chaotic systems can synchronize despite sensitivity
DESCRIPTION
CHAOTIC SYSTEMS CAN SYNCHRONIZE DESPITE SENSITIVITY. two coupled chaotic systems can fall into synchronized motion along their strange attractors when linked through only one variable. z (t). x ’ = (y-x) y ’ = x-y-xz z ’ = - z+xy. y 1 ’ = x-y 1 -x(z 1 ) - PowerPoint PPT PresentationTRANSCRIPT
The Synchronization Approach to Data Assimilation, Parameter Estimation
and Fusion of Climate ModelsGreg Duane
University of Coloradoand
University of Miami
Supported byNSF Grant #0838235
and #0327929
In collaboration with:Joe Tribbia (NCAR)Ben Kirtman (U. Miami)Jeff Weiss (U. Colorado)Ljupco Kocarev (UCSD)Eugenia Kalnay (U. Md.)Josh Hacker (NCAR)
CHAOTIC SYSTEMS CAN SYNCHRONIZEDESPITE SENSITIVITY
• two coupled chaotic systems can fall into synchronized motion along their strange attractors when linked through only one variable
(Pecora and Carroll ’90)
x’= (y-x)y’= x-y-xzz’= -z+xy
y1’= x-y1-x(z1)z1’= -z1)+x(y1)
z (t)
(also works for y-coupling, but not for z-coupling)
SUPPOSE THE WORLD IS A LORENZ SYSTEM AND ONLY X IS OBSERVED
• two coupled chaotic systems can fall into synchronized motion along their strange attractors when linked through only one variable
(Pecora and Carroll ’90)
x’= (y-x)y’= x-y-xzz’= -z+xy
y1’= x-y1-x(z1)z1’= -z1)+x(y1)
z (t)
(also works for y-coupling, but not for z-coupling)
“Truth” “Model”
TWO CHANNEL MODELS SYNCHRONIZE WHEN DISCRETELY COUPLED - makes weather prediction possible
(Duane and Tribbia, PRL ’01, JAS ‘04)
Part I: Treatment of Nonlinearities in the Synchronization Approach
Part II: Synchronization for Parameter Estimation, Model Learning and Fusion of Climate Models
SDE’s: dxA/dt = f (xA) dxB/dt = f (xB) + C (xA- xB + ) is white noise < (t) T(t’) > = R (t- t’)
linearize: de/dt = Fe – Ce + C e xA- xB F Df(xA) Df(xB)
Fokker-Planck eqn for PDF p(e): p/ t + e [p (F-C) e] = ½ (CTRCp)
Gaussian ansatz: p = N exp(-eTKe) pdne = 1 p/ t = 0
Choose C to minimize the spread B (2K)-1 of the distribution.
Fluctuation-Dissipation Relation: B (C-F)T + (C-F) B = CRCT
for C C + dC (dC arbitrary), let dB be such that B B + dB dB=0 if C= Copt = (1/ B R-1
Analysis: Synchronization with Noisy Coupling
Standard Data Assimilation As a Continuous Process (as in Einstein’s treatment of Brownian motion)
Standard methods: xA=xbkd + [B(B+R)-1](xT - xbkd + noise) (perfect model) dxT/dt = f(xT) dxbkd/dt = f(xbkd) + (1/[B(B+R)-1](xT - xbkd + ) + O[ (B(B+R)-1)2] = f(xbkd) + (1/BR-1 (xT - xbkd + )is the time between analyses in incremental data assimilation
The coupling C= (1/ B R-1 = Copt
So, the standard methods of data assimilation (3DVar, Kalman Filtering) are also optimal for synchronization under local linearity assumption!
(Exact treatment of discrete analysis cycle as a map gives Copt= (1/ B (B+R)-1. )
OPTIMAL COUPLING IN FULLY NONLINEAR CASE
de/dt = (F-C)e + Cξ + Ge2 + He3+ ξM ansatz: p=N exp(Ke2+Le3+Me4)
Model error covariance Q=< ξM ξMT>
p/ t + e [(F-C) e + Ge2 + He3 ]p = ½ (CTRCp)
In one dimension, Fokker-Planck eqn (F-C)e + Ge2 + He3 = ½ C2R (-2Ke – 3Le2 – 4Me3) F-C = ½ C2R (-2K) G = ½ C2R (-3L) H = ½ C2R (-4M) background error B= B(K,L,M) = ∫e2p(e)de = B(K(C),L(C),M(C))optimize B as a function of C general correction to KF
If we restrict form of C, e.g. C=F BR-1 cov. inflation factor F
Choose G and H so that the dynamics are those of motion in a two-well potential:
dx/dt = f (x)
e.g. for d1,d2 matching the distances between the fixed points in the Lorenz ’84 system with F=1, one finds G = .15 H = -.75
Minimize background error B as a function of coupling CFind C = 1.51, B=0.145
If C = F B/(R), then we have a covariance inflation factor F= 1.04 (where R=1, = 0.1)
f
B
C
No model error (Q=0):
Model error equal to50% of the resolvedtendency:
The need for inflation is shaped by the nonlinearities, regardless of the amount of model error.
d1
d1
d2
d2
WHAT ABOUT SAMPLING ERROR?
Suppose undersampling uncertainty in estimate of B
multiplicative noise in assimilation
dxT/dt = f(xT) dxbkd/dt = f(xbkd) + (1/[B(B+R)-1+ ξS](xT - xbkd + )
Fokker Planck equation: S
2= =< ξS ξST>
p/ t + e [p (F-C + ½S2) e]
= ½ (CTRC+ S2 e2 - 2 Se/ (CTRC)p
Use change of variables p’= p(CTRC+ S2 e2 - 2 Se (CTRC)
Arguably, effect is small if S R-1
g
Multidimensional Case e.g. D=2
Consider two wells separated in one dimension.
Assume R= (can arrange by rescaling)
Choose a basis such that the dynamical equations are given bya direct product of motion in a two-well potential and simplelinear dynamics.
R is still diagonal.
The FP equation p/ t + e [p (F-C) e] = ½ (CTRCp) separates.
00r
r
Summary: Covariance Inflation in the Synchronization Approach
• In the synchronization approach, the rough magnitudes of covariance inflation factors used in practice might be explained from first principles
• Model error due to unresolved physics makes little difference; the requirement for inflation is shaped by nonlinearities in the dynamics
• Refinements may yield treatments of nonlinearities that improve on covariance inflation
Part I: Treatment of Nonlinearities in the Synchronization Approach
Part II: Synchronization for Parameter Estimation, Model Learning and Fusion of Climate Models
WHAT IF THE MODEL IS IMPERFECT?• can synchronize parameters as well as states
Lorenz system example:
add parameter adaptation laws: 1’= (y-y1) x1
’ = (y1-y) y1
’= y-y1• these augmented equations minimize a Lyapunov function V = ex
2 + ey2 + ez
2 + r2+r
2+r2
where ex = x-x1, ey=……….. r = -1, r=……. since it can be shown that dV/dt < 0, and V is bounded below
So as t→∞, (x1,y1,z1) →(x,y,z) and also 1→ , →1, → 0 i.e. the model “learns”
x’= (y-x)y’= x-y-xzz’= -z+xy
x1’= (y-x1)y1’= x1- y1-x1(z1)+z1’= -z1)+x1(y1)
General Rule for Parameter Estimation, If Systems Synchronize with Identical Parameters
dx/dt = f(x,p)dp/dt=0
dy/dt=f(y,q) + u(y,s) s=s(x) (30)dq/dt=N(y,x-y) (31)
ey-x rq-p h f(y,q)- f(y,p)
Truth:
Model:
(Duane , Yu, and Kocarev, Phys. Lett. A ‘06)
Example: A Column Model With an Unknown Surface Moisture Availability Parameter
Column model summary:
Parameter Adaptation Rule
-interpretation: decrease or increase M in proportion to the covariance between the synchronization (forecast) error and the factor multiplied by M in the dynamical equations
Ý M ~ Kz
f (um0,Tm0,......)(qobs(z) qm (z))
Adapt M according to:
Prognostic equation for humidity:
qm
t
z
{K(qm
z Mf (um0,Tm0,....))} c(qobs qm )
nudging term
soil moisturemoisture availability parameter
RESULTS
-alternating periods of slow convergence to synchronization and rapid ``bursts” away
-apparently can always identify the true value of M
time
M-MT
observations at 7 points in columnnudging at 1 pointnudging coeffiicient = .01
….other configurations show same pattern
…as previous, but with nudging coefficient = .015
observations and nudgingat 7 points,coefficient = .0025
observations and nudgingat 4 points,coefficient = .015
Actual details of model asimplemented in softwarewere unknown!
……..because the state variables also do not convergecompletely in the time interval shown
qm
qT
Single-Realization vs Ensembles
• in principle, should be able to replace ensemble averages with time averages to estimate relatively constant quantities (cf. ergodicity)
• “learn on the fly” → AI view of data assimilation
• compare to “Lagged Average Forecasting” (Hoffman and Kalnay ‘83 ): use a single realization with different initialization times to create an artificial ensemble
Which parameters should we adapt?
TAKE A COLLECTION OF THE BEST MODELS, COUPLE THEM TO ONE ANOTHER, AND ADAPT THE COUPLING COEFFICIENTS
Ki constant: data assimilation
-couple corresponding “model elements” l
adapt Clij:
learning
CONSENSUS
Test Case: Fusing 3 Lorenz Systems With Different Parameters
adaptingnot
adaptingCl
ij=0time time time
z Mav
g -zT
Fused ModelsAverage Output
of Models (Unfused)z from Model
With Best z Eqn
z Mav
g -zT
z Mbe
st-z
T
- Model fusion is superior to any weighted averaging of outputs
dCxij/dt = a(xj-xi)(x – ⅓∑xk)
dCyij/dt=…….
dCzij/dt=…….
Parameter Adaptation in the QG Channel Model
n=0“truth ” A “model” B
foB’=∫(q*-qB)(qA-qB)d2x
timestep n
foB→fo
A
Add terms to FB toassimilate medium scales of A.
Then adapt foB:
What if foB ≠ fo
B ?
Proposed Adaptive Fusion of Different Channel Models
* + *’2
(k-dependence suppressed)
Fo =fo(q-q*)Fo’=fo(q’-q*’)
• If the parallel channelssynchronize, their commonsolution also solves thesingle-channel model withthe average forcing
forcing in Atlantic
forcingin Pacific
To find c adaptively: dc/dt = ∫d2x J(,q’-q)(q-qobs) + ∫d2x J(’,q-q’)(q’-qobs)
Annual Mean SST Temperature
COLA-MOM3
Observations
CAM-MOM3
Longitude
oC
FUSION OF REAL CLIMATE MODELS
typicalscenario:
CAM
MOM
Heat Flux
Momentum Flux
SST
CAM_MOM3
COLA
MOM
Heat Flux
Momentum Flux
SST
COLA_MOM3
CAM COLA
MOM
Heat Flux
Momentum Flux
SST
CAM_COLA_MOM3
“Interactive Ensemble”
CAM COLA
MOM
Heat Flux
Momentum Flux
SST
COLA_CAM_MOM3
“Interactive Ensemble”
Heat Flux: COLA; Momentum Flux: CAM
Heat Flux: CAM; Momentum Flux: COLA
Observations
COLA_MOM3 CAM_MOM3
Heat Flux: COLA; Momentum Flux: CAM Heat Flux: CAM; Momentum Flux: COLA
COLA_MOM3 CAM_MOM3
Heat Flux: COLA; Momentum Flux: CAM Heat Flux: CAM; Momentum Flux: COLA
All Model Error
COLA_MOM3 CAM_MOM3
Heat Flux: COLA; Momentum Flux: CAM Heat Flux: CAM; Momentum Flux: COLA
COLA Heat Flux Errors
COLA_MOM3 CAM_MOM3
Heat Flux: COLA; Momentum Flux: CAM Heat Flux: CAM; Momentum Flux: COLA
Error Amplified by CAM Momentum Flux
COLA_MOM3 CAM_MOM3
Heat Flux: COLA; Momentum Flux: CAM Heat Flux: CAM; Momentum Flux: COLA
CAM Heat Flux Error
INFERENCES ABOUT SOURCES OF ERROR WERE USED TO FORM
A FUSED CAM-COLA MODEL
Guiding principle: For each model element, make the choice of model that reduces truth-model synchronization error
-simplified form of the automated adjustment of coupling coefficients (which need not be binary) proposed here
Adaptive Consensus Formation Approach is Empirical
-reminiscent of learning in neural networks: Hebb’s rule: “Cells that fire together, wire together” here: Model elements “wire together” directionally, until they collectively ``fire” in sync with reality
-Can the role of synchronization in the consensus formation scheme be compared to its proposed role in consciousness, via the highly intermittent synchronization of the 40 Hz oscillation in widely separated regions of the brain?
Conclusion: Adaptive consensus formation among models can likely reduce error in long-range climate forecasts
But what if the dynamical parameters change drastically in the 21st
century as compared to the training period?
Lorenz test case:
Other possible issues: -local vs. global optima in coupling coefficients -climate vs. weather prediction
adaptation
=28 =50 =100
Average of outputs (unfused)Fusion
Attractors
Suggestive of Measure Synchronization……-in jointly Hamiltonian systems, trajectories can become the same, while states differ at any instant of time (Hampton & Zanette PRL ‘99)
-Afraimovich et al. ‘97: “nonisochronic synchronization” of dissipatively coupled systems: