Integrability, neural networks, and the empirical modelling of dynamical systems Oscar Garcia...
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Integrability, neural networks, and the empirical modelling of
dynamical systems
Oscar Garcia
forestgrowth.unbc.ca
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Outline Dynamical Systems,
forestry example The multivariate
Richards model Extensions, Neural
Networks Integrability, phase
flows Conclusions
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An engineer thinks that his equations are an approximation to reality.
A physicist thinks reality is an approximation to his equations.
A mathematician doesn’t care.
Anonymous
Modelling
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All models are wrong, but some are useful.
G. E. P. Box
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Dealing with Time Processes, systems
evolving in time Functions of time Rates (Newton) System Theory
(1960’s) Control Theory,
Nonlinear Dynamics
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Dynamical systems
Instead of
State:Local transition function (rates):
: inputs (ODE)Output function:
Copes with disturbances
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Example (whole-stand modelling)
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3-D
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3-D
SiteEichhorn (1904)
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Integration
No
(Global transition function)
Group:
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3-D
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Equation forms? Theoretical. Empirical. Constraints Simplest, linear:E.g., with Why not
Average spacing? Mean diameter? Volume or biomass? Relative spacing? ... ?
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Multivariate Richards
where
The (scalar) Bertalanffy-Richards:
with
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ExamplesRadiata pine in New Zealand (García, 1984)
t scaled by a site quality parameter
Eucalypts in Spain – closed canopy (García & Ruiz, 2002)
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Parameter estimation
Stochastic differential equation:
adding a Wiener (white noise) process.
Then get the prob. distribution (likelihood function), and maximize over the parameters
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Variations / extensionsMultipliers for site, genetic improvementAdditional state variables: relative closure,
phosphorus concentrationThose variables in multipliers:
with a “physiological time” such that
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Where to from here?
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Transformations to linear
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Transformations to constant
V. I. Arnold “Ordinary Differential Equations”. The MIT Press, 1973.
“Invariants” within a trajectory or flow line
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Integrable systems
Integrable systems?
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Integrability
Diffeomorphic to a constant field <=> Integrable?
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Modelling Assumption: For a “wide enough” class of
systems there exists a smooth one-to-one transformation of the n state variables into n independent invariants
Model (approximate) these transformations “Automatic” ways of doing this?
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Artificial Neural Networks
Problem: Not one-to-one
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The multivariate Richards network
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The multivariate Richards network
Estimation Regularization, penalize overparameterization “Pruning”
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Integration
No
(Global transition function)
Group:
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Modelling the global T.F. (flow)
No
(Global transition function)
Group:
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Arnold
No
(Global transition function)
Group:
“Phase flow”
“one-parameter group of transformations”
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3-D
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In forest modelling... “Algebraic difference equations”, “Self-
referencing functions” Examples (A = age) :
1-D. Often confuse integration constants with site-dependent parameters
But, perhaps it makes sense, after all?
Clutter et al (1983)
Tomé et al (2006)
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Conclusions / Summary Dynamical modelling with ODEs seem
natural, although it is rare in forestry Multivariate Richards, an example of
transformation to linear ODEs, or to invariants
More general empirical transformations to invariants: ANN, etc.
Modelling the invariants themselves, rather than ODEs