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Lecture 5 II Function Approximation

Lecture 5 II Function Approximation. Data driven function approximation MLP network Unconstrained optimization. Objective function. Unconstrained Optimization Non-polynomial, High dimensional function Unknowns: 3M+1. Unconstrained Optimization. Gradient method Gauss-Newton method

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Lecture 5 II Function Approximation

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  1. Lecture 5 II Function Approximation • Data driven function approximation • MLP network • Unconstrained optimization 數值方法

  2. Objective function • Unconstrained Optimization • Non-polynomial, • High dimensional function • Unknowns: 3M+1 數值方法

  3. Unconstrained Optimization • Gradientmethod • Gauss-Newton method • Levenberg-Marquardt method 數值方法

  4. Unconstrained Optimization 數值方法

  5. Iterative approach • Initialize with i=0 • Determine • Update network parameters • If halting condition holds, exit • otherwise i=i+1, go to step 2 數值方法

  6. Flow Chart i=0; guess ~ halting condition 數值方法

  7. Gradient method • Derive the derivative of ES with respect to • Substitute to 數值方法

  8. Install NNSYSID • Install • Download • Set path to recruit the directory where NNSYSID.zip is extracted The NNSYSID Toolbox learn_MLP By Levenberg Marquardt method 數值方法

  9. Data creation Get a function fstr=input('input a function: x.^2+cos(x) :','s'); fx=inline(fstr); range=2*pi; x=linspace(-range,range); y=fx(x);max_y=max(y); plot(x,y/max_y); Function plotting 數值方法

  10. Sampling N=input('keyin sample size:'); x=rand(1,N)*2*range-range; n=rand(1,N)*0.1-0.05; y=fx(x)/max_y+n; plot(x,y,'.'); 數值方法

  11. MLP learning M=5; [a,b,r]=learn_MLP(x',y',M); MLP learning x=linspace(-range,range); y=eval_MLP(x,r,a,b,M); plot(x,y,'r') MLP evaluation 數值方法

  12. Data Driven Function Approximation source code mean_square_error2.m eval_MLP.m 數值方法

  13. Example >> fa1d input a function: x.^2+cos(x) :3*x.^2+2*x+1 keyin sample size:300 keyin the number of hidden units:5 數值方法

  14. Example >> fa1d input a function: x.^2+cos(x) :cos(x)+sin(x) keyin sample size:300 keyin the number of hidden units: 數值方法

  15. High-dimensional function approximation Given Find a parametric function to minimize 數值方法

  16. Network b1 tanh r1 b2 1 a1 tanh r2 a2 x bM ……. rM aM tanh rM+1 1 數值方法

  17. Example Black box plot2d.m observations Create paired data sampling2.m y=cos(x1+x2) DDFA M=input('M:'); [a,b,r]=learn_MLP(x',y',M); 數值方法

  18. D-dimensional function approximation source codes eval_MLP2.m 數值方法

  19. Example >> fa2d input a 2D function: x1.^2+x2.^2+cos(x1) :x1.^2-x2.^2 keyin sample size:300 keyin the number of hidden units:20 數值方法

  20. Numerical results MSE for training data 0.008656 ME for training data 0.065931>> 數值方法

  21. Example 數值方法

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