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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 doesnt 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 (1960s) 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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Site Eichhorn (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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Examples Radiata 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 / extensions Multipliers for site, genetic improvement Additional state variables: relative closure, phosphorus concentration Those 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, 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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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

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