Radial Basis Function Network Using Orthogonal Least Squares-Genetic Algorithm Learning (RBFN using OLSGA) Christopher Katinas February 16, 2016
Overview Problem Definition Training Results Test Results Graphical Approximation of Results Conclusions
Insulated Problem Definition Modeling of heat transfer processes requires thermo-physical properties Experimental setup to determine thermal conductivity (κ) Heat source/sink applied to block of known properties Blocks are insulated on remaining sides Thermally conductive silver grease at interfaces, but resistance exists Redundant thermocouples inserted at interface to measure temperature Very expensive and thermocouple beads modify the results Heating is performed to steady state (based on thermocouples) Utilize resistive network approximation to determine κ Insulated = Thermocouple Applied Heat Source Q Block A (κ A is known) Test Piece Block B (κ B is known) Applied Heat Sink T sp
Mathematical Model of Process Q appl T source RR aa = LL aa kk aa RR tt = LL tt kk tt RR bb = LL bb kk bb T a T b T sink Interface Jump = can exist (random noise 5K) Function Inputs: Q appl & T sink QQ aaaaaaaa RR tt + RR bb = TT aa TT ssssssss Not exciting Thermal Conductivity (W/m-K) Desired Output: k t 18 16 14 12 10 8 6 4 2 0 Temperature Dependence of Thermal Conductivity on Material XYZ 0 200 400 600 800 1000 1200 Temperature (K) QQ aaaaaaaa RR tt (TT aa, TT bb ) + RR bb kk WW mmmm = AATT3 + BBTT 2 + CTT+ DD A 8.00E-08 B -1.20E-04 C 3.00E-02 D 14.8 = TT aa TT ssssssss Added non-linearity to the equation T in [K] TT = TT aa + TT bb 2
OLSGA Pseudocode 1. Create training and verification data 2. Initialize system with zero hidden nodes 3. Invoke GA & Random RBFN a. Determine the error reduction from selected means and standard deviation (fitness function) b. Identify mean and standard deviation with highest effect and store for future use c. Adjust overall response vector by the projection of response vector from found parameters of GA 4. Calculate NDEI for OLSGA and Random RBFN 5. Post-process data to review RBFN function 6. Repeat steps 3-5 until maximum number of nodes is reached (could use error tolerance, if desired)
Network Training Heat Transfer Compare NDEI trends to literature trends Rapid decrease in NDEI when most critical nodes are found! Trained with 500 data points verified with 500 points Trained with 2500 data points verified with 1500 points Lee C. W. and Shin Y. C. Growing Radial Basis Function Networks Using Genetic Algorithm and Orthogonalization. International Journal of Innovative Computing, Information and Control. 2009 Nov 1;5(11):3933-48.
Evolution of the RBFN (n=1 to 12) n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12
Test Results Uniform Grid was created to plot expected function (50 points per variable) with 50 hidden nodes Q appl = 120 to 9980 W, T sink = 291 to 849 K Using verification data, reviewed uncertainty in thermal conductivity Average Error = 0.104 W/m-K with 50 hidden nodes
Graphical Representation of Function Actual Function Hidden Nodes=5 Hidden Nodes=15 Hidden Nodes=50 Temperature in K, Q in W, F(Q,T) in W/m-K
Conclusions Training with OLSGA is very effective for RBFNs Highest error reduction nodes are found first Fewer required nodes when compared to randomly selected hidden nodes Caution needs to be used for systems with excessive noise in the training data. Larger training sets allow for better training GA performance affects number of hidden nodes Size matters Population size drastically affected GA Population 10 Population 25 Population 100
Extra Slides
GA Methodology Genetic algorithm (~180 Matlab lines) Cross-over (0.95) Up to three crossovers per parent pair Mutation Rate (0.005) Population size/generation control Multiple Epochs Allowed
Function ff xx 1, xx 2 = 3(1 xx 1 ) 2 exp xx 2 1 xx 2 + 1 2 WolframAlpha.com Verification of the GA Goal: Find maximum value of the function Verify Functionality of Self-Build Genetic Algorithm 10 xx 1 5 xx 1 3 xx 5 2 exp xx 2 1 xx 2 2 1 3 exp xx 2 2 xx 1 + 1 2 Matlab The maximum of the function was found to be 8.1062. The maximum of the function was found to be at (-0.0095674, 1.5825). Binary conversion caused problems with more than two variables. Thus, Matlab GA toolbox was utilized for actual training.
Evolution of the RBFN for Test Function n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12
Using Exponential Function Compare NDEI from Exponential Function to Literature Trends Rapid decrease in NDEI when most critical nodes are found! Trained with 500 data points verified with 500 points Trained with 1600 data points verified with 841 points Lee C. W. and Shin Y. C. Growing Radial Basis Function Networks Using Genetic Algorithm and Orthogonalization. International Journal of Innovative Computing, Information and Control. 2009 Nov 1;5(11):3933-48.
Evolution of the RBFN (n=1 to 12) n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12