Invariant grids: method of complexity reduction in reaction networks

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Invariant grids method of complexity reduction
in reaction networks

• Andrei Zinovyev
• Institut Curie, Paris
• Institut des Hautes Études Scientifiques

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Stoichiometric equations
as1A1 asnAn ? bs1A1 bsnAn
n number of species, s number of reactions
cn
c1
c2
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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
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Positively Invariant Manifold
fast motion
slow motion
Steady state
W
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
x?RN detailed description y?Rm
macroscopic description. mltltN
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Why for do we need invariant manifold?
Dynamics visualization
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Other useful non-invariant manifolds

• Quasy steady-state
• Fast variables are steady
• Quasi-equilibrium
• Manifolds maximizing entropy
• Intrinsic low-dimensional manifold
• Decomposition of Jacobian fields

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Projector Pc on (some) manifoldinduces new
(reduced) dynamics
induced dynamics
?
tangent space
J
Pc J
W
TxW
? (1-Pc)J - invariance defect
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Quasi-equilibrium manifoldis not necessarily
invariant
entropy S?max
macroscopic (reduced) variables
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Class of dissipative systems
Lyapunov function
G
c
ceq
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Thermodynamic projector
J
Pc J
The induced dynamics is dissipative only if
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Correction of invariance defect
invariant manifold
corrections
C1
invariance equation
1.0
(1-Pc)J 0
0.8
0.6
Newton iterations
equilibrium
0.4
0.2
0.20
0.10
0.15
0.05
C3
initial approximation
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Invariant grid
tangent space
EQUILIBRIUM
J
tangent space
invariance defect is corrected for every node
independently
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Invariant grid
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Growing Invariant Flag
Phase space
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Entropic scalar product
2
1
0
equilibrium
-1
natural parameter entropy
-2
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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
Conservation laws
2cH2 2cH2OcHcOH bH 2cO2cH2OcOcOH bO
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One-dimensional dynamics
equilibrium
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Separation of times
l is the eigen value of symmetrised matrix
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Two-dimensional dynamics
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Visualizing functionsconcentration of H
Fast coordinate
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Visualizing functionsconcentration of H2
Slow coordinate
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Visualizing functionsconcentration of OH
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Visualizing functionsEntropy and entropy
production
Entropy production
Entropy
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Visualizing functionsSeparation of relaxation
times
?3/?2
?2/?1
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Open system as closed system in a flow
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
Construct the invariant manifold W for
W
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First-order approximation
Fast and slow flow
New invariance equation
W
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Conclusions
Invariant grids constructive method for chemical
kinetics class of dissipative systems extension
to open systems Use of thermodynamics metrics
in the phase space unique 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 Kinetics 2004. Physica A, V.333,
pp.106-154
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People
Professor Alexander Gorban University of
Leicester, UK
Doctor Iliya Karlin ETH, Zurich