Andrzej Banaszuk United Technologies Research Center

Andrzej BanaszukUnited Technologies Research Center

Dynamic Network Analysis

for

Approved for Public Release, Distribution Unlimited

Outline•DyNARUM origin

•Dynamics on Graphs

•Beneficial and detrimental interconnection

•and why UTC and DOD care

•Multiscale Complex Dynamics: Barriers and Enablers

•Graph Decomposition

•Learning Models of Coarse Dynamics

•Coarse Temporal Integration

•Efficient Uncertainty Quantification

•Future directions

•Robust Surveillance and Distributed Estimation Networks

•Cyber-Physical Systems


Imposing or breaking symmetry can enforce the desired behavior

flutterrotating stall

aeroacoustics

thermoacoustics

cohe

renc

eincoherence

Coupling strength

Time scale separation

Mistuned => reduced amplitude

Symmetric => high amplitude wave

Pendula analog of engine dynamics

Dynamics on Graphs

Dynamics at node + interconnection strength => emergent behavior


Analysis of symmetry => detrimental and beneficial interactions

Acoustics

0 20 =−+ θθξ papp ttt

-

Heat release

F(p,θ)

Wave equation with skew-symmetric feedback • Positive feedback => detrimental coupling• Creates lightly and heavily damped spinning waves

+ a2(θ)pθθ

Wave Speed Mistuning

Exploit Interconnections

Wave mistuning creates beneficial couplingUnder-damped waves

Heavily damped wavesUtilized as dampers

Re

Im

*** **

*

Use coarse variables (modes +1, -1)=> simple model => analysis tractable

thermoacousticsflutter


Engine problem eliminated by wave speed mistuning

% Wave-Speed Mistuning

Ampl

itude

AmplitudeReduction

Model Pre-Test PredictionsData (Symmetric Design)Data (Mistuned Design)

Design Beneficial Interconnections

•Inspiration: 2004 study of symmetry of DNA molecules (I. Mezic)•From initial concept to engine test in 18 months•Passive solution: no extra hardware necessary

Wave mistuning creates beneficial couplingUnder-damped waves

Heavily damped wavesUtilized as dampers

thermoacousticsflutter


Pratt: thermoacoustics

Multi-scale complex dynamics

HS: electric power systemsCarrier: variable speed

Sikorsky: battlefield simulations

Building systems: air, energy, threat, people dynamics

enabled by networks

Sensitivity Uncertainty

Complexity

Acoustics Heat Release Turbulence

DisturbanceDelays)()),(),(( 2 tntxtxFxaxtt +−=∆− µτ

Uncertain parametersNonlinearityLarge multi-scale

Nontrivial dynamics

fastest

slowest

ττ

610 710310210

Ratio of time scales

410 510

# dynamic of components

210

610

410

510

310

affects United Technologies


Uncertain parametersLarge interconnected network Multiple time scalesBarriers

samplestimestepslinksnodes nnnn •••~Computation cost

Multi-scale dynamical systems

Tractable description 1 2

43

simulation time > real time

Enablers Coarse spatial variables

Network decomposition

Coarse temporal integration

Coarse stochastic variables

Exploit interconnections

•Simulations•Analysis •Design

Learn Model of Coarse Dynamics

barriers and enablers


Uncertainty Propagation Intractable With Current

Methods

Nontrivial Dynamics

Large Scale

Multiscale

Uncertainty

Barriers

Tractable Description

Enablers:

DARPA DSO invests in Mathematics

Program Metrics :

Approach :

1. Decompose networks into components using Spectral Graph Theory.

2. Propagate uncertainty through components using Operator Theory and Geometric Dynamics

3. Iteratively aggregate component uncertainty

Distribution of target detection time

Baseline AlgorithmOptimal/Robust Algorithm

Uncertainty: vehicles, sensors, communications models10,000 Agents Surveillance Network

Objectives:

1. Develop analysis and design tools for Uncertainty Management

2. Demonstrate tools in a surveillance problem with > 10,000 agents

DYNARUM methods and tools

Uncertainty Propagation

ToolsNetwork Design

Tools

Graph Theory

Geometric Dynamics

Opera

tor

Theo

ry

Netw

ork

Deco

mpo

sitio

n

Model Reduction Efficient

Measure

Propagation

Optimal Control

Variational

Integrators Asyn

chro

nous

Com

puta

tions

10x 100x100x + validation on surveillance problem

Year 3

-10x 100x + validation on interacting particles challenge problem

Year 2

--10x + validation on interacting particles challenge problem

Year 1

Reduce standard deviation

Reduce average

Reduce computation time

Multi-scale complex dynamics affects DOD


DyNARUM Team

Raphy Coifman

Yoel Shkolnisky

Amit Singer

Jerry Marsden

Konda Reddy

Phillip Du Toit

Katalin Grubits

Sujit Nair

Sigrid Leyendecker

Igor Mezic

Simeon Grivopoulos

Bryan Eisenhower

Marko Budisic

Gunjan Thakur

Lan Yueheng

University UTRC

Basic Research:Long range technology development

Applied Research:Integrate, mature, and transfer

technology to market opportunities

Andrzej Banaszuk

Marco Arienti

Sorin Costiner

Razvan Florea

Bob LaBarre

George Mathew

Troy Smith

Jose Miguel Pasini

Sergey Shishkin

Sergei Burlatsky

Emrah Biyik (intern, RPI)

Sanjay LallMatt WestSunHwan LeeTsu-Chien LiangLaurent Lessard

Eric Johnson

Mark Lutian

Fred Wagner

Vladimir Fonoberov

Caroline Cardonne

Yannis Kevrekidis (Princeton)

Gleb Oshanin (Univ. P&MCurie)

Sean Meyn (UIUC)

Michael Dellnitz (Paderborn)


DyNARUM Approach

1. Decompose networks into components using Spectral Graph Theory.

Weak connections revealed, strongly connected components identified

Component measure propagation operators reduced, truncation error assessed

2. Propagate uncertainty through components using Operator Theory and Geometric Dynamics

Spectral Graph Theory methods: analysis of spectrum of Markov matrix associated with network reveals interconnection structure, strong and weak connections

Coarse Variables and Symmetry: exploit symmetry todecouple slow and fast scales, enable network decomposition, model reduction, asynchronous parallel computations

Variational Integrators and Asynchronous Computations: preserve energy, probability measures, exploit time scale separation for efficient computing

Geometric Dynamics and Measure Propagation: Efficient choice of basis for measure propagation for networks with nontrivial component dynamics

3. Iteratively aggregate component uncertainty

Probability of vehicles detected in time T

Parallel asynchronous computations, iterations across weak connections

Enabling technologies:Geometric Dynamics and Optimal Control: design beneficial dynamics for robust surveillance with fast detection

Approach:Challenge problem: 10,000 Agents Surveillance Network with uncertain vehicle and sensor models,

Probability distribution of input parameters

Operator Theory and Model Reduction: use eigenfunctionsof Markov matrices associated with components for model reduction while keeping track of truncation error, prove convergence of iterative methods for measure propagation

Analysis

Tools

Design

Tools





•And why UTC and DOD care










into weakly interacting sub-networksDecompose Networks

)(XFX =& XF

∂∂

=> speedup utilizing parallel computations

weak connections

))(,..,1)(,..,)(()(1)( 1k

mxkixkxiftk

ixdtd +=+

iteration onefor timesimulation mixf

m

i ≠∂∂ ,sup

iterations # N)( as decaysError −+KTKTe

i

i

xf

∂∂sup

=> iterate for accuracy


Enabler: Spectral Graph Theory1.Local dynamics + interconnection structure => Markov chain on graph

nn Mµµ =+1 Mxx =λ

Weak connection

Example: analysis of transport of species between coupled rooms

0

11 =λ

2λ

Eigenvalues

2. Small gaps between eigenvaluesreveal weakly connected sub-networks

• Indicates transport bottleneck

+-

2nd Eigenfunction2x

3. Eigenfunctions sign change determines network decomposition

•2nd eigenfunction is the slowest decaying pattern. Sign change is location of bottleneck.

Decomposition enables uncertainty propagation using parallel computations

Decompose Networks


Fine-grained agent-based modelcomputations ~ N2

Graph decomposition

Aggregation to zones

Enable model-based estimation of occupancy

14

Identify strong and weak coupling to zone systemand track movement between zones

Application: analysis of people dynamics in buildingsDecompose Networks


Learn Model of Coarse DynamicsChoose local observablemeasure local symmetry => average => phase

Average observable

Simulation with 2000 atoms

nn M µ=µ + 1

Learn its probability transition matrix

−0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real(λi(T)

Imag

(λi(T

)

Eigenvalues of transition matrix; Temperature = 3.0625K

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Invariant Measure

Z

µ(Z)

Eigenvector = asymptotic probability distribution

observable

Eigenvalues of M

rapidly decaying patterns

slowly decaying patterns

Real λi

fluidsolid

Imλ i

Learning of second eigenvalue of M

Learning completed = > 10x speedup of phase determination

to reveal important time scales in the system

helium

carbon

Problem: determine phase


Equation-Free Coarse Temporal Integrationcontinuity of time scales reduces acceleration

order parameter Ψ captures order-disorder transition

a1

a2

Procedure

coarse stepper

restriction

atomistic simulator

lifting

Density (commensurate monolayers)

Tem

pera

ture

(K)

0 0.2 0.4 0.6 0.8 1

70

80

90

100

110

120 10,000 atoms (baseline)Equation Free

calculated phase diagram

T0 1 2 3 4 5 6

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

T=50 KT=65 KT=90 KT=100 K

equation-freeorder parameter evolution

at given density

0 500 1000 1500 2000 2500 3000 3500 4000

0.855

0.86

0.865

0.87

0.875

0.88

coarse step

10 atomistic runs

average of atomistic runs

Ψ

a1a2

a1a2

50 100 150 200 250 300 350 400 450 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Midterm and final exams

2006 2007 2008 2009

10x reduction of computation time

Quantify quality, stability, robustness

Analysis Tools

Design Tools

100x reduction of computation time over Monte-Carlo

10x better performance 100x better performance

10x better robustness

Phase diagram (10K particles)

Self-assembly (~100 particles)

Distribution of performance, stability

measure

Baseline Algorithm Improved/Robust Algorithm

Distribution of target detection time

Baseline AlgorithmOptimal/Robust Algorithm

Surveillance network (10K agents)

Coarse VariablesModel ReductionNetwork Decomposition

Coarse VariablesModel ReductionNetwork Decomposition

Stochastic Galerkin =>1000x faster than MC (Brown)Equation-Free UQ => 3x faster than MC (Princeton)

Optimal Control

Geometric Dynamics Chaotic search => exponentially fast coverage

(and why we think we will pass)


Speed-up wrt baseline

%rm

sdi

stan

cefro

mba

selin

e

1001011021030

1

2

3

4

5

6

7

8

xx x

Milestone 1B: learning dynamics demonstrated on Phase Diagram problem

Fluid

Commensurate + Fluid

Solid

Baseline:Phase diagrams from 10,000 Helium atoms on C

2000 simulations, total 100 hours (12 CPUs)

Phase diagrams obtained with RUM tools

Baseline

Metastability from Markov model (UTRC)

reduced-size system (Aimdyn)

Bayesian Markov model extraction

(Stanford)


Milestone 1A: new potential design method demonstrated in Self-Assembly problem

232 )(40

1100

12 4.05)( arra eeara

rrV −−− −+−=

Particle interaction potentials

Baseline potential

Goal: Design interaction potential so that 100 particles self-assemble to a honeycomb lattice

Baseline (Princeton 2005):• Final lattice still has defects• Computationally expensive (Simulated Annealing gets stuck in local minima of a defect measure)

DyNARUM:

•Repulsive potentials avoids defects produced by local minima

•Trend-based optimization 100x faster than Simulated Annealing

Defect measure

# Molecular Dynamics runs

Defect measure





•and why UTC and DOD care










Future directions: distributed estimation Distributed Estimators of Multiscale Phenomena

Multiscale

Single scale Multiscale 10x faster

Dynamics of consensus algorithm

Estimator of physical dynamicsEstimate contaminant

concentration in buildings


Input parameter probability distribution

Output parameter probability distribution

Challenge: Robust Cyber-Physical Systems

Power grid control City evacuation support Surveillance networks

Reduced uncertainty with robust design

Building emergency response support

communications

physics

computations

No time scale separation

Uncertainty

Multiscale dynamics

Codesign of Physical and IT Dynamics necessary


Summary•DyNARUM origin



•And why UTC and DOD care









Andrzej Banaszuk United Technologies Research Center

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