toward predictive digital twins - kiwi.oden.utexas.edu · interpretable data-driven adaptation of...
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Toward Predictive Digital Twinsvia component-based reduced-order models and interpretable machine learning
Michael Kapteyn*, Dr. David Knezevic, Prof. Karen WillcoxAIAA SciTech | Paper AIAA-2020-0418 | January 6, 2020
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Outline 1 Motivation
Predictive digital twins inform critical decision-making
2
Results
Enabling a self-aware UAV: progress and outlook
3
Methodology
Interpretable data-driven adaptation of scalable reduced-order models
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An aircraft that can sense changes in its own internal state, and adapt accordingly
Prior work has shown that this provides [Kordonowy 2011, Singh 2017]
– Increased survivability– Increased utilization
sense structural
data
estimate structural
state
predictflight
capabilities
dynamically replan
mission
1
Motivation: Enabling a self-aware aircraft
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predictive digital twin
We create a digital twin that adapts to the evolving structural health of the UAV,providing near real-time capability predictions to enable dynamic decision-making.
sense structural
data
estimate structural
state
predictflight
capabilities
dynamically replan
mission
Motivation: Enabling a self-aware aircraft
2
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Customized 12ft Telemaster aircraft:
Complex structure with multiple materials
Custom wing sets: pristine & damaged
Custom sensor suite
3
3 axis accelerometer
3 axis gyro
Dual high-frequency dynamic strain and vibration sensors
Temperature, pressure and humidity sensors
*One of the authors has a family member who is co-founder of Divinio. Purchase of the sensors for use in the research was reviewed and approved in compliance with all applicable MIT policies and procedures.
Flight test vehicle
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High-consequence decisions require digital twins that arepredictive • reliable • explainable
predictive digital twin
component-based reduced-order models
interpretable machine learning
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Our approach: data-driven adaptation of component-based reduced-order models
Offline:
Online:
Use model library to train a classifier that predicts asset state based on sensor data
Construct library of reduced-order models representing different asset states
sensor data
Analysis,Prediction,Optimization
updated digital twincurrent digital twin
5
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Component-based reduced-order model library
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Example component: section of a wing
component interior
component port
𝑐
governing PDE(in our case linear elasticity)
computational meshdamageparameters
reduced stiffnessmaterial losscrack length delamination size…
Young’s modulusPoisson’s rationumber of pliesply angles…
geometricparameters
non-geometricparameters
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From components to systems
system parameters𝜇 = [𝜇%, 𝜇', 𝜇(]
component parameters 𝜇%
Instantiate and Assemble Apply Loads
+ assembly parameters 𝜇' + load parameters 𝜇( =
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Start with the usual finite element problem statement:
Find𝑢, ∈ 𝑉, such that 𝑎 𝑢,, 𝑣; 𝜇 = 𝑓 𝑣; 𝜇 ∀𝑣 ∈ 𝑉,𝐴5,5 𝐴5,67 𝐴5,68𝐴5,679 𝐴67,67 0𝐴5,689 0 𝐴68,68
𝕌𝑢67𝑢68
=𝑓5𝑓67𝑓68
Express interior DOFs in terms of port DOFs
𝐴6<,6<𝑢67 = 𝑓67 − 𝐴5,679 𝕌
Substitute to get a system involving only port DOFs:𝕊 𝜇 𝕌 𝜇 = 𝔽(𝜇)
Issue: Schur complement 𝕊(𝜇) is large (𝐌×𝐌), and expensive to compute
Solving a component-based model
Mport DOFsN interior DOFs
ΩG ΩH
𝑃
Solve on each componentindependently
port DOFs
Interior DOFs
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Model reduction strategy
Static-condensation reduced-basis-element (SCRBE) method:[Huynh 2013]
i. Port Reduction:Retain only the first 𝑚 dominant modes at component ports
▸ Reduces the size of 𝕊:M×M 𝑚×𝑚
ii. Component Interior Reduction: Replace the finite element space inside each component witha reduced basis (RB) space of dimension 𝑛
▸ Reduces the size of matrices required to compute entries of 𝕊:N×N 𝑛×𝑛
Mport DOFsN interior DOFs
ΩG ΩH
𝑃
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How does SCRBE meet the demands of a digital twin?
1. Model training can be performed using only small groups of components▸ Never have to solve full-system FE model
2. Component-wise RB admits a modest number of parameters per component▸ System may have many spatially distributed parameters
3. Component instantiation and replacement offers more flexible parametrization▸ Allows for expressive adaptation: changes to topology, meshes etc.
• Cloud-based parallel solvers• Equipped with a posteriori error indicators• Extends to both modal and dynamic analysis [Vallaghé 2015]
• Hybrid solver incorporates local non-linearities• Recourse to full non-linear FEA if required 10
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Performance:FEA: 387,906 dof 55 seconds SCRBE: 694 dof 0.03 seconds
▸ 1000x speedup, solve in near real-time
How does SCRBE meet the demands of a digital twin?
root top skin
top skin
bottom skin
spar caps
shear web
ribs
flaps
aileron linkages
circular rods
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From component-based model to digital twin: Constructing a model library
increasing effective damage(reduction in stiffness)
0%
80%
20%
40%
60%
damage region
Offline: Construct a library of damage states for each component1. Create multiple copies of each component 2. Train components for parameter ranges
of interest (local + interactions)
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Interpretable machine learning
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Data-driven digital twin: Onboard sensors are used to select a reduced-order model from the library
• Use predictive models to generate training data• Use machine learning to train an interpretable, explainable reactive model
asset state noisy sensor data
Forward (predictive) model
Inverse (reactive) model
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Onboard sensors inform which model is used in the digital twin
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From component-based model to digital twin: Interpretable machine learning
0%
80%
20%40%
60%
60 70 80 90 100
80
90
100
110
120
Sensor 16
Sensor24
80%60%40%20%0%
Component1
Component2
300 350 400 450 500 550 600 65080
90
100
110
Sensor 22
Sensor8
80%60%40%20%0%
sensor22< 429?
80%
𝑛𝑦
20%0%
𝑛𝑦
sensor22< 495?sensor22< 383?
sensor22< 351?
𝑛𝑦
60%40%
𝑛𝑦
sensor24< 103?
80%
𝑛
𝑦
20%0%
𝑛
𝑦
sensor16+1.6365*(sensor24)< 237?
sensor16+0.3675*(sensor24)< 112?
sensor16< 73?
𝑛𝑦
60%40%
𝑛𝑦
14
• Highly interpretable• Natural framework for
sensor selection• Rapid online classification• As expressive as
standard neural networks
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Goal: Find a partitioning of the space of possible sensor measurements, and assign to each partition the library model that best explains the measurements
Optimal Classification Trees [Bertsimas, 2019] uses mixed-integer optimization techniques to find a partition in the form of an optimal binary tree, T:
min^R T + 𝛼|T|
• Globally optimal• Scalable• Naturally extends to hyperplane splits
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error on training data complexity of the tree
tradeoff parameter
From component-based model to digital twin: Interpretable machine learning
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Recall our approach: data-driven adaptation of component-based reduced-order models
Offline:
Online:
Use model library to train a classifier that predicts asset state based on sensor data
Construct library of reduced-order models representing different asset states
sensor data
Analysis,Prediction,Optimization
updated digital twincurrent digital twin
16
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Test with experimental data
Incorporate multimodal observations
Flight demonstration
Future Work
Open challengesImproving damage models
Accounting for model uncertainty and inadequacy
Combining component-based reduced-order models and interpretable machine learning enables predictive digital twins
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For a project overview, slides, and the full paper, visit https://kiwi.oden.utexas.edu/research/digital-twin
High-consequence decisions require digital twins that arepredictive • reliable • explainable
Funding acknowledgements:• Air Force Office of Scientific Research (AFOSR) Dynamic Data-Driven Application Systems (DDDAS) • The Boeing Company• SUTD-MIT International Design Centre 19
predictive digital twin
component-based reduced-order models
interpretable machine learning
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References[Kordonowy 2011]
[Singh 2017]
[Vallaghé 2015]
[Huynh 2013]
[Bertsimas 2019]
Kordonowy, D., and Toupet, O., "Composite airframe condition-aware maneuverability and survivability for unmanned aerial vehicles." Infotech@ Aerospace 2011, 2011-1496.
Singh, V., and Willcox, K.E., "Methodology for Path Planning with Dynamic Data-Driven Flight Capability Estimation." AIAA Journal (2017): 2727-2738.
Vallaghé, S., et al. "Component-based reduced basis for parametrized symmetric eigenproblems." Advanced Modeling and Simulation in Engineering Sciences 2.1 (2015): 7.
Huynh, D.B.P., D.J. Knezevic, and A.T. Patera. "A static condensation reduced basis element method: approximation and a posteriori error estimation." ESAIM: Mathematical Modelling and Numerical Analysis 47.1 (2013): 213-251.
Bertsimas, D., and Dunn, J., "Machine Learning under a Modern Optimization Lens." Dynamic Ideas (2018).