Improved Gene Expression Programming to Solve the Inverse Problem for Ordinary Differential Equations Kangshun Li Professor, Ph.D Professor, Ph.D College of Information, South China Agricultural University, China Hong Kong December 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)

1. Introduction Inverse problems  How to establish the ODEs based on previous data. Canonical problem Inverse problem

1. Introduction Example

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

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. 3. Improved GEP for the inverse problem of ODEs

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* chromosome ： *+-1q* *-*1+*+* *+*1q* population size ： 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 +-sinabcdef +- abc

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

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 Stock Market Weather ForecastPopulation Trends

Thank you! Q&A