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Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Reverse Engineering Dynamical Systems FromTime-Series Data Using `1-Minimization
Robert Thompson
ASU School of Mathematical & Statistical Sciences
November 2011
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
1 Brief Basis Pursuit Review
2 Sparse Polynomial Identification
3 ODE System Identification
4 PDE System Identification
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Basis Pursuit
Find the sparsest vector that satisfies a set of constraints:
Basis Pursuit
minx‖x‖1 s.t. y = Ax (LP)
Find the sparsest vector that comes close to satisfying a set ofconstraints:
Relaxed Basis Pursuit
minx‖x‖1 s.t. ‖y −Ax‖2 < ε (SOCP)
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Sparse Polynomial Identification: the problem
1 Given a polynomial:
f(t) =N∑
i=1
αiφi(t) t ∈ (−1, 1)
where f is K-sparse: # {i : αi 6= 0} = K < N ,and the αi unknown.
2 Measure:y = (f(t1), f(t2), . . . , f(tm))T
K < m < N
3 Determine x = (α1, . . . , αN ).
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Sparse Polynomial Identification: Basis Pursuit
f(t1)f(t2)
...f(tm)
︸ ︷︷ ︸
y∈Rm
=
φ1(t1) φ2(t1) . . . φN (t1)φ1(t2) φ2(t2) . . . φN (t2)
...φ1(tm) φ2(tm) . . . φN (tm)
︸ ︷︷ ︸
A∈Rm×N
α1
α2......αN
︸ ︷︷ ︸x∈RN,
K−sparse
Apply Basis Pursuit:
minx‖x‖1 s.t. y = Ax → f =
∑Ni=1 αiφi(t)
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
“Sparse Identification”?
Interpolation: given N + 1 points, find the consistent polynomial ofdegree N .
Regression: given P � N points, force data to fit pre-selectedmodel of degree N .
“Sparse Identification”: given m� N points, find the sparsestmodel that fits the data.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Sparse Polynomial Identification in Natural Basis
φi(t) = ti−1
A is Vandermonde matrix.
Badly conditioned!
y =
f(t1)f(t2)
...f(tm)
=
t01 t11 . . . tN−1
1
t02 t12 . . . tN−12
...t0m t1m . . . tN−1
m
α1
α2......αN
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Sparse Polynomial Identification Experiments
1 Select random K-sparse x ∈ RN
2 Measure:y = (f(t1), . . . , f(tm))T
tm uniformly distributed in (−1, 1).
3 Construct:
Aij =(tj−1i
), i ∈ (1 . . .m), j ∈ (1, N)
4 Find:
x∗ = minx′∈RN
‖x′‖ s.t. Ax′ = y
5 Measure accuracy of recovery: e = ‖x∗ − x‖26 e < µ = success
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Discovering 2D Sparse Polynomials - Recovery Example
−1 0 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
← Recovered from 25 samples on(0, 1)
−1 0 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x x*
--------------------
-0.0000 0
0.0000 0
0.0000 0
-0.0000 0
0.0000 0
-0.0000 0
-0.0000 0
-0.0000 0
-0.1168 -0.1168
1.2525 1.2525
0.0000 0
0.5675 0.5675
0.0000 0
-0.0000 0
... ...
0.0000 0
-0.0000 0
0.0000 0
-1.5505 -1.5505
0.4812 0.4812
-0.0000 0
0.0000 0
... ...
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
m/N
k/m
1d polynomial recovery for N = 36, uniform sampling
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0
Each pixel plots recovery rate after 50 experiments, dark → betterrecovery rate.Increasing m doesn’t make as much difference as might be hoped.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Increasing m doesn’t help
Many sets of columns of V are nearly dependent.
Many K-sparse vectors x ∈ RN become indistinguishable oncetransformed by A to y ∈ Rm, even with m = N .
Increasing m can’t solve this problem.
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Comparison With Chebyshev-Sparse Polynomials
Consider same scenario, but now using Chebyshev polynomials:
φi(t) = Ti(t) = cos((i− 1) arccos(t))
y =
f(t1)f(t2)
...
...f(tm)
=
T1(t1) T2(t1) . . . TN (t1)T1(t2) T2(t2) . . . TN (t2)
......
T1(tm) T2(tm) . . . TN (tm)
α1
α2......αN
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
Vandermonde
m/N
k/m
1d polynomial recovery for N = 36, Chebyshev sampling
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0
ChebyshevSparse 1d Chebyshev polynomial recovery, N = 36
m/Nk/m
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0
Using Chebyshev basis functions, we realize improvement as mincreases.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Increasing m helps
Columns of C are orthogonal.
All vectors y = Ax will be distinguishable if we use full C.
If we use less than full C, orthogonality is lost, some vectors start tobecome indistinguishable.
V
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
C
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
The Role of the Nullspace of A
Theorem
2K-sparse vectors in nullspace of A⇒ exist K-sparse x that can’t berecovered by any algorithm.
Proof.
Let z = x1 − x2 be 2K-sparse ⇒ x1, x2 are K-sparse.Az = A(x1 − x2) = 0⇒ Ax1 = Ax2
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
2-D Sparse Polynomial Identification
What about 2-D polynomials?
In natural basis: f(t, u) =∑
i+j=0..Q
αijtiuj
(tm, um) according to some distribution in (−1, 1)× (−1, 1).
y =
f(t1, u1)f(t2, u2)
...
...f(tm, um)
=
1 t1 u1 t1u1 t21 u2
1 . . .1 t2 u2 t2u2 t22 u2
2 . . ....
...1 tm um tmum t2m u2
m . . .
α00
α10
α01
α11
α20...
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Discovering 2D Sparse Polynomials - Recovery Example
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−15
−10
−5
0
5
10
← Recovered from 25 samples on(0, 1)× (0, 1)
0
5
10
15
20
25
0
5
10
15
20
25
−15
−10
−5
0
5
10
x x*
--------------------
0 -0.0000
0 0.0000
0 0.0000
0 0.0000
-9.1652 -9.1652
-3.2926 -3.2926
0 -0.0000
0 -0.0000
.
.
0 0.0000
0 -0.0000
2.4264 2.4264
0 -0.0000
0 -0.0000
.
.
0 0.0000
0 -0.0000
4.2605 4.2605
0 0.0000
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
2d polynomial recovery, N = 36
m/N
k/m
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0
Similar to 1-d results.
Again increasing m doesn’t change much.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Dimension vs Order
1-d: f(t) =∑N
i=0 xiti. Goes up to order N .
2-d: f(t) =∑
i+j=0..Q
xijtiuj . N = (Q+ 1)(Q+ 2)/2.
For Q = 7, N = 36.
Increasing dimension reduces required order of terms
m/N
k/m
1d polynomial recovery for N = 36, Chebyshev sampling
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0
2d polynomial recovery, N = 36
m/N
k/m
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Discovering 3d Sparse Polynomials
Did someone say 3-D?
In natural basis: f(t, u, v) =∑
i+j+k=0..Q
αijktiujvk
y =
266664f(t1, u1, v1)f(t2, u2, v2)
.
.
.f(tm, um, vm)
377775 =
2666666664
1 t1 u1 v1 t1u1 t1v1 u1v1 . . .1 t2 u2 v2 t2u2 t2v2 u2v2 . . .
.
.
.
.
.
.
.
.
.
.
.
.1 tm um vm tmum tmvm umvm . . .
3777777775
2666666666664
α000α100α010α001α110α101α011
.
.
.
3777777777775
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Discovering 3D Sparse Polynomials - Recovery Example
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−4
−2
0
2
4
6
8
← Recovered from 25 samples on(0, 1)× (0, 1)× (0, 1)
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−4
−2
0
2
4
6
8
x x*
--------------------
0 0.0000
0 -0.0000
.
.
0 0.0000
0 -0.0000
-1.2378 -1.2378
0 0.0000
0 -0.0000
0 -0.0000
0 -0.0000
0 0.0000
0 0.0000
0 0.0000
0 -0.0000
6.0409 6.0409
0 -0.0000
3.5132 3.5132
-1.3987 -1.3987
0 -0.0000
0 -0.0000
-2.3806 -2.3806
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
2d polynomial recovery, N = 36
m/N
k/m
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0
m/Nk/m
3d polynomial recovery, N = 35
0.25 0.5 0.75 1
0.75
0.5
0.25
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Similar to 2d case.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Summary of Sparse Polynomial Identification
Can recover very sparse polynomials with a handful of data points.
In comparison with more ideal bases, Vandermonde has limitationsthat preclude increasing K = number of non-zero entries.
For higher dimensions, can avoid use of higher order terms inVandermonde matrix, but improvement is marginal (if at all).
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
ODE System Identification: the idea
Measure an unknown dynamical system at m points in time.
Determine the system of equations governing the behavior, includingparameter values.
Originally proposed by Wang, Yang, Lai, Kovanis, and Grebogi.
Wen-Xu Wang, Rui Yang, Y.-C. Lai, Vassilios Kovanis, and CelsoGrebogi, ”Predicting catastrophes in nonlinear dynamical systems bycompressive sensing”, Physical Review Letters 106, 154101 (2011).
Similar to polynomial discovery, but A and y will come from systemmeasurements.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
ODE System Identification: the idea
Measure an unknown dynamical system at m points in time.
Determine the system of equations governing the behavior, includingparameter values.
Originally proposed by Wang, Yang, Lai, Kovanis, and Grebogi.
Wen-Xu Wang, Rui Yang, Y.-C. Lai, Vassilios Kovanis, and CelsoGrebogi, ”Predicting catastrophes in nonlinear dynamical systems bycompressive sensing”, Physical Review Letters 106, 154101 (2011).
Similar to polynomial discovery, but A and y will come from systemmeasurements.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
ODE System Identification - 1d
Autonomous 1d discrete dynamical map xn+1 = f(xn)
Assume f has form f(x) =N∑
i=0
αixi
Measure f(xnj ), f(xnj+1), j = 1..mTry to recover the αi. Sparse coefficient vector: f is “simple”.
Similar to 1d polynomial discovery setup.
Note: m now really means taking 2m points, because we must takexnj and xnj+1.
Each (xi, xi+1) pair is a single sample in this setup.
y =
xj1+1
xj2+1
...xjm+1
=
1 xj1 . . . xN
j1
1 xj2 . . . xNj2
...1 xjm
. . . xNjm
α1
α2
...
...αN
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
ODE System Identification - 1d
Autonomous 1d discrete dynamical map xn+1 = f(xn)
Assume f has form f(x) =N∑
i=0
αixi
Measure f(xnj ), f(xnj+1), j = 1..mTry to recover the αi. Sparse coefficient vector: f is “simple”.
Similar to 1d polynomial discovery setup.
Note: m now really means taking 2m points, because we must takexnj and xnj+1.
Each (xi, xi+1) pair is a single sample in this setup.
y =
xj1+1
xj2+1
...xjm+1
=
1 xj1 . . . xN
j1
1 xj2 . . . xNj2
...1 xjm
. . . xNjm
α1
α2
...
...αN
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example - Logistic Map
xn+1 = f(xn) = rxn(1− xn)Coefficient vector: (0, r,−r, 0, . . . )We can recover the system equation in chaotic regime taking about10 sample pairs or more.
0 5 10 15 20 25 30 350.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n
xn
Sampl ing the logi sti c map, m = 10
0 5 10 15 20 25 30 3510
−8
10−6
10−4
10−2
100
102
104
106
m
||c∗−
c||2
Recovery error for logistic map, r = 3.7
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
5 10 15 20 25 30 35
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Sensitive to the dynamics determined by r.
(Bifurcation diagram: Wikipedia).
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
5 10 15 20 25 30 35
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
r=2.5
r=3.0
r=3.2
r=3.5
3
13
3
17
Right: number of unique system values (to 2 digits).
When system is too uniform, taking more samples is just morecopies of the same data, doesn’t improve the decoding.
Chaotic regime is best in this case.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Parameter Estimation vs. System Identification
In the context of the previous example:
“Parameter Estimation”:
Assume a-priori that f(xn) = rxn − rx2n, then estimate r.
“System Identification”:
No a-priori assumption of non-zero terms.
Assume f(x) =N∑
i=0
αixi
Find non-zero αi and their values.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
ODE System Identification - 2d
Autonomous 2d discrete dynamical map
[xn+1
yn+1
]=[f1(xn, yn)f2(xn, yn)
]Assume fk have form
∑i+j=0..Q
αkijx
iyj
Measure system values at ni, i = 1..mTry to recover the αk
ij .
Analogous to the setup of the 2d sparse polynomial discovery, butwe repeat the process twice: X → f1, Y → f2
X =
xi1+1
xi2+1...
xim+1
=
1 xi1 yi1 xi1yi1 . . .1 xi2 yi2 xi2yi2 . . ....1 xim yim ximyim . . .
α100
α110
α111
α120
α102...
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
ODE System Identification - 2d
Analogous to the setup of the 2d sparse polynomial discovery, butwe repeat the process twice: X → f1, Y → f2
X =
xi1+1
xi2+1
...
...xim+1
=
1 xi1 yi1 xi1yi1 . . .1 xi2 yi2 xi2yi2 . . .......1 xim yim ximyim . . .
α100
α110
α111
α120
α102...
Y =
yi1+1
yi2+1
...
...yim+1
=
1 xi1 yi1 xi1yi1 . . .1 xi2 yi2 xi2yi2 . . .......1 xim
yimxim
yim. . .
α200
α210
α211
α220
α202...
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example - Tinkerbell Map
xn+1 = x2n − y2
n + axn + byn
yn+1 = 2xnyn + cxn + dyn
−1.5 −1 −0.5 0 0.5−2
−1.5
−1
−0.5
0
0.5
1
x
y
Tinkerbell map in chaotic regime
Coefficient vectors:
For x: (0, a, b, 1, 0,−1, 0, . . . )For y: (0, c, d, 0, 2, 0, . . . )
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
0 5 10 15 20 25 30 35 4010
−10
10−5
100
105
||α∗ x−
αx||
2
Recovery of system equations, Tinkerbe l l map, a = 0.9, b = −0.6, c = 2, d = 0
0 5 10 15 20 25 30 35 4010
−10
10−5
100
105
||α∗ y−
αy||
2
m
For these system parameters, recovery works well.
With various other parameters not shown, similar results, as long assystem is in a stable regime.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
ODE System Identification - 3d
Autonomous 3d dynamical systemdf
(dx,dy,dz)(x, y, z) =∑
i+j+k=0..Q
(x,y,z)αijk x
iyjzk
Now a continuous system.
Measure system values and derivatives at ti, i = 1..mEssentially the same problem setup as 3d polynomial discovery.
Perform recovery for each X → dfdx , Y → df
dy , Z → dfdz
X =
266666666664
dfdx
(x1, y1, z1)dfdx
(x2, y2, z2)
.
.
.
.
.
.dfdx
(xN , yN , zN )
377777777775=
26666641 x1 y1 z1 x1y1 x1z1 y1z1 x2
1 y21 . . .
1 x2 y2 z2 x2y2 x2z2 y2z2 x22 y22 . . .
.
.
.
.
.
.
1 xm ym zm xmym xmzm ymzm x2m y2m . . .
3777775
26666666664
α000α100α010α001α110α101
.
.
.
37777777775
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example - Rossler System
dx
dt= −y − z
dy
dt= x+ ay
dz
dt= b+ z(x− c)
Coefficient vectors:
For x: (0, 0,−1,−1, 0, 0, 0, . . . )For y: (0, 1, a, 0, 0, 0, . . . )For z: (b, 0, 0,−c, 0, 1, . . . )One difference: x, y, z not in (−1, 1), columns of A get really bad.
Normalize each column of Φ by√∑m
j=1 φ2i (tj), rescale α
accordingly after reconstruction.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example - Rossler System
−40 −20 0 20 40−30
−20
−10
0
10
20
30
x
y
Sampling the Rossler System, m = 25
−40 −20 0 20 400
10
20
30
40
50
60
x
z
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
How Well Does It Work?
||α∗ − α||2
0 5 10 15 20 25 30 3510
−10
100
1010
dx/d
t
0 5 10 15 20 25 30 3510
−10
100
1010
dy/d
t
0 5 10 15 20 25 30 3510
−10
100
1010
dz/d
t
m
Recovery works well here in the chaotic regime:a = 0.1, b = 0.1, c = 18.Similar results in stable periodic regimes.Large number of samples required for z recovery due to the briefnessof excursions away from z ≈ 0 (see figure on previous page).
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Summary of ODE System Identification
Can recover sparse dynamical system equations with a handful ofdata points.
Highly stable periodic behaviors may cause issues.
Number of non-zero terms in power series expansion needs to be lowdue to Vandermonde basis.
Can exploit dimension of system to avoid use of higher order terms.
Noise is an issue that needs to be explored.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
PDE System Identification: the idea
Measure a system at m points in time and space.
Determine the system of equations governing the behavior, includingparameter values.
Inspired by Prof Platte’s prompt “pode fazer com PDE’s?”
(in reference to the ODE version of the problem: Wang, Yang, Lai,Kovanis, Grebogi)
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
PDE Problem Setup
Will work with 2 spatial dimensions.
Assume u(x, y, t) solves a 2nd order PDE and satisfies:
uxx = α1ut + α2ux + α3uy + α4uyy + α5uxy + α6uyt + α7utt
Want to measure u(xi, yi, ti)mi=1 and determine the αj .
Will assume instead the ability to measure the derivatives of u.
y =
266664uxx(x1, y1, t1)uxx(x2, y2, t2)
.
.
.uxx(xm, ym, tm)
377775 =
@(x1, y1, t1)@(x2, y2, t2)
.
.
.@(xm, ym, tm)
266664ut ux uy uyy uxy uyt uttut ux uy uyy uxy uyt utt
.
.
.ut ux uy uyy uxy uyt utt
377775266664α1α2
.
.
.α7
377775
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example: Heat Equation
uxx = ut − uyy, (x, y) ∈ (0, 1)× (0, 1)u(0, y, t) = u(1, y, t) = u(x, 0, t) = u(x, 1, t) = 0Identification of heat system:
0 10 20 30 40 50 60
−20
−15
−10
−5
0
5
# samples
log(||
x∗−
x||
2)
Recovery error for heat equation with 3 modes in initial conditions100 experiments per data point
@(xi,yi,ti)mi=1
ux, uxx, uy, uyy, ...
min `1
uxx = ut − uyy
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example: Wave Equation
uxx = utt − uyy, (x, y) ∈ (0, 1)× (0, 1)u(0, y, t) = u(1, y, t) = u(x, 0, t) = u(x, 1, t) = 0Identification of wave system:
0 10 20 30 40 50 60
−20
−15
−10
−5
0
5
# samples
log(||
x∗−
x||
2)
Recovery error for wave equation with 3 modes in initial conditions100 experiments per data point
@(xi,yi,ti)mi=1
ux, uxx, uy, uyy, ...
min `1
uxx = utt − uyy
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example: Euler-Tricomi Equation
uxx = xuyy, (x, y) ∈ (0, 1)× (0, 1)No time in this system, so leave time derivatives out of the process.
Coefficient vector now includes terms of form xux:
uxx = α1ux + α2uy + α3uyy + α4uxy
+ x(α5ux + α6uy + α7uyy + α8uxy)+ y(α9ux + α10uy + α11uyy + α12uxy)
+ x2(α13ux + α14uy + α15uyy + α16uxy)
+ y2(α17ux + α18uy + α19uyy + α20uxy)
And adjust A accordingly.
Outline Brief Basis Pursuit Review Sparse Polynomial Identification ODE System Identification PDE System Identification
Example: Euler-Tricomi Equation
uxx = xuyy, (x, y) ∈ (0, 1)× (0, 1)Identification of ET system:
0 10 20 30 40 50 60
−20
−15
−10
−5
0
5
# samples
log(||
x∗−
x||
2)
Recovery error for Euler−Tricomi equation100 experiments per data point
@(xi,yi,ti)mi=1
ux, uxx, uy, uyy, ...
min `1
uxx = utt − uyy