1 / 31

Integrability, neural networks, and the empirical modelling of dynamical systems

Oscar Garcia. Integrability, neural networks, and the empirical modelling of dynamical systems. forestgrowth.unbc.ca. Outline. Dynamical Systems, forestry example The multivariate Richards model Extensions, Neural Networks Integrability, phase flows Conclusions. Modelling.

doane
Download Presentation

Integrability, neural networks, and the empirical modelling of dynamical systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Oscar Garcia Integrability, neural networks, and the empirical modelling of dynamical systems forestgrowth.unbc.ca

  2. Outline • Dynamical Systems, forestry example • The multivariate Richards model • Extensions, Neural Networks • Integrability, phase flows • Conclusions

  3. Modelling 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

  4. All models are wrong, but some are useful.G. E. P. Box

  5. Dealing with Time • Processes, systems evolving in time • Functions of time • Rates (Newton) • System Theory (1960’s) • Control Theory, Nonlinear Dynamics

  6. Dynamical systems Instead of • State: • Local transition function (rates): : inputs (ODE) • Output function: Copes with disturbances

  7. Example (whole-stand modelling)

  8. 3-D

  9. 3-D Site Eichhorn (1904)‏

  10. Integration No (Global transition function) Group:

  11. 3-D

  12. Equation forms? Theoretical. Empirical. Constraints Simplest, linear: E.g., with Why not Average spacing? Mean diameter? Volume or biomass? Relative spacing? ... ?

  13. Multivariate Richards where The (scalar) Bertalanffy-Richards: with

  14. Examples Radiata pine in New Zealand (García, 1984) t scaled by a site quality parameter Eucalypts in Spain – closed canopy (García & Ruiz, 2002)

  15. Parameter estimation Stochastic differential equation: adding a Wiener (white noise) process. Then get the prob. distribution (likelihood function), and maximize over the parameters

  16. Variations / extensions • Multipliers for site, genetic improvement • Additional state variables: relative closure, phosphorus concentration • Those variables in multipliers: with a “physiological time” such that

  17. Where to from here?

  18. Transformations to linear

  19. Transformations to constant V. I. Arnold “Ordinary Differential Equations”. The MIT Press, 1973. “Invariants” within a trajectory or flow line

  20. Integrable systems Integrable systems?

  21. Integrability Diffeomorphic to a constant field <=> Integrable?

  22. 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?

  23. Artificial Neural Networks Problem: Not one-to-one

  24. The multivariate Richards network

  25. The multivariate Richards network • Estimation • Regularization, penalize overparameterization • “Pruning”

  26. Integration No (Global transition function)‏ Group:

  27. Modelling the global T.F. (flow) No (Global transition function) Group:

  28. Arnold No “Phase flow” (Global transition function) Group: “one-parameter group of transformations”

  29. 3-D

  30. 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)

  31. 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

More Related