Data-driven reduced model construction with the time-domain Loewner - - PowerPoint PPT Presentation

Data-driven reduced model construction with the time-domain Loewner framework and operator inference

Benjamin Peherstorfer Courant Institute of Mathematical Sciences New York University in collaboration with Serkan Gugercin and Karen Willcox

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Many-query applications

• ptimization

control inference multi-discipline coupling model calibration uncertainty quantification visualization

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Intro: Surrogate model types

Surrogate models

◮ Approximate QoI of high-fidelity model ◮ Often significantly reduced costs

Simplified surrogate models

◮ Drop nonlinear terms, average ◮ Early-stopping schemes

Data-fit surrogate models

◮ Learn input-QoI map induced by hi-fi model ◮ SVMs, Gaussian processes, neural networks

(Projection-based) reduced models

◮ Approximate state xk in low-dim space ◮ Project E, A, B, C onto low-dim space ◮ POD, interpolatory model reduction, RBM RN x1 x2 xK

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Intro: Surrogate model types

Surrogate models

◮ Approximate QoI of high-fidelity model ◮ Often significantly reduced costs

Simplified surrogate models

◮ Drop nonlinear terms, average ◮ Early-stopping schemes

Data-fit surrogate models

◮ Learn input-QoI map induced by hi-fi model ◮ SVMs, Gaussian processes, neural networks

(Projection-based) reduced models

◮ Approximate state xk in low-dim space ◮ Project E, A, B, C onto low-dim space ◮ POD, interpolatory model reduction, RBM RN x1 x2 xK

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Intro: Dynamical systems

Consider linear time-invariant (LTI) system Σ :

• Exk+1 = Axk + Buk ,

k ∈ N , yk = Cxk

◮ Time-discrete single-input-single-output (SISO) LTI system ◮ System matrices E, A ∈ RN×N, B ∈ RN×1, C ∈ R1×N ◮ Input uk and output yk at time step tk, k ∈ N ◮ State xk at time step tk, k ∈ N ◮ Asymptotically stable

Deriving QoI from outputs yk and states xk, k ∈ N

◮ High-dimensional state xk makes computing QoI expensive ◮ Repeated QoI computations can become prohibitively expensive

◮ Uncertainty propagation, statistical inference, control, . . . 4 / 37

Intro: Classical (intrusive) model reduction

Construct n-dim. basis V = [v 1, . . . , v n] ∈ RN×n

◮ Proper orthogonal decomposition (POD) ◮ Interpolatory model reduction ◮ Reduced basis method (RBM) ◮ ... RN x1 x2 xK

Project full model operators E, A, B, C onto reduced space ˜ E = V T

N×N

• E

V

• n×n

, ˜ A = V T

N×N

• A V
• n×n

, ˜ B = V T

N×p

• B .
• n×p

, ˜ C =

q×N

• C V

q×n

Construct reduced model ˜ Σ : ˜ E ˜ xk+1 = ˜ A˜ xk + ˜ Buk , k ∈ N ˜ yk = ˜ C ˜ xk with n ≪ N and yk − ˜ yk small in appropriate norm In general, projection step requires full-model operators E, A, B, C

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Intro: Classical (intrusive) model reduction

Construct n-dim. basis V = [v 1, . . . , v n] ∈ RN×n

◮ Proper orthogonal decomposition (POD) ◮ Interpolatory model reduction ◮ Reduced basis method (RBM) ◮ ... RN x1 x2 xK

Project full model operators E, A, B, C onto reduced space ˜ E = V T

N×N

• E

V

• n×n

, ˜ A = V T

N×N

• A V
• n×n

, ˜ B = V T

N×p

• B .
• n×p

, ˜ C =

q×N

• C V

q×n

Construct reduced model ˜ Σ : ˜ E ˜ xk+1 = ˜ A˜ xk + ˜ Buk , k ∈ N ˜ yk = ˜ C ˜ xk with n ≪ N and yk − ˜ yk small in appropriate norm In general, projection step requires full-model operators E, A, B, C

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Intro: Black-box models and model reduction

Full model often given as black box

◮ Operators E, A, B, C unavailable ◮ Time-step model Σ with inputs to obtain outputs and states

u =

• u1

u2 . . . uK T y = y1 y2 . . . yK T X =   | | | x1 x2 · · · xK | | |   Goal: Learn reduced model from data u, y, and X ⇒ Learn reduced equations (in contrast to data-fit surrogates) ⇒ Discover the “dynamics” that govern full model input Exk+1 = Axk + Buk yk = Cxk

• utput

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Intro: Black-box models and model reduction

Full model often given as black box

◮ Operators E, A, B, C unavailable ◮ Time-step model Σ with inputs to obtain outputs and states

u =

• u1

u2 . . . uK T y = y1 y2 . . . yK T X =   | | | x1 x2 · · · xK | | |   Goal: Learn reduced model from data u, y, and X ⇒ Learn reduced equations (in contrast to data-fit surrogates) ⇒ Discover the “dynamics” that govern full model input Exk+1 = Axk + Buk yk = Cxk black-box dynamical system

• utput

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Intro: Black-box models and model reduction

Full model often given as black box

◮ Operators E, A, B, C unavailable ◮ Time-step model Σ with inputs to obtain outputs and states

u =

• u1

u2 . . . uK T y = y1 y2 . . . yK T X =   | | | x1 x2 · · · xK | | |   Goal: Learn reduced model from data u, y, and X ⇒ Learn reduced equations (in contrast to data-fit surrogates) ⇒ Discover the “dynamics” that govern full model input Exk+1 = Axk + Buk yk = Cxk black-box dynamical system

• utput

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Literature overview

System identification

◮ System identification [Ljung, 1987], [Viberg, 1995], [Qin, 2006], . . . ◮ Eigensystem realization [Kung, 1978], [Mendel, 1981], [Juang, 1985], [Kramer & Gugercin,

2016], . . .

◮ Finite impulse response estimation [Rabiner et al., 1978], [Mendel, 1991],

[Abed-Meraim, 1997, 2010], . . .

Learning in the frequency domain

◮ Loewner framework [Antoulas, Anderson, 1986], [Lefteriu, Antoulas, 2010], [Mayo, Antoulas,

2007], [Beattie, Gugercin, 2012], [Ionita, Antoulas, 2012], . . .

◮ Vector fitting [Drmac, Gugercin, Beattie, 2015a], [Drmac, Gugercin, Beattie, 2015b], . . .

Learning from states

◮ Learning models with dynamic mode decomposition [Tu et al., 2013],

[Proctor, Brunton, Kutz, 2016], [Brunton, Brunton, Proctor, Kutz, 2016], . . .

◮ Learning models [Chung, Chung, 2014], [Xie, Mohebujjaman, Rebholz, Iliescu, 2017], . . . ◮ Sparse identification [Brunton, Proctor, Kutz, 2016], . . .

Machine learning

◮ Gaussian process regression, support vector machines, . . .

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Data-driven reduced model construction with time-domain Loewner models

Inferring frequency-response data from time-domain data joint work with Serkan Gugercin and Karen Willcox

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Loewner: Transfer function

Transfer function of LTI system Σ H(z) = C(zE − A)−1B , z ∈ C Consider reduced model ˜ Σ with ˜ H(z) = ˜ C(z ˜ E − ˜ A)−1 ˜ B , z ∈ C Measure error of reduced transfer function ˜ H as H − ˜ HH∞ = sup

|z|=1

|H(z) − ˜ H(z)| Relate to error in quantity of interest y − ˜ yℓ2 ≤ H − ˜ HH∞uℓ2 If ˜ H approximates well H, then know that ˜ y approximates well y

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Loewner: Interpolatory model reduction

Select m = 2n interpolation points z1, . . . , zm ∈ C Construct bases as V = (z1E − A)−1B . . . (znE − A)−1B ∈ RN×n W =

• (zn+1E T − AT)−1C T

. . . (zn+nE T − AT)−1C T ∈ RN×n Project (Petrov-Galerkin) to obtain operators ˜ E = W TEV , ˜ A = W TAV , ˜ B = W TB, ˜ C = CV Then obtain reduced model ˜ Σ with ˜ H H(zi) = ˜ H(zi) , i = 1, . . . , m Requires full-model operators E, A, B, C

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Loewner: Interpolatory model reduction with Loewner

Loewner framework derives ˜ Σ directly from H(z1), . . . , H(zm) with Lij = H(zi) − H(zn+j) zi − zn+j , Mij = ziH(zi) − zn+jH(zn+j) zi − zn+j , i, j = 1, . . . , n Reduced operators of ˜ Σ are ˜ E = −L, ˜ A = −M, ˜ B = H(z1) . . . H(zn)T , and ˜ C = H(zn+1) . . . H(zn+n) Data-driven (nonintrusive) construction of ˜ Σ

◮ No access to E, A, B, C required ◮ Requires transfer function values (frequency-response data)

[Antoulas, Anderson, 1986] [Lefteriu, Antoulas, 2010] [Mayo, Antoulas, 2007] 11 / 37

Loewner: Problem formulation

Can time-step full LTI model Σ for K ∈ N time steps

◮ Given inputs u = [u0, u1, . . . , uK−1]T ∈ CK ◮ Compute outputs y = [y0, y1, . . . , yK−1]T ∈ CK via time stepping ◮ Transfer function H unavailable (E, A, B, C, xk unavailable as well)

Our goal is approximating transfer function values from y

◮ Given are interpolation points z1, . . . , zm ◮ Perform single time-domain simulation of Σ to steady state ◮ Derive approximate ˆ

H(z1), . . . , ˆ H(zm) values from output y

◮ Construct ˆ

Σ that approximates (classical) Loewner ˜ Σ H(zi)

full model

= ˜ H(zi)

classical Loewner model

≈ ˆ H(zi)

time-domain Loewner model

, i = 1, . . . , m

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Loewner: Output of LTI system

Define points on the unit circle qi = e

2πj K i ,

i = 0, . . . , K − 1 Represent input in discrete Fourier coefficients U = [U0, . . . , UK−1]T uk =

K−1

• i=0

Uiqk

i ,

k = 0, . . . , K − 1 W.l.o.g. have set Ir = {1, . . . , r} of non-zero Fourier coefficients uk =

r

• i=1

Uiqk

i ,

k = 0, . . . , K − 1 Output is convolution of impulse response hk and input uk yk =

k

• l=0

hluk−l =

k

• l=0

hl

r

• i=1

Uiqk−l

i

• uk−l

=

r

• i=1

Uiqk

i k

• l=0

hlq−l

i

• =Hk(qi)

, k = 0, . . . , K−1

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Loewner: Asymptotic properties of Hk(z)

Relationship between output yk and Hk(qi) yk =

r

• i=1

Uiqk

i k

• l=0

hlq−l

i

• =Hk(qi)

=

r

• i=1

Hk(qi)Uiqk

i ,

k = 0, . . . , K − 1 Transfer function H is z-transform of impulse response H(z) =

∞

• l=0

hlz−l , z ∈ D Sequence (Hk(z)) converges to H(z) for z ∈ D |H(z) − Hk(z)| ≤ cρk Decay of error |H(z) − Hk(z)| depends on spectral radius ρ of E −1A

◮ Problem-dependent rate of decay of error |H(z) − Hk(z)| ◮ Slow decay of error if many time steps to reach steady state

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Loewner: Regression problem

Relationship between output yk and Hk(qi) yk =

r

• i=1

Hk(qi)Uiqk

i ,

k = 0, . . . , K − 1 Solve for approximate transfer function values ˆ H1, . . . , ˆ Hr ∈ C arg min

ˆ H1,..., ˆ Hr ∈C K−1

• k=kmin
• yk
• utput

−

r

• i=1

ˆ Hi Uiqk

i

• non-zero

Fourier component

2 ⇒ Note that dim of optimization problem grows with r For tolerance ǫ > 0, select value kmin ∈ N such that |H(qi) − Hkmin(qi)| < ǫ , i = 1, . . . , r

◮ Controls the time step from which on Hk(qi) sufficiently accurate ◮ Asymptotic analysis confirms that kmin is problem-dependent ◮ If kmin ≤ K − r, then system overdetermined

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Loewner: Time-domain Loewner algorithm

Time-domain Loewner approach

• 1. Time-step full model Σ for input u to obtain output y
• 2. Select value kmin
• 3. Determine indices {i1, . . . , ir} of non-zero Fourier coefficients of u
• 4. Solve for approximate transfer function values ˆ

H1, . . . , ˆ Hr

• 5. Select interpolation points z1, . . . , zm ⊂ {qi1, . . . , qir }
• 6. Use Loewner with ˆ

H1, . . . , ˆ Hm to derive ˆ Σ with H(zi) ≈ ˆ H(zi) , i = 1, . . . , m Choice of interpolation points

◮ Restricted by non-zero Fourier coefficients of input ◮ Number of time steps K determines frequency range

2π K , 2π(K − 1) K

• ⊂ R

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Loewner: Numerical results

1e-14 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 1e+00 0.01 0.1 1 relative error spectral radius ρ freq ω1 = 0.12566 freq ω2 = 0.37699 freq ω3 = 1.00531 freq ω4 = 2.51327 freq ω5 = 6.15752 1e-14 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 1e+00 0.01 0.1 1 relative error spectral radius ρ kmin = ⌊3/4K⌋ kmin = ⌊1/2K⌋ kmin = ⌊1/4K⌋ (a) dependence on ρ, K = 50 (b) dependence on kmin, K = 50 ◮ Synthetic example where we can control ρ ◮ Relative error of approximate transfer function values

errrel( ˆ Hl) = |H(qil) − ˆ Hl| |H(qil)| , l = 1, . . . , m

◮ A large spectral radius leads to larger error for fixed K ◮ Large kmin avoids early, inaccurate transfer function approximations ◮ Setting kmin too large, leads to ill-conditioned least-squares problem

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Loewner: Numerical results

1e-14 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 1e+00 0.01 0.1 1 relative error spectral radius ρ freq ω1 = 0.12566 freq ω2 = 0.37699 freq ω3 = 1.00531 freq ω4 = 2.51327 freq ω5 = 6.15752 1e-14 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 1e+00 0.01 0.1 1 relative error spectral radius ρ kmin = ⌊3/4K⌋ kmin = ⌊1/2K⌋ kmin = ⌊1/4K⌋ (a) dependence on ρ, K = 100 (b) dependence on kmin, K = 100 ◮ Synthetic example where we can control ρ ◮ Relative error of approximate transfer function values

errrel( ˆ Hl) = |H(qil) − ˆ Hl| |H(qil)| , l = 1, . . . , m

◮ A large spectral radius leads to larger error for fixed K ◮ Large kmin avoids early, inaccurate transfer function approximations ◮ Setting kmin too large, leads to ill-conditioned least-squares problem

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Loewner: Eady example

Eady LTI system

◮ Order of system is N = 598 ◮ Discretize with 4th-order scheme ◮ Time step size δt = 10−1 and K = 103

Time-domain Loewner reduced model

◮ Dimension of reduced model n = 5 ◮ Set kmin = ⌊1/4K⌋ ◮ Select m logarithmic interpolation pts

qi1, . . . , qim ⊂ {q0, . . . , qK−1}

◮ Input uk at time tk, k = 0, . . . , K − 1

uk = 1 K

m

• l=1

(1 + j) qk

il ◮ Simulate full model Σ once

1e+01 1e+02 1e+03 1e+04 1e-02 1e-01 1e+00 magnitude frequency ω full model

(a) magnitude

• 3e+00
• 2e+00
• 2e+00
• 2e+00
• 1e+00
• 5e-01

0e+00 1e-02 1e-01 1e+00 phase frequency ω full model

(b) phase

http://slicot.org/20-site/126-benchmark-examples-for-model-reduction 18 / 37

Loewner: Eady example: Transfer function

1e+01 1e+02 1e+03 1e+04 1e-02 1e-01 1e+00 magnitude frequency ω [rad/s] full model classical Loewner time Loewner −3 −2.5 −2 −1.5 −1 −0.5 1e-02 1e-01 1e+00 phase frequency ω [rad/s] full model classical Loewner time Loewner (a) magnitude (b) phase

H− ˜ HH2 HH2 H− ˆ HH2 HH2 ˜ H− ˆ HH2 HH2 H− ˜ HH∞ HH∞ H− ˆ HH∞ HH∞ ˜ H− ˆ HH∞ HH∞

1.41 × 10−1 1.11 × 10−1 5.42 × 10−2 3.63 × 10−1 2.42 × 10−1 1.85 × 10−1

◮ Construct time-domain Loewner from single trajectory ◮ Magnitude of transfer function matched well; slight difference in

phase

◮ Time-domain (& classical) Loewner model are asymptotically stable

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Loewner: Penzl example

Penzl LTI system

◮ Order of system is N = 1006 ◮ Discretize in time with implicit Euler ◮ Time step size δt = 10−4 ◮ Number of time steps K = 106

Time-domain Loewner reduced model

◮ Dimension of reduced model n = 10 ◮ Set kmin = ⌊1/4K⌋ ◮ Select m logarithmic interpolation pts ◮ Construct input as in Eady example ◮ Simulate full model Σ once

1e-01 1e+00 1e+01 1e+02 1e-04 1e-03 1e-02 1e-01 1e+00 magnitude frequency ω full model

(a) magnitude

• 3e+00
• 2e+00
• 1e+00

0e+00 1e+00 2e+00 3e+00 1e-04 1e-03 1e-02 1e-01 1e+00 phase frequency ω full model

(b) phase

[Penzl, 2006], [Ionita, 2013] 20 / 37

Loewner: Penzl example: Transfer function

1e-01 1e+00 1e+01 1e+02 1e-04 1e-03 1e-02 1e-01 1e+00 magnitude frequency ω [rad/s] full model classical Loewner time Loewner −3 −2 −1 1 2 3 1e-04 1e-03 1e-02 1e-01 1e+00 phase frequency ω [rad/s] full model classical Loewner time Loewner (a) magnitude (b) phase ◮ Number of interpolation points m = 64 ◮ Test points logarithmically distributed in range [10−4, 1] ◮ Time-domain Loewner matches classical Loewner model well

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Loewner: Penzl example: Poles

−0.04 −0.03 −0.02 −0.01 0.01 0.02 0.03 0.04 0.98 0.99 1 imaginary part real part classical L. time L. 0.940 0.950 0.960 0.970 0.980 0.990 1.000 1.010 1 2 3 4 5 6 7 8 9 10 magnitude of eigenvalue index of eigenvalue classical Loewner time Loewner (a) eigenvalues (b) magnitude of eigenvalues

H− ˜ HH2 HH2 H− ˆ HH2 HH2 ˜ H− ˆ HH2 HH2 H− ˜ HH∞ HH∞ H− ˆ HH∞ HH∞ ˜ H− ˆ HH∞ HH∞

5.88 × 10−1 5.88 × 10−1 1.07 × 10−4 2.67 × 10−3 2.67 × 10−3 9.97 × 10−6

◮ Time-domain Loewner model matches poles of classical Loewner ◮ Time-domain (& classical) Loewner model are asymptotically stable

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Loewner: Beam example

Cantilever beam

◮ Full 3D finite-element model of beam ◮ Force applied at tip of beam ◮ Implicit Euler, δt = 10−4, K = 106

Time-domain Loewner

◮ Dimension of reduced model n = 8 ◮ Select m = 150 interpolation points ◮ Same kmin and input as in Eady ◮ Simulate full model Σ once force

x3 x1 x2

(a) geometry of beam

[Panzer et al., 2009]

(b) beam at time step 4452 (c) beam at time step 5061

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Loewner: Beam example: The kmin value

1e+00 1e+02 1e+04 1e+06 1e+08 1e+10 1e+12 1e+14 1e+16 1e+18 2 . 5 e + 5 5 e + 5 7 . 5 e + 5 9 e + 5 9 . 9 e + 5 9 . 9 9 e + 5 condition number kmin

◮ The kmin significantly influences the condition number ◮ Conservative choice seems sufficient in practice

24 / 37

Loewner: Beam example: Transfer function

1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1e-04 1e-03 1e-02 1e-01 1e+00 magnitude frequency ω [rad/s] full model classical Loewner time Loewner −3 −2 −1 1 2 3 1e-04 1e-03 1e-02 1e-01 1e+00 phase frequency ω [rad/s] full model classical Loewner time Loewner (a) magnitude (b) phase ◮ Time-domain Loewner model matches transfer function well ◮ Differences can be seen for high frequencies

25 / 37

Loewner: Beam example: Error

1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1e+00 abs error frequency ω [rad/s] classical Loewner time Loewner

• 1e+02

0e+00 1e+02 2000 4000 6000 8000 10000

• 1e+02

0e+00 1e+02 2000 4000 6000 8000 10000

• utput

time step classical Loewner

• utput

time step time Loewner (a) absolute error (b) output

H− ˜ HH2 HH2 H− ˆ HH2 HH2 ˜ H− ˆ HH2 HH2 H− ˜ HH∞ HH∞ H− ˆ HH∞ HH∞ ˜ H− ˆ HH∞ HH∞

2.51 × 10−2 1.28 × 10−2 2.12 × 10−2 2.26 × 10−4 2.16 × 10−4 1.21 × 10−4

◮ Absolute error is low for low frequencies ◮ Perform time-domain simulation of reduced model ◮ Output of time-domain Loewner matches output of classical Loewner

26 / 37

Loewner: Beam example: Input signals

−2 −1.5 −1 −0.5 0.5 0e+00 2e+05 4e+05 6e+05 8e+05 1e+06 input time step k 1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1e-04 1e-03 1e-02 1e-01 1e+00 magnitude frequency ω [rad/s] full model classical Loewner time Loewner (a) chirp signal (b) magnitude ◮ Extract input u from “chirp” signal (non-zero Fourier coefficients) ◮ Simulate Σ at u and construct time-domain Loewner model ◮ Time-domain Loewner shows similar behavior as for synthetic input

27 / 37

Data-driven operator inference for nonintrusive model reduction

joint work Karen Willcox

28 / 37

Inference: Nonlinear dynamical system

Consider dynamical system with polynomial nonlinear term Σ :

• ˙

x(t) = Ax(t) + F(x(t) ⊗ x(t)) + . . . + Bu(t) y(t) = Cx(t)

◮ Time-continuous multiple-input-multiple-output (MIMO) system ◮ System matrices A ∈ RN×N, B ∈ RN×p, C ∈ Rq×N ◮ Input u(t) ∈ Rp and output y(t) ∈ Rq at time t ∈ [0, ∞) ◮ State x(t) ∈ RN at time t ∈ [0, ∞) ◮ Nonlinear term with matrix F ∈ RN×N2

Given as black box, only data available at time steps t1, . . . , tK

◮ State trajectory

X =   | | x1 . . . xK | |  

T

∈ RK×N

◮ Outputs Y =

y 1 . . . y K T ∈ RK×q

◮ Inputs U =

• u1

. . . uK T ∈ RK×p

29 / 37

Inference: Learning from states

full-model trajectories reduced space construct reduced model project full model full-model

• perators

assemble full-model trajectories reduced space construct infer

30 / 37

Inference: Fit operators to projected states

Construct n-dim. basis from state trajectory X V = [v 1, . . . , v n] ∈ RN×n Project state trajectory X onto reduced space ˆ x1 . . . ˆ xK

• ∈ Rn×K

Find operators ˆ A, ˆ B, ˆ C, and ˆ F that satisfy ˙ ˆ xk ≈ ˆ Aˆ xk + ˆ F(ˆ xk ⊗ ˆ xk) + ˆ Buk, k = 1, . . . , K y k ≈ ˆ C ˆ xk, k = 1, . . . , K Measure error in L2 norm

• ˙

ˆ xk − ˆ Aˆ xk − ˆ F(ˆ xk ⊗ ˆ xk) − ˆ Buk

• 2 ,

k = 1, . . . , K

31 / 37

Inference: Optimize for operators

Fit operators ˆ A, ˆ B, and ˆ F by solving optimization problem min

ˆ A,ˆ F, ˆ B K

• k=1
• ˙

ˆ xk − ˆ Aˆ xk − ˆ F(ˆ xk ⊗ ˆ xk) − ˆ Buk

• 2

2 ◮ Transform into n independent least-squares problem ◮ Can be solved efficiently with standard solvers

Find operator ˆ C with min

ˆ C K

• k=1

y k − ˆ C ˆ xk2

2

Inferred operators converge in · F to “intrusive operators”

◮ Need sufficient data ◮ Need that V spans RN for n → N

In the linear case, the inferred operators are the DMD operators

[Tu et al., 2013], [Chung, Chung, 2014], [Proctor et al., 2016], [Xie, Mohebujjaman, Rebholz, Iliescu, 2017] 32 / 37

Inference: Burgers’ equation

Consider viscous Burgers’ equation ∂ ∂t x(ω, t; µ) + x(ω, t; µ) ∂ ∂ω x(ω, t; µ) − µ ∂2 ∂ω2 x(ω, t; µ) = 0

◮ Spatial domain ω ∈ [0, 1] and time domain [0, 1] ◮ Parameter µ in parameter domain D = [0.1, 1] ◮ Dirichlet boundary conditions x(0, t; µ) = −x(1, t; µ) = u(t)

Discretize with finite differences d dt x(t; µ) = A(µ)x(t; µ) + F(x(t; µ) ⊗ x(t; µ)) + Bu(t) , Derive outputs and state trajectories with semi-implicit Euler

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Inference: Burgers’: Results

1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1e+00 2 4 6 8 10 12 14 avg error of states dimension n intrusive inference-based 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1e+00 2 4 6 8 10 12 14 avg error of states dimension n intrusive inference-based (a) training parameters (b) test parameters ◮ Inferred from µ1, . . . , µ10 and tested on µ11, . . . , µ20 ◮ Error measured with respect to full model ◮ Inferred operators lead to similar error as intrusive operators ◮ For n > 10, inference problem becomes ill-conditioned

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Inference: Limit cycle oscillation

Tubular reactor

◮ Nonlinear convection-diffusion-reaction ◮ Third-order nonlinear term ◮ Damköhler µ ∈ D controls behavior ◮ Implicit Euler, δt = 10−4, K = 5 × 105

Operator inference

◮ Reduced model dimension n = 5 ◮ Inference for parameters µ1, . . . , µ15 ∈ D ◮ Error on parameters µ16, . . . , µ23 ∈ D 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 10 20 30 40 50 temperature time [s] Damköhler µ = 0.163 (a) steady-state 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 10 20 30 40 50 temperature time [s] Damköhler µ = 0.1655 (b) LCO [Heinemann & Poore, 1981], [Zhou, 2012]

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Inference: LCO: Amplitude

0.004 0.008 0.012 0.016 0.02 0.024 0.163 0.164 0.165 LCO amplitude Damköhler number µ intrusive inference-based 0.004 0.008 0.012 0.016 0.02 0.024 0.163 0.164 0.165 LCO amplitude Damköhler number µ intrusive inference-based (a) training parameters (b) test parameters ◮ Inference-based model enters LCO at same µ as intrusive model ◮ Similar behavior of inference-based and intrusive reduced model

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Conclusions

1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1e-04 1e-03 1e-02 1e-01 1e+00 magnitude frequency ω [rad/s] full model classical Loewner time Loewner ◮ Nonintrusive construction of reduced model directly from data ◮ Can be seen as data-driven model construction ◮ More details in

• 1. P., Gugercin & Willcox. Data-driven model construction with

time-domain Loewner reduced models. submitted.

• 2. P. & Willcox. Data-driven operator inference for nonintrusive

projection-based model reduction. Computer Methods in Applied Mechanics and Engineering, 306:196-215, 2016.

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