andrzej banaszuk united technologies research center
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Andrzej BanaszukUnited Technologies Research Center
Dynamic Network Analysis
for
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
Approved for Public Release, Distribution Unlimited
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
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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
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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
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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
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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
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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)
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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)
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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
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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
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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
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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
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Summary•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
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