Opened 22 months ago

Last modified 2 months ago

## #2925 new feature request

# Support for AutoDiff-based optimization of coefficients for symbolic regression models with auto-regressive variables

Reported by: | gkronber | Owned by: | gkronber |
---|---|---|---|

Priority: | medium | Milestone: | HeuristicLab 4.x Backlog |

Component: | Problems.DataAnalysis.Symbolic | Version: | |

Keywords: | Cc: |

### Description

We aim to model dynamical systems using a set of differential equations for observed variables of the system. The proposed approach is to simulate the system forward in time using numerical integration (e.g. Runge-Kutta). If we want to combine this with our gradient-based optimization of coefficients then it is necessary to calculate gradients using AutoDiff over all steps of the numeric integration. For this it is probably necessary to extend the class for optimization of coefficients (aka ConstantsOptimizer) and to provide a symbol for auto-regressive variables where it is possible to set a gradient value.

E.g. for a uni-variate model: y'(t) = a * y(t)

we have:

- y'(0) = a y(0)
- y'(1) = a y(1) = a (y(0) + a y'(0)) = a y(0) + a²y'(0)

using a naive integration step

and:

dy'(0)/da = y(0)

which we need to calculate the dy'(1) / da

i.e.

dy'(1)/da = y(0) + 2 a dy'(0)/da = y(0) + 2 a y(0)

and so on for each time step.

### Change History (56)

### comment:1 Changed 22 months ago by gkronber

### comment:2 Changed 22 months ago by gkronber

r15964: first exploratory implementation of AutoDiff for modelling of dynamical systems

### comment:3 Changed 22 months ago by gkronber

### comment:4 Changed 22 months ago by gkronber

r15970: added expressions for latent variables and allow parameterization of the number of integration steps

### comment:5 Changed 19 months ago by lkammere

r16126: added constants values to tree in result view and minor bugfix when initializing problem.

### comment:6 Changed 19 months ago by gkronber

### comment:7 Changed 19 months ago by gkronber

- added support for multiple training episodes,
- added simplification of models,
- fixed a bug in the normalization based on target variable variance

### comment:8 Changed 19 months ago by gkronber

r16154: bug fix for the extraction of target values (missed when introducing episodes)

### comment:9 Changed 19 months ago by gkronber

r16155: allow tuning individual parameter vectors for episodes (using the same structure)

### comment:10 Changed 18 months ago by gkronber

TODO: show line chart of integrated latent variables.

### comment:11 Changed 18 months ago by gkronber

r16214: added datasets from supplementary material for "distilling free-form laws ..." (oscillator, pendulum, double pendulum, ...)

### comment:12 Changed 18 months ago by gkronber

r16215 added sin and cos functions

### comment:13 Changed 18 months ago by gkronber

- added comments about parameter identification for differential equation models
- added source code of cvodes library (part of sundials) which provides functionality to calculate gradients for the parameters of partial differential equation models efficiently using the 'adjoint state method'.
- added compiled version of cvodes

### comment:14 Changed 18 months ago by gkronber

r16225: first working version of CVODES integration

### comment:15 Changed 18 months ago by gkronber

r16226: added vector access methods

### comment:16 Changed 18 months ago by gkronber

r16227: changed test case to linear oscillator

### comment:17 Changed 18 months ago by gkronber

r16245: worked on integration of CVODES library

### comment:18 Changed 18 months ago by gkronber

r16246: working example with forward sensitivity analysis

### comment:19 Changed 18 months ago by gkronber

r16248: extracted CVODES external methods into a separate class

### comment:20 Changed 18 months ago by gkronber

r16249: extracted Vector into a separate class

### comment:21 Changed 18 months ago by gkronber

r16250: preparations to switch between HL ODE solver and CVODES

### comment:22 Changed 18 months ago by gkronber

r16251: implemented interface to CVODES solver with forward sensitivity calculation.

### comment:23 Changed 18 months ago by gkronber

r16253: fixed memory leak and fixed bug in determination of integration time span

### comment:24 Changed 18 months ago by gkronber

r16254: made usage of x86 native library explicit.

### comment:25 Changed 18 months ago by gkronber

r15256: added x64 version of CVODES as well as dispatching code for x86 and x64.

### comment:26 Changed 17 months ago by gkronber

r16268: created a separate algorithm (similar to NLR but for parameter identification of ODE systems) for debugging of the parameter optimization and fixed a problem with the order of items in the checkedItemCollection for TargetVariables

### comment:27 Changed 17 months ago by gkronber

r16329: made several extensions in relation to blood glucose prediction

- added sqr function,
- added outputs for latent variables (linechart and model),
- added optimization of initial values for latent variables (for each episode separately)
- TODO: test with CVODES (so far only our own integration scheme has been tested)

### comment:28 Changed 17 months ago by gkronber

TODO: for testing: estimate parameters for model (including initial values for hidden variables) based on the immediately previous data points.

### comment:29 Changed 16 months ago by gkronber

r16386: merged changes r15972:16382 (HEAD) from trunk to branch

### comment:30 Changed 16 months ago by gkronber

r16395: removed unnecessary usings

### comment:31 Changed 16 months ago by gkronber

r16398: small changes

### comment:32 Changed 16 months ago by gkronber

r16399: solution class and solution view

### comment:33 Changed 16 months ago by gkronber

r16400: small gui improvement

### comment:34 Changed 14 months ago by gkronber

r16597: made some adaptations while debugging parameter identification for dynamical models

### comment:35 Changed 14 months ago by gkronber

r16599: changed code to use LM instead of LBFGS and removed standardization

### comment:36 Changed 14 months ago by gkronber

r16600: made some simplifications (Vector) to aid debugging.

### comment:37 Changed 14 months ago by gkronber

r16601: refactored dynamical modelling code

### comment:38 Changed 14 months ago by gkronber

r16602: write back optimized constants to trees

### comment:39 Changed 14 months ago by gkronber

r16603: scaling of residuals to target variance and update constant values directly in the tree

### comment:40 Changed 14 months ago by gkronber

r16604: efficiency improvements

### comment:41 Changed 14 months ago by gkronber

- added crossover probability (important for multi-encoding in this case)
- re-added scaling of targets

### comment:42 Changed 13 months ago by gkronber

r16652: allow arbitrary number of arguments to +,-,*,/ operators, fixed bug in single-argument division

### comment:43 Changed 13 months ago by gkronber

r16653: added support for fixed numeric parameters as "constant variables"

### comment:44 Changed 13 months ago by gkronber

r16660: re-introduced partial support for latent variables

### comment:45 Changed 13 months ago by gkronber

### comment:46 Changed 13 months ago by gkronber

r16663: adapted to work with new persistence

### comment:47 Changed 13 months ago by gkronber

r16664: generate individual solutions for each of the trees to allow individual analysis of functions e.g. using PDP (only works without latent variables so far)

### comment:48 Changed 13 months ago by gkronber

r16665: fixed a problem which prevents loading persistence files

### comment:49 Changed 12 months ago by gkronber

r16785: fixed a bug in AQ evaluation

### comment:50 Changed 12 months ago by gkronber

r16786: fixed a problem with solutions for latent variables

### comment:51 Changed 11 months ago by gkronber

r16892: merged r16661:16890 from trunk to branch

### comment:52 Changed 11 months ago by gkronber

r16893: allow separate configuration of const opt steps for pre-tuning and the full ODE. Allow weighted combination of fitness using pretuning NMSE and the full ODE NMSE

### comment:53 Changed 11 months ago by gkronber

### comment:54 Changed 11 months ago by gkronber

- Add problem instance provider and instances.
- Use penalized regression splines for calculation of numeric differences (for pre-tuning).

### comment:55 Changed 7 months ago by gkronber

r17246: merged r17037:17242 from trunk to branch

### comment:56 Changed 2 months ago by gkronber

r17409: merged r17255:17402 from trunk to branch

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r15962: created branch for ticket