Integrability, neural networks, and the empirical modelling of

dynamical systems

Oscar Garcia

forestgrowth.unbc.ca

Outline Dynamical Systems,

forestry example The multivariate

Richards model Extensions, Neural

Networks Integrability, phase

flows Conclusions

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

All models are wrong, but some are useful.

G. E. P. Box

Dealing with Time Processes, systems

evolving in time Functions of time Rates (Newton) System Theory

(1960’s) Control Theory,

Nonlinear Dynamics

Dynamical systems

Instead of

State:Local transition function (rates):

: inputs (ODE)Output function:

Copes with disturbances

Example (whole-stand modelling)

3-D

3-D

SiteEichhorn (1904)

Integration

No

(Global transition function)

Group:

3-D

Equation forms? Theoretical. Empirical. Constraints Simplest, linear:E.g., with Why not

Average spacing? Mean diameter? Volume or biomass? Relative spacing? ... ?

Multivariate Richards

where

The (scalar) Bertalanffy-Richards:

with

ExamplesRadiata pine in New Zealand (García, 1984)

t scaled by a site quality parameter

Eucalypts in Spain – closed canopy (García & Ruiz, 2002)

Parameter estimation

Stochastic differential equation:

adding a Wiener (white noise) process.

Then get the prob. distribution (likelihood function), and maximize over the parameters

Variations / extensionsMultipliers for site, genetic improvementAdditional state variables: relative closure,

phosphorus concentrationThose variables in multipliers:

with a “physiological time” such that

Where to from here?

Transformations to linear

Transformations to constant

V. I. Arnold “Ordinary Differential Equations”. The MIT Press, 1973.

“Invariants” within a trajectory or flow line

Integrable systems

Integrable systems?

Integrability

Diffeomorphic to a constant field <=> Integrable?

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?

Artificial Neural Networks

Problem: Not one-to-one

The multivariate Richards network

The multivariate Richards network

Estimation Regularization, penalize overparameterization “Pruning”

Integration

No

(Global transition function)

Group:

Modelling the global T.F. (flow)

No

(Global transition function)

Group:

Arnold

No

(Global transition function)

Group:

“Phase flow”

“one-parameter group of transformations”

3-D

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)

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