Do global sensitivity metrics for parameterized dynamical systems admit their own dynamical systems?

Do global sensitivity metrics for parameterized dynamical systems admit their
own dynamical systems? PAUL CONSTANTINE Assistant Professor Department of Computer Science University of Colorado Boulder activesubspaces.org! @DrPaulynomial! SLIDES AVAILABLE UPON REQUEST DISCLAIMER: These slides are meant to complement the oral presentation. Use out of context at your own risk. IZZY AGUIAR PhD Student Stanford University

ABOUT PAUL ’S GROUP

Jeff Hokanson Postdoc Izzy Aguiar MS 2018 Zachary Grey PhD
2019 Andrew Glaws PhD 2018

Local sensitivity (derivatives) f = f(p) = f(p1, . .
. , pd) <latexit 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inputs, parameters, independent variables, … How do small changes in each parameter change the output? @f @pi = @f @pi (p) <latexit 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<latexit sha1_base64="WHyWVTfEMp6WL72UR3K345XnKRE=">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</latexit> <latexit sha1_base64="WHyWVTfEMp6WL72UR3K345XnKRE=">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</latexit> <latexit sha1_base64="WHyWVTfEMp6WL72UR3K345XnKRE=">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</latexit> depends on parameter value, “local”

E  @f @pi = Z @f @pi (p) dp
<latexit 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<latexit 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Some global metrics Many metrics; measusure different properties of input/output relationship Used to rank “importance” of inputs; one number per input Metric = 0 implies associated input not “important” at all w/r/t probability measure on parameters =) <latexit 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parameter reduction! =) <latexit 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insight!

Parameterized dynamical systems x = x( t, p) 2 Rk
satisﬁes ˙ x = F (x , p) x(0 , p) = x0(p) for t 2 [0 , T ] p 2 P ✓ Rd <latexit 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<latexit 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<latexit 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<latexit 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assume nothing crazy happens over the parameter space output of interest: f(t, p ) = ( x (t, p )) 2 R <latexit 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<latexit 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functional of states

Local sensitivity analysis for parameterized dynamical systems x = x(
t, p) 2 Rk satisﬁes ˙ x = F (x , p) x(0 , p) = x0(p) for t 2 [0 , T ] p 2 P ✓ Rd <latexit 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<latexit 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<latexit 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<latexit 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r ˙ x = A( x , p ) (r x ) + b ( x , p ) r x (0, p ) = 0 <latexit 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<latexit 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<latexit 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“Linear” dynamical system for parameter gradients [See, eg, Gronwall (1919)]

Global sensitivity analysis for parameterized dynamical systems output of interest:
f(t, p ) = ( x (t, p )) 2 R <latexit 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<latexit 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<latexit 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<latexit 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functional of states ⌫i(t) = E  @f @pi (t, p) , i = 1, . . . , d <latexit 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GSA metrics create a continuous time process example GSA metric not parameter-dependent but is measure-dependent one process per input parameter

⌫(ti) ⇡ X j wj |rf(ti, pj)| <latexit sha1_base64="YOkbiRTp3qTnPqjByAjH/tCCINM=">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</latexit> <latexit
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rf(ti, pj) = r ( x (ti, pj)) <latexit 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A numerical algorithm Integrate several dynamical systems at different inputs and store time grid numerical integration nodes Compute functional of interest at each point in time Numerically integrate for global sensitivity metrics numerical integration weights

Constantine, P. and Doostan, A., Time-dependent global sensitivity analysis with
active subspaces for a lithium ion battery model, SAM (2017) input parameters time history of global sensitivity metrics (= time, in this case)

Do the GSA metrics admit a dynamical system? ˙ ⌫
= G(⌫), ⌫(0) = 0 <latexit 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<latexit 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<latexit 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Why do we care? If this system exists and we can compute / estimate it, then we can reduce lots of high-dimensional integrals into a few one-dimensional integrals (h)uge computational savings! =) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Do the GSA metrics admit a dynamical system? ˙ ⌫
= G(⌫), ⌫(0) = 0 <latexit 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Does this system exist? We have very partial answers!

Do the GSA metrics admit a dynamical system? If …
linear dynamical system for states parameters are initial conditions or independent forcings output of interest is linear functional of states sensitivity metrics are standardized regression coefficients Then … sensitivity metrics admit a linear dynamical system x <latexit 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Aguiar, I. P., Dynamic active subspaces: A data-driven approach to computing time-dependent active subspaces in dynamical systems, MS Thesis (2018)

Do the GSA metrics admit a dynamical system? A data-driven
approach … Given pairs: Fit a model: TONS of open questions about this promising approach! Aguiar, I. P., Dynamic active subspaces: A data-driven approach to computing time-dependent active subspaces in dynamical systems, MS Thesis (2018) Brunton, S. L. et al., Discovering governing equations from data by sparse identification of nonlinear dynamical systems, PNAS (2016) (⌫i, ˙ ⌫i), i = 1, . . . , N <latexit 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˙ ⌫i ⇡ ˆ G(⌫i) <latexit 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Do the GSA metrics admit a dynamical system? Other ideas
(that have yet to bear fruit): Take the time derivative of the global sensitivity metrics and see if a function of the metrics pops out Transform and integrate dynamical system for derivatives to get an expression for the time derivative of the global sensitivity metrics

A Matlab demo! Enzyme kinetics system 4 states, 3 parameters

Remind me again why this is interesting / important? That
looks neat! Does your group have openings? That looks neat! Would you like to write a proposal? PAUL CONSTANTINE Assistant Professor University of Colorado Boulder activesubspaces.org! @DrPaulynomial! QUESTIONS? Active Subspaces SIAM (2015)

Do global sensitivity metrics for parameterized...

Do global sensitivity metrics for parameterized dynamical systems admit their own dynamical systems?

Paul Constantine

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Do global sensitivity metrics for parameterized dynamical systems admit their

ABOUT PAUL ’S GROUP

Jeff Hokanson Postdoc Izzy Aguiar MS 2018 Zachary Grey PhD

Local sensitivity (derivatives) f = f(p) = f(p1, . .

E  @f @pi = Z @f @pi (p) dp

Parameterized dynamical systems x = x( t, p) 2 Rk

Local sensitivity analysis for parameterized dynamical systems x = x(

Global sensitivity analysis for parameterized dynamical systems output of interest:

Constantine, P. and Doostan, A., Time-dependent global sensitivity analysis with

Do the GSA metrics admit a dynamical system? ˙ ⌫

Do the GSA metrics admit a dynamical system? ˙ ⌫

Do the GSA metrics admit a dynamical system? If …

Do the GSA metrics admit a dynamical system? A data-driven

Do the GSA metrics admit a dynamical system? Other ideas

A Matlab demo! Enzyme kinetics system 4 states, 3 parameters

Remind me again why this is interesting / important? That