Lagrangian Coherence, Braids, and Machine Learning

7/23/2019 Lagrangian Coherence, Braids, and Machine Learning

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Department of Mathematics

68th Annual Meeting of the

APS Division of Fluid DynamicsBoston, Nov 2015

Marko Budi!i"

Jean-Luc Thiffeault

LAGRANGIANCOHERENCE,BRAIDSANDMACHINELEARNING

Supported by NSF grants

DMS-0806821 (JLT) and

CMMI-1233935 (MB, JLT)
http://www.math.wisc.edu/


2/14

M. Budisic: Lagrangian Coherence, Braids and Machine LearningNov 22, 2015 2

Similarity: entanglement of loops bythe braid of trajectories.

Goal is to aggregatetopologically-similarLagrangian trajectories into coherent structures.

Aggregation:a machine-learning problem.


3/14

M. Budisic: Lagrangian Coherence, Braids and Machine LearningNov 22, 2015

Many faces of transport-coherent structures

3

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Lagrangian Coherent

Structures

Haller et al.doi:10.1146/annurev-fluid-010313-141322

Almost-Coherent Sets

Froyland et al.doi:10.1016/j.physd.2009.03.002

Mesochronic Analysis

Mezi"; MB et al.arxiv.org:1506.05333

Ergodic Quotient

MB; Mezi"doi:10.1016/j.physd.2012.04.006

Finite-Time Curvature

Bollt et al.doi:10.1137/130940633

Trajectories/initial conditions

aggregatedby a common property.

Trajectories

separatedby boundaries.


4/14


Braids: good for sparsely sampled trajectory sets

4

60 40 20

40

50

60

70

Latitude

Longitude

Spaghetti plot of planar

trajectories

050

4050

607060

40

20

TimeLongitude

Latitude

Trajectories in space-time,

physical braid

0 50 10040

50

60

70

Time

Latitude

Project onto a plane

and monitor exchanges

Braid encodes

trajectory crossings as

symbols (generators).

k

1

k

k

k+ 1

k

k + 1

2 21

1 2 1

1

Crossin

Topology retained,

geometry discarded.[Thiffeault, 2010]


5/14


Where else do braids show up?

5

Braiding of solar coronal loops:

Random braids, tightness leads tocoronal reconnection

Braiding of non-abelian anyons:Topological Quantum Computing

[Berger, 2009]doi:10.1088/0004-637X/705/1/347

[Caussin, Bartolo, 2015]doi:10.1103/PhysRevLett.114.258101

Braiding of flocks:

Use braiding to detect breaking of

symmetry, associated with an

external stimulus

Microsoft Q Station, [Hormozi, et al. 2007]doi:10.1103/PhysRevB.75.165310
http://www.apple.com/http://dx.doi.org/10.1103/PhysRevLett.114.258101http://dx.doi.org/doi:10.1088/0004-637X/705/1/347


6/14


Braids act on loops

6

2 21

1 2

1

1

Loops are tightened (rubber bands) and cannot be broken.

Reduced modelFull model

BraidsODEs, PDEs,Dynamics

Loops pulled tightDetailed curvesMaterial

Maps on integer vectorsFront/interface trackingAdvection

1

1

2

|`| = 2

|11 `| = 4

|21

1 `| = 6

Punctures


7/14M. Budisic: Lagrangian Coherence, Braids and Machine LearningNov 22, 2015

Detecting coherence (Allshouse, Thiffeault, 2012)

7

Separation strategy:

Direct search for

non-growing loops.

Impractical: Numberof

subsets to test grows

exponentiallywith numberof punctures (the more

data, the worse off we are).

Trajectories moveeither in left half or in right half,

e.g., two side-by-side vortices.

Imagine two non-interacting structures

Time

Loops surrounding either of the halves

do not grow (topological analog to LCS definitions).


8/14M. Budisic: Lagrangian Coherence, Braids and Machine LearningNov 22, 2015 8

Pair-loop

Before braid action

After braid action

Entangled punctures:

caused the pair-loop to grow

Aggregation strategy:

Dynamically-similar trajectories

will deform the same loops.

Test all pair-loops (quadratic in N) and

aggregate punctures that entangle same

pair-loops.

Alternative method from (A., T., 2012)

Instead of binary entangled /unentangled, to each puncture

assign entanglement depth.

New refinement:

Already-seen pair-loop:

Some other pair-loop:



Pair-loop entanglement is a good measure of similarity

9

Pair-loop

anchors

Pair-loop

anchors

Pair-loop

anchors

3 different pair-loops:

Color is the depth of entanglement

There are many more pair-loops:

how do we simultaneously use

information from all of them?{Nearby trajectories commonly

get buried to similar depths,

regardless of the chosen pair-loop.Loop anchored within avortex is entangled only by

vortex trajectories.

Loops between vortices

entangled by all trajectories.

Simple steady system to test the idea.

Think of each pair-loop as a different

measurement/dimension

in the description of a trajectory



Detecting regularities in data (machine learning)

10

Element (i,j)is the

entanglement of

puncture iby the pair-loopj.

Stack entanglement

vectors into rows

Sort columnsso that

neighbors look alike.

Group trajectorieswith

similar entanglements.

d(k, j) =kckcjk2

Look alike?

Column-wise distance

All-to-all distances.

Diagonal blocks are

clusters of tightly-

packed trajectories.

Loops from

previous

slide



Extract Large-Scale Structure using Diffusion Maps

11

Each vertex (trajectory)

is a samplefrom an unknown

underlying structure. {

Distance between

vertices is the differencein entanglement vectors.

Diffusion Maps

constructs a random

walk on the graph.

Paths between

large features

are less likely

than paths inside

them.

Close-to-invariant

densities are

concentrated on the

large-scale features.

Using density

values as

coordinates for

points maps

large features to

clusters.

Distance matrix defines a graph:

[Coifman, Lafon, et al. cca 2005]



Diffusion Map of Entanglement Coherence

12

Sorting by D.C.1: Neighbors are similar

Separation of structures

No blobs at ends:

Structures are 1-parameter

families.

Branches indeed correspond to vortices:Block diagonals correspond to clusters


13/14

M. Budisic: Lagrangian Coherence, Braids and Machine LearningNov 22, 2015 13

Apply to time varying flows

Multi-scale flows may require

adapting row-differencesExperimental flows have their

own challenges

(missing, incomplete data)

Structural analysis of Diffusion Map

embedding (persistent homology)

How many trajectories do we need?

Is there a better way to choose test

loops (instead of pair-loops)?

Hackborn Rotor-Oscillator (w/ Margaux Filippi, Tom Peacock)

Future work:


14/14


References

14

Previous work on coherence in braids:

Allshouse, Michael, and Jean-Luc Thiffeault. 2012. Detecting Coherent Structures Using

Braids. Physica D. Nonlinear Phenomena, 95105.

doi:10.1016/j.physd.2011.10.002.

Previous work using Diffusion Maps to detect coherence: Budi!i", Marko, and Igor Mezi". 2012. Geometry of the Ergodic Quotient Reveals Coherent

Structures in Flows. Physica D. Nonlinear Phenomena 241 (15): 125569.

http://dx.doi.org/10.1016/j.physd.2012.04.006.

Diffusion Maps:

Coifman, Ronald, and Stphane Lafon. 2006. Diffusion Maps. Applied and Computational

Harmonic Analysis 21 (January): 530.

doi:10.1016/j.acha.2006.04.006.

Lee, Ann B, and Larry Wasserman. 2010. Spectral Connectivity Analysis. Journal of the

American Statistical Association 105 (491): 124155.

doi:10.1198/jasa.2010.tm09754.

Braids of trajectories:

Thiffeault, Jean-Luc. 2010. Braids of Entangled Particle Trajectories. Chaos: An

Interdisciplinary Journal of Nonlinear Science 20 (1): 017516017514.

doi:10.1063/1.3262494.
http://dx.doi.org/10.1063/1.3262494http://dx.doi.org/10.1198/jasa.2010.tm09754http://dx.doi.org/10.1016/j.acha.2006.04.006http://dx.doi.org/10.1016/j.physd.2011.10.002

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