Invariant grids: method of complexity reduction in reaction networks

Andrei ZinovyevInstitut Curie, Paris

Institut des Hautes Études Scientifiques

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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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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