Diffusion Maps as Invariant Functions of Dynamical Systems
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Transcript of Diffusion Maps as Invariant Functions of Dynamical Systems
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7/28/2019 Diffusion Maps as Invariant Functions of Dynamical Systems
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Mezi Research Group
SIAM Conference on Applications of Dynamical Systems
Snowbird, UT, May 2013
Marko Budii
Igor Mezi
DIFFUSION MAPS AS
INVARIANT FUNCTIONS OF
DYNAMICAL SYSTEMS
Funding:
ONR MURI
Ocean 3D+1
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Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013 2
Invariant functions define barriers to dynamical transport
and simplify dynamics.
Classical mechanics:
Integrals of motion foliate the state space.
We can reduce system to each of the leaves of the
foliation and reduce order of the system.
Fluid dynamics:
Boundaries of (sub-)level sets of
invariant functions arebarriers to material transport.
Invariant functions: functions on the state space whose values are constant along trajectories.
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xp = u(xp), xp(0) = p
(p, t) 7! xp(t)
f(p) := limT!1
1
T
ZT0
f(xp())d
: !
p! g(xp(t)) p! g p
p! f(xp(t)) p! f(p)
f(xp(t)) f(p)
Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013
Every invariant function is an
average of a scalar function along trajectories.
3
xp(t)p
PTrajectory:
Time-average:
Vector field Initial state
Time averages are invariant functions:
Different initial functions may not yield independent
time-averages.
Level sets of time-averages are invariant sets.
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R
Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013
Ergodic Quotient is a static representation of
dynamical invariants in the space of averaged functions.
4
p 7!
264f1(p)
f2(p)...
375
Functions selected from a basis
xp = u(t, xp)
Entire trajectories are represented
by single points.
Axes are averagedfunctions.
f2(p)
f1(p)
f(p) := limT!1
1
T
ZT0
f(xp())d
Ergodic Quotient (EQ)
maps initial conditions to
infinite vectors
of time averages.
Ergodic Quotient
Any function with the domain in EQ
is again an invariant function.
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Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013
Sobolev space topology on Ergodic Quotient
captures families of orbits as continuous segments.
5
State space
portrait
Reeb Graphs
(ideal EQ structure)
Fixed point onseparatrices preventsegments fromconnecting.
d(p1, p2)2 =
X
k2Zd
fk(p1) fk(p2)2
(1 + |k|2)s Acts as a low-pass filter:
de-emphasizes
small scale differences.
Wavevectors
Ergodic Quotientf2(p)
f1(p)
Desired topology:
Time-averages of Fourier harmonics are
spatial Fourier coefficients of
averaging distributions, supported on orbits.
P
fk(x) = ex
Required choice of observables:
Sobolev space norm does the trick:
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Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013
Diffusion coordinates reduce ambient dimension,
while preserving intrinsic geometry of the EQ.
6
Ergodic Quotient
f2(p)
f1(p)
The ambient dimension is the number of
functions averaged (axes).
It is ideally large, to achieve
a high resolution.
It is known a priori.
The intrinsic dimension depends on the
complexity of dynamics, but can be
very small, even zero (ergodicity).
It is unknown a priori.
To disentangle the wire (EQ),sort the points by time it takes them to heat up.
Heat sources are placed onan entangled wire (EQ).Diffusion Maps:
[Coifman, Lafon,
ACHA, 2006]
Diffusion coordinates
pulled back onto the state space
are invariant functions,
sensitive to similarity in dynamics.
p
!
264
f1(p)
f2(p)...
375!
264
1((p))2((p))
..
.
375
Averaging DiffusionMaps
Practical matters:Bubacarr Bah,MSc Thesis,Oxford, 2008.
http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/ -
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Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013
Coloring the state space by values of dominant
diffusion coordinates reveals large scale features.
7
21 3
Diffusion pseudocolors identify invariant sets
Different colors indicate there is no
material transport between regions.
Coordinates of higher order
distinguish between finer features.
State space portraitEQ in Diffusion Coordinates Coloring of the state space
1
2
Averaging
+Diffusion Maps
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R
z
0
2
0.2
0.3
0.1
0.1
Cluster 3
Cluster 2
Cluster 1
Cluster 0
2
1
1 0.5 0 0.5 11
0.8
0.6
0.4
0.2
0
0.2
0.4
0.
6
0.8
1
Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013
Clustering in diffusion coord. identifies large-scale invariant sets.
8
0.00
0.50
1.00
0
2
4
Cluster
x
z
y
3
1
0.00
0.50
1.00 0.00
0.50
1.00
6
5
Index
6
Index 4 Index 20.15
0.1
0.05
00.
05
0.1
0.08
0.04
0
0.
04
0.080.03
0.01
0.01
0.03
0.05
Vortices identified as clusters ofpoints in diff. coord. space.
z
R
Secondary vortex appears asa bifurcation in the primary segment
Steady ABC flow:
Time-periodic Hills vortex ring:
Time-periodic perturbation splits
the primary vertex core.
State space
State space
Ergodic Quotient
Ergodic Quotient
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Future: method resolves structures in finite-time, unsteady flows.
9by Drew Poje (CUNY)
HYCOM simulation of 2D Gulf Stream (along surface of equal potential density of water)
Averaging Time = 20 days
Averaging Time = 10 days
1 2 3
1 2 3
12
3
1Color:
12
3
1Color:
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To achieve automatic analysis,
there are several issues that need resolving:
10
Tuning the diffusion bandwidth.
Incorrect bandwidth can artificially
connect or disconnect regions.
Tuning is heuristic: Lafon (min. distance),
Lee (neighborhood size), Singer (linear sensitivity).
Detection of intrinsic dimensionand independent coordinates.
Cluster 3
Cluster 2
Cluster 1
Cluster 0
2
1
1 0.5 0 0.5 11
0.8
0.6
0.4
0.
2
0
0.2
0.4
0.6
0.8
1
Due to highly
non-convex structures,
K-means clustering
is not a good choice.
Alternative segmenting
would be preferable.
Diffusion eigenvalues depend on the tuning
Where is the
spectral gap?
k = 5
k = 4
k = 3
k = 2
0.3 0.35 0.4 0.45 0.51
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
k
Dependence of diffusion coordinates
Post-processing:
K-means, proximity graphs, etc.
What is
relevant for
dynamical
systems?
Ergodic quotient for Hills ring vortex
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Take away points:
Budii and Mezi, Geometry of the ergodic quotient reveals coherent structures in flows,Physica D, 241, (2012).
Budii,Ergodic Quotients in Analysis of Dynamical Systems, PhD Thesis,UC Santa Barbara, (2012).
Budii, Mohr, and Mezi,Applied Koopmanism, Chaos 22, (2012).
Ergodic Quotient (EQ) is a geometric representation
of the space of invariant functions.
Time-averaged functions provide computable axes for EQ.
Diffusion Maps provide efficient axes for EQ,
while preserving intrinsic geometry.
EQ was successfully used to extract invariant sets
in the state space.
Even in finite-time, method recovers significant structures.