improved gene expression programming to solve the inverse problem for ordinary differential...
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Improved Gene Expression Programming to Solve the Inverse Problem for Ordinary Differential Equations
Kangshun Kangshun LiLi Professor, Ph.DProfessor, Ph.D
College of Information, College of Information,
South China Agricultural University, ChinaSouth China Agricultural University, China
Hong KongHong Kong
December 6, 2014December 6, 2014
Outline of My Talk
Introduction
Inverse problems for ODEs
Improved GEP for the inverse problem of ODEs
Experiments
Conclusions and future research
Outline of My Talk
Introduction
Inverse problems for ODEs
Improved GEP for the inverse problem of ODEs
Experiments
Conclusions
1. Introduction
Dynamic systems Their dominant features are complicated or non-linear.
They often change over time.
How to predict them?
Stock MarketWeather Forecast Population Trends
Features of such dynamic systems It’s difficult to find the functional relations among variables in the
complicated changing processes. It’s possible to find out the change rate or differential coefficient of
some variables.
1. Introduction
Ordinary Differential Equations (ODEs)
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1. Introduction
Inverse problems How to establish the ODEs based on previous data.
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0.00 1.000000 1.000000 1.000000
0.01 1.010050 1.030403 1.040555
0.02 1.020201 1.061627 1.082236
0.03 1.030455 1.093692 1.125074
0.04 1.040811 1.126619 1.169095
Canonical problem
Inverse problem
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1. Introduction
Example
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1. Introduction
Challenges of solving inverse problems With a few observed data, it’s difficult to create ODEs. It’s difficult to determine the model structure. It’s difficult to adjust parameters.
Outline of My Talk
Introduction
Inverse problems for ODEs
Improved GEP for the inverse problem of ODEs
Experiments
Conclusions
A dynamic system can be expressed by: , and t denotes time.
A series of observed data collected at times . .
2. Inverse problems for ODEs
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Approaches to solving inverse problems of ODEs: Linear modeling Autoregressive model
Moving Average model
Autoregressive Moving Average model
Pre-selected based on experience
Faced with complex data, it’s hard to select the right differential equation model.
Evolutionary modeling Genetic Programming (GP)
Gene Expression Programming (GEP)
2. Inverse problems for ODEs
Non-linear dynamical systems
Outline of My Talk
Introduction
Inverse problems for ODEs
Improved GEP for the inverse problem of ODEs
Experiments
Conclusions
GEP Based on genome and phenomena.
Refer to the gene expression rule in the genetics.
Have advantages of both GP and GA.
GEP chromosome Q ×+×a×Q a a ba b b a a b a b a a b
× stands for the multiplication operation.
Q represents square root operation.
Segment without underline belongs to the Head.
Underlined segment is the Tail.
3. Improved GEP for the inverse problem of ODEs
An example of GEP coding Each gene describes an ODE.
A chromosome describes an ODE group.
- + * x1 ^ 2 x3 x2 2 x1 x2 + + x3 x1 x2 2 4 x1 x2 x3 8 + + 3 x1 ^ t 3 6 x1 x2 x3
|————gene1———| ————gene2——— | ———gene3——— |
3. Improved GEP for the inverse problem of ODEs
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+ *
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The flowchart of GEP algorithm for the ODEs inverse problem
3. Improved GEP for the inverse problem of ODEs
Share the same evolution framework with other evolutionary algorithms!
Initialize population Set control parameters
termination symbol
Functional set head
head length: 8
tail lengths : 9
gene number: 3
Create initial population
genetic : *+-1q*+3201321023
chromosome: *+-1q*+3201321023*-*1+*+*202312032*+*1q*+3210301323
population size : 50
3. Improved GEP for the inverse problem of ODEs
Fitness evaluation and chromosomes ranking
3. Improved GEP for the inverse problem of ODEs
Calculate genes Traditional method
Convert the chromosome into the expression tree, and then solve it
via stacks.
Our approach
Gene Read & Compute Machine (GRCM) algorithm.
The procedure of converting the chromosome into the expression
tree can be avoided.
3. Improved GEP for the inverse problem of ODEs
An example of GRCM algorithm
3. Improved GEP for the inverse problem of ODEs
+ - sin a b c d e f
+ - sin a b c
+ - sin a b c
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+ - sin(c) a b
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+ a-b sin(c)
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(a-b)+sin(c)
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Generate training data and prediction data The Runge-Kutta is adopted in this phase, which is a iteration method for
simulating the ODE solutions. The RK4 formula is shown:
3. Improved GEP for the inverse problem of ODEs
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Construction of fitness function It is constructed by the differences between X and X*, i.e. ∆=‖X-X* ‖
3. Improved GEP for the inverse problem of ODEs
Genetic operators Selection
The Roulette selection is adopted, which means that he better the fitness, the greater probability an individual is reproduced to the next generation.
Mutation
The Head can be mutated into any function or terminal symbol, while the Tail can only be mutated into the terminal symbol
Transportation
Insertion Sequence Transposition
Root Insertion Sequence Transposition
Gene Transposition
3. Improved GEP for the inverse problem of ODEs
Reconstruction
Single point restructuring
Double-point restructuring
Gene restructuring
Termination conditions
The maximum number of generations is reached.
The fitness of the best individual reaches a predefined value, or it is unchanged for a predefined number of generations.
3. Improved GEP for the inverse problem of ODEs
Outline of My Talk
Introduction
Inverse problems for ODEs
Improved GEP for the inverse problem of ODEs
Experiments
Conclusions
Four different datasets are used.
GP and the basic GEP are involved in the comparison.
Three different metrics are compared. Training standard deviation
Prediction
Running time
4. Experiments
Four different datasets are used.
GP and the basic GEP are involved in the comparison.
Three different metrics are compared. Training standard deviation
Prediction
Running time
4. Experiments
Datasets
4. Experiments
Results
4. Experiments
Running time
Almost performs similar with the standard GEP.
Significantly less than the GP algorithm.
Stability
Better than using the GP algorithm, particularly for complex problems.
Prediction accuracy
Better than standard GEP for each dataset
Also be superior to the standard deviation of GP algorithm
4. Experiments
An improved GEP is proposed to solve the inverse problem of ODE
Overcome the shorting of evolution operations in the recessive segment.
Provide a better way to model dynamic systems.
5. Conclusions and future research
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Weather Forecast Population Trends
Thank you!Thank you!Q&AQ&A