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![Page 1: Invariant grids: method of complexity reduction in reaction networks Andrei Zinovyev Institut Curie, Paris Institut des Hautes Études Scientifiques.](https://reader034.fdocuments.us/reader034/viewer/2022042821/56649f455503460f94c6747c/html5/thumbnails/1.jpg)
Invariant grids: method of complexity reduction in reaction networks
Andrei ZinovyevInstitut Curie, Paris
Institut des Hautes Études Scientifiques
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2 - European conference on complex systems, Paris, November 2005 -
Stoichiometric equations
Jc
s1A1 + … + snAn s1A1 + … + snAn
n – number of species, s – number of reactions
si
n
jjs
n
jjsss
jj ckckJ..1 11
)(
c1
c2
cn
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What is Model Reduction?
•1 Shorten list of species– eliminate some– create “integrated” components•2 Shorten list of reactions– eliminate some– “freeze” fast reactions•3 Decompose motion into fast and slow
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Approaching steady state
Jc
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Positively Invariant Manifold
Steady state
fast motion
slow motion
Why Invariant? once the point on the manifold, the trajectory will stay on it until the equilibrium
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Why for do we need invariant manifold?
Model reduction:“Macroscopic” system description
xRN – detailed description yRm – “macroscopic” description. m<<N
)(xJdt
dx )(yJ
dt
dy
niyyxx mii ...1),...( 1
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Why for do we need invariant manifold?
Dynamics visualization
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Other useful non-invariant manifolds
Quasy steady-stateFast variables are steady
Quasi-equilibriumManifolds maximizing entropy
Intrinsic low-dimensional manifoldDecomposition of Jacobian fields
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Projector Pc on (some) manifoldinduces new (reduced) dynamics
J
induced dynamics
Pc J
tangent space
= (1-Pc)J - invariance defect
x
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Class of dissipative systems
cG
ceq
n
ieqi
ii c
ccG
1
1ln
Lyapunov function
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Thermodynamic projector
GJ
Pc J
The induced dynamics is dissipative only if
0))(),(( cJPcG C
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Correction of invariance defect
0.2
0.4
0.6
0.8
1.0
0.05 0.10 0.15 0.20
C1
C3
equilibrium
invariant manifold
initial approximation
corrections
(1-Pc)J = 0
invariance equation
Newton iterations
0
0))()(1(
xP
xx
JxJP
C
C
xxx '
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Invariant grid
EQUILIBRIUM
tangent space
tangent space
J
invariance defect is corrected for every node independently
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Invariant grid
Jc
J
Tx
x0+ker PC
)()()( 000 xxxx
JxJ
)( 0xJ)()1( 0xJPc
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Growing Invariant Flag
Phase space
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Entropic scalar product
),(
),(,
2
2
Hxxx
Hyxyxc
GH
0 1
2
-1-2
equilibrium
natural parameterentropy
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Hydrogen burning model reaction
1: H2 2H
2: O2 2O
3: H2O H + OH
4: H2 + O H + OH
5: O2 + H O + OH
6: H2 + O H2O
21 k12 k13 k
34 10k
35 10k
26 10k
Conservation laws:
2cH2 +2cH2O+cH+cOH = bH
2cO2+cH2O+cO+cOH = bO
H2
H
O2 H2O
O
OH
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One-dimensional dynamics
equilibrium
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Separation of times
is the eigen valueof symmetrised
matrixc
J
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Two-dimensional dynamics
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Visualizing functions:concentration of H
“Fast” coordinate
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Visualizing functions:concentration of H2
“Slow” coordinate
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24 - European conference on complex systems, Paris, November 2005 -
Visualizing functions:concentration of OH
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Visualizing functions:Entropy and entropy production
Entropy Entropy production
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Visualizing functions:Separation of relaxation times
2/1 3/2
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Open system as “closed system in a flow”
),()( txJxJdt
dxex
closed system
flow
• Entropy does not increase everywhere
• Non-uniqueness of stationary states, auto-oscillations, etc.
• inertial manifold often exists
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Zero-order approximation
),()( txJxJdt
dxex
)(xJdt
dxConstruct the invariant manifold for
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First-order approximation
),()( txJxJdt
dxex
fastex
slowexexxexxex JJJPJPJ )1(
0
0)()1(
xP
JxxJP
x
fastexx
New invariance equation:
“Fast” and “slow” flow:
’
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Conclusions
Invariant grids: constructive method for chemical kineticsclass of dissipative systemsextension to open systems
Use of thermodynamics: metrics in the phase spaceunique thermodynamic projector
Possibility to visualize and explore system dynamics globally
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Papers
Gorban A, Karlin I, Zinovyev A.
Constructive Methods of Invariant Manifolds for Kinetic Problems 2004. Physics Reports 396, pp.197-403.
Gorban A, Karlin I, Zinovyev A.
Invariant Grids for Reaction Kinetics2004. Physica A, V.333, pp.106-154
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People
Doctor Iliya Karlin ETH, Zurich
Professor Alexander GorbanUniversity of Leicester, UK