Deep learning to diagnose
the neutron star
Kenji Fukushima
The University of Tokyo
Based on work with Yuki Fujimoto, Koichi Murase— Deep Learning and Physics 2018 —June 2, 2018 @ Osaka!1

Disclaimer
I am a “user” of the deep learning…

June 2, 2018 @ Osaka!2

From the point of view of physics users…
Sounds fancy is not enough…

Useful ?
Advantageous?
!3Conventional Physics Approach
MODEL

Input Data

Very Uncertain

Easy Hard
Nonlinear Mapping

Compare…
Exclude?

Output Data
Limited Information

Exp.
TheoryJune 2, 2018 @ Osaka

EXPERIMENT!4

One another to infinity, so what ?

!5“Model Independent” Analysis
Not unique…
What is the“most likely”
one?

Input Data
Easy Hard
Nonlinear Mapping

Output Data
Inverse Problem

Limited Information

Exp.

EXPERIMENT
June 2, 2018 @ Osaka

!6

!7

Neutron Star EoS
RESEARCH LETTER
Figure 1 | Shapiro delay measurement for PSR
J1614-2230. Timing residual—the excess delay
not accounted for by the timing model—as afunction of the pulsar’s orbital phase. a, Fullmagnitude of the Shapiro delay when all othermodel parameters are fixed at their best-fit values.
The solid line shows the functional form of the
Shapiro delay, and the red points are the 1,752
timing measurements in our GBT–GUPPI data set.
The diagrams inset in this panel show top-down
schematics of the binary system at orbital phases of
0.25, 0.5 and 0.75 turns (from left to right). The
neutron star is shown in red, the white dwarf
companion in blue and the emitted radio beam,
Received
7 July;accepted
Septemberpointing towards
Earth,
in yellow.1At
orbital phase 2010.
of 0.25 turns, the Earth–pulsar line of sight passes
to the companion(,240,000M.
km),
1.nearest
Lattimer,J. M. & Prakash,
The physics of neutron stars. Science 304, 536–542
producingthe sharp peak in pulse delay. We found(2004).
for J.
anyof pulse intensity
2.no evidence
Lattimer,
M.kind& Prakash,
M. Neutron star observations: prognosis for equation ofvariations, as from an eclipse, near conjunction.
state
constraints.
Phys.
Rep.
442, 109–165 (2007).
b, Best-fit residuals obtained using an orbital model
3.that Glendenning,
N.
K.
&
Schaffner-Bielich,does not account for general-relativistic effects. J. Kaon condensation and dynamical
nucleons
inofneutronstars.
Phys.
In this case, somethe Shapiro
delay
signalRev.
is Lett. 81, 4564–4567 (1998).
4.absorbed
Lackey,B. D., Nayyar,
M. & Owen,by covariant
non-relativistic
model B. J. Observational constraints on hyperons inparameters.
ThatthesePhys.
residuals
neutronstars.
Rev.deviate
D 73, 024021 (2006).
fromrandom,5.significantly
Schulze,
H.,aPolls,
A.,Gaussian
Ramos,distribution
A. & Vidaña, I. Maximum mass of neutron stars. Phys.
of zeromean
showsthat the (2006).
Shapiro delay must be
Rev.
C 73,
058801included to model the pulse arrival times properly,
6.especially
Kurkela,A., Romatschke, P. & Vuorinen, A. Cold quark matter. Phys. Rev. D 81,
at conjunction. In addition to the red
105021(2010).
GBT–GUPPI points,
the 454 grey points show the
7.previous
Shapiro,I. I. Fourth
general relativity. Phys. Rev. Lett. 13, 789–791 (1964).
‘long-term’
data set.test
Theofdrastic
8.improvement
Jacoby, B.
A., Hotan,
Bailes, M.,
Ord, S. & Kulkarni, S. R. The mass of a millisecondin dataquality A.,
is apparent.
c, Post-fit
residuals
for thefully relativistic
timingmodel
pulsar.
Astrophys.
J. 629,L113–L116(2005).
Shapirohave a roottiming
mean of PSR J0437–4715: an accurate pulsar
9.(including
Verbiest,
J. P.delay),W. etwhichal. Precision2
value
of on the variation of Newton’s gravitationalsquaredresidualaofhigh
1.1 mspulsar
and a reduced
distance,
mass, xanda limit
1.4 with 2,165 degrees of freedom. Error bars, 1s.

a 30
20

Demorest et al. (2010)

10
0
–10

Precise determination LETTER
of RESEARCH
NS mass using Shapiro delay

–20
–30
–40

b 30

Timing residual (μs)

20

GR

2.510

P

0

<∞

AP3

us

Ca

–10

2.0

SQM3
FSU
J1903+0327

Mass (M()

SQM1–40 J1909-3744
1.5
30
20

MS2

(slightly changed in 2016)

J1614-2230

–20
–30

c

1.928(17) Msun (J1614-2230)

PAL1

ENG

y
alit AP4

MS0

MPA1

MS1

GM3

PAL6

GS1

Double neutron stars sysystems10

1.0

Antoniadis et al. (2013)

0

–10
–20

0.5
–30

Rota

2.01(4) Msun (PSRJ0348+0432)

tion

constant. Astrophys. J. 679, 675–680 (2008).
10. Hessels, J. et al. in Binary Radio Pulsars (eds Rasio, F. A. & Stairs, I. H.) 395 (ASP Conf.
Orbital phase (turns)
Ser. 328, Astronomical Society of the Pacific, 2005).
0.0
7
8
9
10
11
12
13
14
15
11.
Crawford,
et al. A survey
of 56
midlatitude EGRET error boxes for radio pulsars.
systemtoberemarkablyedge-on,with anF.inclinationof 89.17u
6 0.02u.
parameters, with MCMC error estimates, are given in Table 1. Owing to
Astrophys.
J.
652,
1499–1507(2006).
This
is
the
most
inclined
pulsarbinarysystemknown
atpresent.
The
Radius(km)
the high significance of this detection, our
MCMC
procedure and a
June
2,
2018
@
Osaka
!8 pulsar PSR
2
12.
Özel,
F.,
Psaltis,D.,
Ransom,S.,
Demorest,
P. & Alford, M. The massive
amplitude
andsharpnessoftheShapirodelay
increase
rapidly
withstandard x fit produce similar uncertainties.
Figure3 the| Neutron
star mass–radius
diagram.
The mass
plot ofshows
non-rotating
chromodynamics,
gamma-ray bursts, andincreasing
binary inclinationJ161422230:
and the overalllinking
scaling quantumof the signal
is
Fromdetected Shapiro
delay, we measure
a companion
27
gravitational
wave
astronomy.
Astrophys.
J.
(in
the
press).
linearly
proportional
to
the
mass
of
the
companion
star.
Thus,
the
: blue, nucleons; pink,mass
versusphysicalradiusfor thatseveraltypical EOSs
impliesthe companion
is a helium–(0.50060.006)M[, which
–40
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Neutron Star EoS
Equation of State Input Data
Pressure : p
Mass density : r

p = p(⇢)

Nonlinear Mapping
Tolman-Oppenheimer-Volkoff (TOV) Eqs(Energy density : " = ⇢c2 )pressure diff

M-R Relation Output Data
NS mass : M
NS radius : R

M = M (⇢max )R = R(⇢max )gravity

Mathematically one-to-one correspondence
June 2, 2018 @ Osaka!9Neutron Star EoS
Lindblom (1992)

Brute-force solution of the inverse problem
Test data put by handPressureThanks to Y. Fujimoto

Solve TOV

Mass

Reconstructed
DensityRadius

The answer exists!
June 2, 2018 @ Osaka!10

No magic box…
Only “solvable” problem can be solved…

!11

Neutron Star EoS (Side Remark)R is fixed by TOV with p(R)=0 (“surface” condition)

dp/dr(r = R) = 0
d2 p/dr2 (r = R) / M 2 /R2Very uncertain“by definition”

Determination of R is very uncertain, and on top of that,
R itself is anyway very uncertain…

People do not care assuming that NS mass > 1.2 Msun
June 2, 2018 @ Osaka!12

Model Independent Analysis
Bayesian Analysis(Bayes’ theorem)

B : M-R Observation
A : EoS Parameters

Normalization

P (A|B)P (B) = P (B|A)P (A)Want to know

Likelihood
prior
Calculable by TOV Model
Model must be assumed.
EoS parametrization must be introduced.
Integration in parameter space must be defined.

If infinite observations, prior dependence should be gone.
June 2, 2018 @ Osaka!13

Model Independent Analysis
Raithel-Ozel-Psaltis (2017)
4

Mock data (SLy + Noises)

Prior
Dep.

Black curve
True EoS
Magenta curve
Guessed EoSGray band
68% credibility

Fig. 2.— (Top) Inferred equation of state and mass-radius curve from a sample of mock data, assuming a uniform prior distributionof pressures. The mock data are drawn from the nucleonic EoS SLy (Douchin & Haensel 2001) and are dithered with Gaussian noisecorresponding to M = 0.1M , R = 0.5 km. The actual curves for SLy are shown in black. The magenta curve represents the most likely
EoS inferred via our Bayesian method. The 68% credibility region is shown in gray. (Middle) Identical to top panel, but with our Gaussian
regularizer included in the inversion. (Bottom) Identical data to the top two panels, but assuming a prior distribution that is uniform
in the logarithm of pressure and including a Gaussian regularizer. Assuming a uniform distribution leads to a preference towards highpressures in the regions where there are few data to constrain the inversion, while assuming that the pressures are distributed uniformly
in the logarithm leads to a preference towards lower pressures. Including the Gaussian regularizer reduces the sensitivity to the choice ofprior.

June 2, 2018 @ Osaka!14

Model Independent Analysis
Bayesian Analysis
Supervised Learning
Several M-R
observation points
with errors

Nonlinear
Mapping

Several parameters
to characterize EoS{Mi , Ri }{Pi } = F ({Mi , Ri }){Pi }~ 15 Points (observations)~ 5 Points

Too precise parametrization of EoS is useless(beyond the uncertainty from observations)
June 2, 2018 @ Osaka!15

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Deep Learning{M , R }

~ 15

{P }

~5

about fitting functions but the multi layer structures areDeep Learning
Our Neural Network Design
Layer index Nodes Activation
1
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30
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N/A
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TABLE I. Our neural network design in this work. In the 1st
Probably
don’t need
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hiddenlayerslayer
30 nodeswecorrespond
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pointsof themass andtheand such
nodes…
thistois5 one
radius.
In themany
5th layer
5 nodesanyway,
correspondparametersof theEoS. example…
workingJune 2, 2018 @ Osaka
M , we always take larger
R discarding thermodynami-!17

Deep Learning
For good learning, the “textbook” choice is important…
Training data (200000 sets in total)
Randomly generate 5 sound velocities → EoS × 2000 sets
Solve TOV to identify the corresponding M-R curve
Randomly pick up 15 observation points × (ns = 100) sets(with

M = 0.1M ,R = 0.5 km)

The machine learns the M-R data have error fluctuations
Validation data (200 sets)
Generate independently of the training data
June 2, 2018 @ Osaka!18

Deep Learning
0.18

Loss function (msle)“Loss Function"= deviation from thetrue answers (msle)

ns = 100
ns = 1

0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

1

10

100

1000

10000

Monotonically decrease
for the training data, but
not necessarily so forthe validation data

EpochsWith fluctuations in the training data, the learning goes quickly

Once the over-fitting occurs,
the model becomes more stupid…
June 2, 2018 @ Osaka!19

For other E
M -R curves a
discussed abo
accuracy, weradius deviat
masses as show
deviations bet
the genuine a
Fujimoto-Fukushima-Murase (2017)tances betwee
FIG. 3. Two examples of the randomly generated EoSs
that is, R(M(dashed lines) and the machine learning outputs (solid lines)
ues in Tab. II
reconstructed from one observation of 15 M -R points [seeindicates that
Fig. 4 for actual (Mi , Ri )].
4 member thatR ⇠ 0.5 km
neutron stars
Mass (M ) 0.6 0.8 1.0 1.2 1.4 1.6 1.8
RMS of the
RMS (km) 0.16 0.12 0.10 0.099 0.11 0.11 0.12
3
within the acc

Deep Learning

Test with the validation data(parameters not optimized to fit the validation data)

Two Typical Examples (not biased choice)100

TABLE II. Root mean square of radius deviations for fixedmasses.

M / M⊙

p / ρc2

Finally, letanalysis using2
10-1
parametrized
For other EoSs in validation data, the corresponding rameter space
1 curves are reconstructed well similarly to examples distribution P
M -R
discussed above. To quantify the overall reconstruction by an observa
10-2
accuracy, we calculated the root mean square (RMS) of The neural ne0
radius10deviations12using 19614validation 16
data for several data points, i
0.1
0.2
0.5
1
masses as shown in Tab. II. We defined the RMS from the fitting param2
3[km]
ρc [GeV/fm ]deviations between not Rtheobservational data points but minimize thethe genuine and reconstructed M -R relations (i.e. disFIG. tances
4. Randomly
sampled15 dataand theM -R
betweenthe solid
and points
the dashedlines
inrelaFig. 4),h`[f ]iwith the reconstructed(rec)EoS (solid lines)(0) and the original
FIG. 3. Two examples of the randomly generated EoSstionsthat
is, R(M ) = R(M ) R (M ). The RMS vallines). The red and blue colors correspond to two(dashed lines) and the machine learning outputs (solid lines)EoS (dashedues
in Tab.
II arearound⇠ Fig.
0.1 kmshown
with thesamecolor in
3. for all masses! This Here, let usreconstructed from one observation of 15 M -R points [seeEoSsindicatesthat our method works surprisingly good; relanguage. In
Fig. 4 for actual (Mi , Ri )].
member that data points have random fluctuations by
the EoS is ass
June 2, 2018 R@⇠Osaka
0.5 km. It should be noticed that, even without!20
observation
deviations
from the genuine
M -R the tribution is oneutronwith
starserror
aroundM = 0.6–0.8M
in our setup,Pr(✓) Pr(D|✓)
relation (which is shown by the dashed lines). Thus, each: randomly generated original EoS: reconstructed EoS and associated M-R

Deep Learning
Overall performance test

4

Mass (M ) 0.6 0.8 1.0 1.2 1.4 1.6 1.8
RMS (km) 0.16 0.12 0.10 0.099 0.11 0.11 0.12
(with
M of
= radius
0.1M , deviations
R = 0.5 km)
TABLE II. Root meansquarefor fixed
masses.

Very promising!

Credibility estimate has not been done for simplicity, but
Forit other
EoSs ininvalidation
can be included
the learning data,process.the corresponding

M -R curves are reconstructed
June 2, 2018 @well
Osaka similarly to examples!21

Usefulness confirmed,
easy implementationbut
advantageous ?
Bayesian or NN,
which to choose?
!22

Here,
let ususingtranslate
BayesiananalysisIninto
the
above
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symbolic
notations.
ouranalysisuracyof language.
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sis
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✓
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us comment
on istheBayesian
metrized
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{cand
},
which
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rameter
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generated
EoSs
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14

Conclusion

Yes, useful!
Maybe, less biased ?
Fig. 10.— The 68% confidence contours in mass and radius for the quiescent neutron star in ! Cen, inferred by Heinke et al. (2014; H14)
and by Guillot & Rutledge (2014; G14) using di↵erent assumptions regarding the interstellar extinction (wabs: Morrison & McCammon
1983; tbabs: Wilms et al. 2000), the presence of a power-law spectral component, and for di↵erent distances to the globular cluster (4.8 kpc
vs. 5.3 kpc).

Developing a toolkit for real data likenot discrete data with error,
but regions of credibilityωError analysis (credibility estimate)in the output side
June 2, 2018 @ Osaka

Fig. 11.— The combined constraints at the 68% confidence level over the neutron star mass and radius obtained from (Left) all neutronstars with thermonuclear bursts (Right) all neutron stars in low-mass X-ray binaries during quiescence.

!24