4)

in which $x(k)$, $x(k+1)$ are any sampled states in traces $w_{i}$,
$i=1,\ldots,L$ defined by Definition 1. Therefore, $\mathcal{W}_{q}$ contains
all the sampled state traces evolving within cell $\mathcal{X}^{(q)}$.

To train neural network $\Phi_{q}$ to model local system behaviors of
dynamical system (1) evolving in $\mathcal{X}^{(q)}$, the training input-
output pairs need to be abstracted from trace set $\mathcal{W}_{q}$.

###### Definition 5

Given sampled trace $w_{i}$ in set $\mathcal{W}_{q}$, an input-output pair is
defined as

$\displaystyle p_{\ell,i,q}=\\{d_{\ell,i},~{}x_{i}(k_{\ell+1},i)\\}$ (5)

where
$d_{\ell,i}=[x^{\top}_{i}(k_{\ell,i}),~{}u^{\top}_{i}(k_{\ell,i})]^{\top}$ is
given in Definition 1, and $d_{\ell,i}\in\mathcal{W}_{q}\times\mathcal{U}$,
$x_{i}(k_{\ell+1},i)\in\mathcal{X}_{q}$. The training data set out of
$\mathcal{W}_{q}$ includes all the input-output pairs and is given as below:

$\displaystyle\mathcal{P}_{q}=\\{p_{0,1,q},\ldots,p_{\ell,i,q},\ldots,p_{M_{L},L,q}\\}$
(6)

where $\ell=0,1,\ldots,M_{i}$, $i=1,\ldots,L$.

Under Assumption 1, we assume that there exist a sufficient number of training
input-output pairs in each $\mathcal{P}_{q}$ out of segmented data set
$\mathcal{W}_{q}$ to train neural network $\Phi_{q}$ for modeling local system
behaviors in cell $\mathcal{X}^{(q)}$.

With training set $\mathcal{P}_{q}$ for each cell $\mathcal{X}^{(q)}$, the
neural networks can be trained respectively for location $q$. The set of
dynamical description $\mathcal{F}$ which is a collection of neural networks
$\Phi_{q}$, $q=1,\ldots,N$ can be trained, which can be summarized as the
following problem:

$\displaystyle\min\nolimits_{W_{q},b_{q}}\left\|\Phi_{q}(D_{q})-\hat{X}_{q}\right\|,~{}q=1,2,\cdots,N$
(7)

where $W_{q}$ and $b_{q}$ are weight matrices and bias vectors to determine
neural network $\Phi_{q}$, $D_{q}$ are input data matrix and $\hat{X}_{q}$ is
output data matrix from input-output pair $\mathcal{P}_{q}$, respectively.

###### Remark 3

In general, the learning processes can be viewed as an optimization procedure
to find optimized weights and biases that minimize the error between the
output of the trained neural network and output training data, as described in
(7). Instead of using one single neural network for modeling, a collection of
neural networks $\Phi_{q}$, $q=1,\ldots,N$ are used to model local system
behaviors in each cell $\mathcal{X}^{(q)}$, $q=1,\ldots,N$. We have the
following advantages if we use multiple neural networks:

  * •

Compared with modeling the complex global system behavior using a single
large-size neural network $\Phi$, a number of neural networks $\Phi_{q}$,
$q=1,\ldots,N$ with much smaller sizes are able to abstract the local system
behaviors with the same or even higher modeling precision.

  * •

Since $\mathcal{W}_{q}$, $q=1,\ldots,N$ are independent of each other, the
training processes for small-size neural networks $\Phi_{q}$, $q=1,\ldots,N$
can be conducted in a parallel manner which would be more computationally
efficient than training a large scale neural network.

  * •

Even though there are multiple neural networks in the model, only one small-
size neural network is activated at each time step $k$. Therefore, the
computation effort at each step is only determined by the active small-size
neural network. This feature is extremely helpful for executing
computationally expensive tasks based on the model such as reachability-based
safety verification in which we only need to compute the reachable set of a
small-size neural network at each step.

The above benefits of using multiple small-size neural networks enable
efficiently verifiable neural-network-based models of dynamical system (1). A
detailed evaluation will be presented in the evaluation section.

After obtaining the collection of neural networks as the set of dynamical
descriptions in the hybrid automaton, the transition between two neural
networks of dynamical descriptions is defined as follows.

###### Definition 6

Transitions between two topologies are automatically generated by the
dynamical description of hybrid automaton, such that $\forall
p,q=1,2,\ldots,N,\ p\neq q$, $(p,q):x\in\mathcal{X}^{(p)}$,
$\Phi_{p}(x,u)\in\mathcal{X}^{(q)}$.

An illustration of neural-network-based hybrid automaton model ${\mathcal{H}}$
is given in Fig. 1. After modeling the dynamical system (1) in the framework
of hybrid automaton model ${\mathcal{H}}$, we can evaluate ${\mathcal{H}}$
using the following Mean Square Error (MSE) performance out of $L$ test traces
which is defined as

$\displaystyle\mathrm{MSE}=\frac{1}{\sum\limits_{\ell=1}^{L}M_{i}}\sum_{\ell=1}^{L}\left\|\sum_{k=1}^{M_{i}-1}(\Phi_{q}(x_{i}(k),u(k))-x_{i}(k+1))\right\|$

in which $M_{i}$ denotes the length of $i$th trace. In the evaluation example,
this MSE performance will be used for evaluating model precision.

Figure 1: A neural-network-based hybrid automaton model ${\mathcal{H}}$ is
constructed with $6$ topologies valid partitions
${\mathscr{X}}=\\{\mathcal{X}^{(1)},\mathcal{X}^{(2)},\mathcal{X}^{(3)},\mathcal{X}^{(4)},\mathcal{X}^{(5)},\mathcal{X}^{(6)}\\}$
and one state trajectory (in Brown) evolves in $\mathcal{X}$.

###  III-C Reachable Set Analysis

In this subsection, Problem 2, i.e., safety verification, will be addressed in
the framework of reachability. The reachable set of hybrid automaton model
$\mathcal{H}$ is defined as follows.

###### Definition 7

Given a neural-network-based hybrid automaton model $\mathcal{H}$ with initial
set $\mathcal{X}_{0}$ and input set $\mathcal{U}$, the reachable set at a time
instant $k$ is $\mathcal{X}_{k}\triangleq\\{x(k)\mid
x(k)~{}\mathrm{satisfies}~{}\mathcal{H}~{}\mathrm{and}~{}x(0)\in\mathcal{X}_{0}\\}$
and the reachable set over time interval $[0,k_{f}]$ is defined by
$\mathcal{X}_{[0,k_{f}]}=\bigcup\nolimits_{s=0}^{k_{f}}\mathcal{X}_{s}$.

Reachable set analysis for a neural-network-based hybrid automaton can be
referred for safety verification in [13]. Based on Definition 7,
$\mathcal{X}_{k}$ may intersect multiple elements from $\mathscr{X}$, which
means there will be a split computation of reachable set for each
intersection. This process is called _Split_ and is defined by

###### Definition 8

For a reachable set $\mathcal{X}_{k}$ of $\mathcal{H}$ intersects with $l$
elements of $\mathscr{X}$, given a subspace $\mathcal{V}_{i,k}$ from
$\mathcal{X}_{k}$ in which
$\mathcal{V}_{i,k}:\mathcal{V}_{i,k}=(\mathcal{X}_{k}\cap\mathcal{X}^{(m)});~{}\cup_{i=1}^{l}\mathcal{V}_{i,k}=\mathcal{X}_{k},~{}\forall
i=1,\cdots,l,~{}\exists m=1,\cdots,N$, the process of splitting analysis of
the output space $\mathcal{V}_{i,k+1}$ is given by

$\displaystyle\mathcal{V}_{i,k+1}\triangleq\\{\eta_{i,k+1}\mid\eta_{i,k+1}=\Phi_{m}({\eta_{i,k}}),~{}\eta_{i,k}\in\mathcal{V}_{i,k}\\}$
(8)

where the process of obtaining $\mathcal{V}_{i,k+1},~{}\forall i=1,2,\cdots,N$
is called _Split_.

After Split, the _Combine_ process is needed to obtain a complete reachable
set for the next step.

###### Definition 9

For $\mathcal{V}_{i,k+1},~{}i=1,\cdots,l$ the output reachable set
$\mathcal{X}_{k+1}$ for a neural-network-based hybrid automaton model
$\mathcal{H}$ at time step $k+1$ is given by
$\mathcal{X}_{k+1}\triangleq\bigcup_{i=1}^{l}\mathcal{V}_{i,k}$ by which the
_Combine_ derives the reachable set at $k+1$ time instance.

With the _Split and_ _Combine_ defined above, the reachable set of
$\mathcal{H}$ can be paralleling analyzed at time instance $k$ if
$\mathcal{X}_{k}$ intersects with multiple elements from $\mathscr{X}$. The
_Split_ and _Combine_ is given in Algorithm 1.

Figure 2: Sketch map for _Split_ while $\mathcal{X}_{k}$ intersects with $l=4$
valid partitions

Input : Reachable set $\mathcal{X}_{k}$; Neural-network-based hybrid automaton
model $\mathcal{H}$.

Output : Output Reachable Set $\mathcal{X}_{k+1}$.

/* Initialization */

1 $i\leftarrow 1$

/* _Split_ */

2 while  _$i\leq N$_ do

3

4 if  _$\mathcal{X}^{(i)}\cap\mathcal{X}_{k}\neq\emptyset$_ then

5 $\mathcal{V}_{i,k}\leftarrow\mathcal{X}^{(i)}\cap\mathcal{X}_{k}$

Computing $\mathcal{V}_{i,k+1}$ ;

// Under Definition 8

6

7 else

8 $\mathcal{V}_{i,k}\leftarrow\emptyset$

9 end if

10 $i\leftarrow i+1$

11 end while

/* _Combine_ */

Computing $\mathcal{X}_{k+1}$ ;

// Under Definition 9

12

return _$\mathcal{X}_{k+1}$_

Algorithm 1 Pseudo Code for _Split and_ _Combine_

###### Remark 4

Note that if the output reachable set $\mathcal{X}_{k}$ intersects with
multiple valid partitions, the process of Split & Combine may compute the
output reachable sets for more times than modeling with a single neural
network and the conservatism may increase because of Combine. However,
according to [8, 13, 9], the computational cost for the set-valued analysis of
the neural network is mainly affected by the scale of the neural network
model, e.g., the layers and neurons of the neural network model. In our case,
due to parallel training of shallow neural network models, the neural-network-
based hybrid automaton model may have less computational cost compared with
traditional methods.

##  IV Evaluation

In this section, a numerical example of the limit cycle borrowed from [16] is
used for evaluation in the form of

$\displaystyle r(k+1)$ $\displaystyle=(1+\tau)r(k)-\tau r^{3}(k)+\tau u(k)$
$\displaystyle\theta(k+1)$ $\displaystyle=\theta(k)+\tau\omega$ (9)
$\displaystyle u(k)$ $\displaystyle=\mu+\delta\zeta(k)$

where $\omega=2\pi/3$ and $\tau=0.1$ are the angular velocity and time step
width, respectively and the uniform random number $\zeta(k)\sim U(-1,1)$.
Namely, the input $u(k)\sim U(\mu-\delta,\mu+\delta)$ ($\mu=0.2$ and
$\delta=1.5$) in which $U$ denotes uniform distribution.

Figure 3: 50 trajectories of the limit cycle with random initial condition
$r(0)\in[-4,4]$, $\theta(0)\in[-\pi,\pi]$ each of which contains 150 samples
and the input $u\sim U(-1.3,1.7)$.

(a) Single neural network with 200 hidden neurons (b) Hybrid automaton
$\mathcal{H}$ with 12 neural networks (20 Neurons)

Figure 4: 50 test trajectories of single neural network model (a) and neural-
network-based hybrid automaton model (b) with 12 neural networks.

Given dynamical system (9), 50 training traces
$\emph{w}=\\{w_{1},w_{2},\cdots,w_{50}\\}$ with $h_{i}=50,~{}\forall
i=1,2,\cdots,50$ are generated with random initial output where
$\theta(0)\in[-\pi,\pi]$, $r\in[-4,4]$. To derive the neural-network-based
hybrid automaton ${\mathcal{H}}$ which aims to model the dynamics of system
(9). Firstly, the state region $\mathcal{X}$ defines $x_{1}\in[-4,4]$ and
$x_{2}\in[-3,3]$. Based on $\mathcal{X}$, valid partitions representing the
_Topologies_ of ${\mathcal{H}}$ are obtained by desired segments with
$M_{1}=4$, $M_{2}=3$ for each dimension, totaling 12 topologies. The
_Transitions_ between topologies are obtained by the relationship between
samples and regions.

TABLE I: Training Time and MSE Method | MSE | Training Time
---|---|---
Single Neural Network Model $\Phi$ | $0.0817$ |  $4.35$ s
Hybrid Automaton Model $\mathcal{H}$ | $0.0346$ |  $0.32$ s

Then, by training a set of neural networks with each $\Phi_{i}$ containing
$20$ hidden neurons for each $q_{i}$, the hybrid automaton model
${\mathcal{H}}$ is obtained. Moreover, for the sake of comparison, a neural
network model $\Phi$ with $200$ hidden neurons with similar MSE is trained as
well. Fig. 4 shows that both single neural network and hybrid automaton models
can capture the system’s behaviors well. However, in Table I, it can be
observed that hybrid automaton model ${\mathcal{H}}$ is with lower MSE which
implies higher modeling precision. In addition, the training time can be
significantly reduced by training for 12 neural networks parallelly.

Set-valued analysis using NVV in [9] and _Split and_ _Combine_ compared with a
single neural-network-based dynamical model is given in Fig. 5, and hybrid
automaton model can produce tighter reachable set other than single neural
network does. Moreover, the reachable set computation time has been
significantly reduced compared with a single neural network model as shown in
Fig. 6.

Figure 5: $200$-time-step reachable set computation of $\mathcal{H}$ (red) and
a single neural network $\Phi$ (blue) with initial output
$x_{1,0}\in[-3.02,-3],x_{2,0}\in[-2.603,-2.5]$ and input $u(k)\in[-1.3,1.7]$
Figure 6: Time for reachable set computation, with a single neural network
with 200 hidden neurons (blue), and hybrid automaton $\mathcal{H}$ with 12
neural networks (red).

In summary, the evaluation results show that the developed neural-network-
based hybrid automaton model can enable computationally efficient training and
reachability analysis processes with better modeling accuracy and reachability
analysis results for complex dynamical systems.

##  V Conclusions

A data-driven neural-network-based hybrid automaton is developed to model
complex dynamical systems. First, sampled data is generated by the dynamical
system given random initial conditions and input series. Then, the region of
output is divided into valid partitions, based on which the topologies and
guards of the proposed model are obtained. Neural networks are trained as the
dynamical description for their corresponding topologies. The set-valued
reachability of the proposed model is analyzed by the reachable set estimation
method and _Split and_ _Combine_. Modeling a numerical example of the limit
cycle with a neural-network-based hybrid automaton is given. Compared with the
traditional model method with one single neural network, the computational
cost can be significantly reduced for computationally expensive tasks such as
reachable set computation.

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