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Differential Equations

Genetic Programming: Finding Approximate Analytic Solution to First-Order Ordinary Differential Equations. Differential Equations. Initial Value Problem: Find y=f(x) if y’ = f( x,y ) at y(a)= b (1) Closed-form solution: explicit formula -y’ = y at y(0)=1 (Separable)

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Differential Equations

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  1. Genetic Programming:Finding Approximate Analytic Solution to First-Order Ordinary Differential Equations

  2. Differential Equations Initial Value Problem: Find y=f(x) if y’ = f(x,y) at y(a)=b (1)Closed-form solution: explicit formula -y’ = y at y(0)=1 (Separable) Ans: y = e^x (2) Numerical solution: y’ = (x*y)/(x+y) + y^2 at y(0)=1 Can’t solve explicitly… So we’ll use numerical methods: (a) Dormand-Prince  (b) ode45: returns [x,y] values

  3. Regression / Curve-fitting • Given data points, find best fitting curve • Normal regression finds coefficients of some predefined function (ie linear regression) • Symbolic Regression finds function and coefficients • Since normal requires human intuition, we’ll use symbolic regression using genetic programming

  4. Genetic Programming Basics Tree Structure Function Nodes from a Function Set Terminal Nodes (numbers, variables) from a Terminal Set Create Random Function Trees Making the function: y = x*sin(3.63/(2+x)) 

  5. Function Tree = Individual • Use Polish Notation • Operator comes first. Ex: 5 + x – (x * 8))  + 5 – x * x 8 Quiz: + 3 * 6 – 2 1 = ? • Syntax-preserving: binary or unary functions? • Binary: takes two nodes • Unary: takes one node

  6. Fitness Evaluation Evaluate error at each point Fitness= -Total Sum of Error If we want greater accuracy, we will want to evaluate more points in domain We may also want to limit the maximum error between any sample point and the function

  7. Reproduction • Random tournament to select who mates • Randomly select two individuals in population • The one with the best fit is parent 1 • Randomly select crossover or mutation • If mutate, mutate parent 1 • Else find parent 2 and crossover • Repeat reproduction until Population size is filled

  8. Reproductive Operator: Crossover • Crossover (GP) Crossover (DNA analogy) Parent 1: Parent 2: (a+b*sqrt(b)) / sqrt(b+b) a*(((b/b)/a) * sqrt(b+b)) Child: a+(b*(b/b)/a) / sqrt(b+b) ( Throw away second child)

  9. Reproductive Operator: Mutation Randomly select node in tree to mutate Randomly select a node of the same type Mutate node Mutation gets us out of pitfalls

  10. Parameters • Determine Number of Data Points in Domain (Matlab) • Step size: 0.001 • Determine Function Set • F = {sin, cos, e^, /, *, +, -} • Determine Terminal Set • T = {x, pi/2, 100 random numbers between -5 and 5} • Create a Random Population of Function Trees • 10,000 indiv. w/ initial depth of 5 • Set Minimum Tolerance (error) • Tolerance: 0.01 • Set Maximum Num of Generations • Max num of generations: 600 • Set Crossover and Mutation rate • Crossover: 80% Mutation: 1-Crossover = 20%

  11. Results: Finding a Closed-form Solution • y’=-y*cos(x)  Solvable  Ans: y=1 / e^(sin(x)) • Using Genetic Programming • Pop size: 1000 • Domain [0 , 5] • Problem • Protected division

  12. Program Evaluation: Finding: y=1 / e^(sin(x))

  13. Results: Finding an Approximation • y’=cos(2*x*y^2) • No closed form solution • Polyfit fails in matlab

  14. Results: y’ = cos(2*x*y^2) …the explicit solution is…

  15. Program Evaluation …the explicit solution is…

  16. The Approximate Solution y = e^( ((((x / cos( sin( ((cos( -1.3977331148909333) / (x * ((x - (sin( e^( 0.5839073910575703)) / ((0.5839073910575703 + (((cos( -1.0768808474282068) / (((x - e^( (sin( (x + (-2.1183334622298835 + cos( -1.3977331148909333)))) * (sin( (x + 4.404029716618364)) / (x + x ))))) / -3.9294994016950726) * cos( cos( cos( ((x + -2.1183334622298835) / (((-3.488290785681106 / 4.429086745576107) - (4.64613256685335 + (2.032700600098037 / (cos( x ) / (3.951688988841374 / cos( sin( (((-1.6691938170386078 / cos( ((2.032700600098037 / -1.3977331148909333) / cos( cos( sin( (0.5839073910575703 + (0.5839073910575703 + (cos( 0.21588984462299265) / cos( 0.5839073910575703)))))))))) + 0.5839073910575703) + (0.5839073910575703 + (cos( 0.21588984462299265) / cos( 0.5839073910575703))))))))))) + -2.1183334622298835))))))) / -1.3977331148909333) + (cos( ((0.5839073910575703 / (0.5839073910575703 * 0.5839073910575703)) * (sin( (x + e^( 0.5839073910575703))) / (x - cos( e^( 0.5839073910575703)))))) / cos( 0.5839073910575703)))) / (3.951688988841374 / cos( sin( (0.5839073910575703 + (0.5839073910575703 + (cos( 0.21588984462299265) / cos( 0.5839073910575703)))))))))) * -1.3977331148909333))) / (sin( -4.09932580082895) / -1.3977331148909333))))) / cos( sin( ((cos( -1.3977331148909333) / (x * ((x - (sin( e^( 0.5839073910575703)) / ((0.5839073910575703 + (((cos( -1.0768808474282068) / (((0.5839073910575703 - e^( (sin( (x + -2.1183334622298835)) * (sin( (x + 4.404029716618364)) / (x + x ))))) / -3.9294994016950726) * cos( cos( cos( 0.5839073910575703))))) / -1.3977331148909333) + (cos( (e^( 0.5839073910575703) * (sin( (x + e^( 0.5839073910575703))) / (x - cos( e^( 0.5839073910575703)))))) / cos( 0.5839073910575703)))) / (3.951688988841374 / cos( sin( (0.5839073910575703 + (0.5839073910575703 + (x / cos( 0.5839073910575703)))))))))) * -1.3977331148909333))) / (sin( -4.09932580082895) / -1.3977331148909333))))) - e^( (sin( (cos( (cos( (((0.5839073910575703 / (0.5839073910575703 * 0.5839073910575703)) * (sin( (x + sin( (((2.032700600098037 / -1.3977331148909333) / 0.5839073910575703) + -2.1183334622298835)))) / (x - cos( e^( 0.5839073910575703))))) + -1.6691938170386078)) / (((e^( x ) + sin( e^( (x / cos( cos( (x / e^( 0.5839073910575703)))))))) - ((x / cos( sin( (((0.5839073910575703 - cos( -1.3977331148909333)) / (x * (3.141592653589793 * -1.3977331148909333))) / (x / -1.3977331148909333))))) / -1.3977331148909333)) + ((0.5839073910575703 / ((x / cos( cos( x ))) - (0.5839073910575703 * cos( cos( (0.5839073910575703 / (((-3.9294994016950726 * sin( x )) - cos( (0.5839073910575703 - (0.5839073910575703 * cos( (x / 0.5839073910575703)))))) + -2.1183334622298835))))))) - cos( -1.3977331148909333))))) + (x + sin( (((2.032700600098037 / -1.3977331148909333) / sin( 0.5839073910575703)) + -2.1183334622298835))))) * sin( (cos( (x / (cos( (x - (((cos( -1.0768808474282068) / (((x - e^( (sin( (x + (x - ((x - (sin( e^( 3.9169117826293363)) / ((0.5839073910575703 + (((cos( -1.0768808474282068) / (((3.141592653589793 - e^( (sin( (x - (sin( (x - 0.5839073910575703)) + 0.5839073910575703))) * (sin( (x + 4.404029716618364)) / (x / 4.5366847258566025))))) - -3.9294994016950726) + cos( cos( cos( ((x - -2.1183334622298835) / (((-3.488290785681106 / -3.423767380013757) / (4.496363738665307 + (2.032700600098037 + (cos( -2.9602495684565455) / (3.951688988841374 / (x - cos( e^( 0.5839073910575703)))))))) + -2.1183334622298835))))))) * -1.3977331148909333) * (cos( ((0.5839073910575703 / (0.5839073910575703 * 0.5839073910575703)) * (sin( (-1.7908824571411053 + e^( 0.5839073910575703))) / (x - cos( e^( x )))))) + e^( 0.5839073910575703)))) / (3.951688988841374 / cos( sin( (0.5839073910575703 + (-2.8354922153845505 + (cos( 0.21588984462299265) / cos( 0.5839073910575703)))))))))) * -1.3977331148909333)))) * sin( (cos( cos( 0.5839073910575703)) + (sin( 0.5839073910575703) / (x - ((((2.032700600098037 / -1.3977331148909333) / sin( ((sin( (cos( (x - (x + e^( 0.5839073910575703)))) + sin( (x + cos( sin( e^( (0.5839073910575703 * (sin( (x + 4.404029716618364)) / (x + x )))))))))) / cos( x )) / x ))) + -2.1183334622298835) + 0.5839073910575703)))))))) / -3.9294994016950726) * cos( 0.5839073910575703))) / -1.3977331148909333) / ((x - (cos( 0.5839073910575703) + (sin( e^( 0.5839073910575703)) / (((x - (sin( e^( (((0.5839073910575703 / ((x / cos( cos( x ))) - (0.5839073910575703 * cos( (cos( 4.404029716618364) / (x + cos( (-1.6691938170386078 / x )))))))) - cos( (sin( (x - cos( ((0.5839073910575703 / (0.5839073910575703 * (sin( 0.5839073910575703) * cos( 0.5839073910575703)))) * (0.5839073910575703 / (0.5839073910575703 * 0.5839073910575703)))))) * -1.3977331148909333))) / -4.09932580082895))) / (4.5366847258566025 / (3.951688988841374 / cos( -1.3977331148909333))))) * -1.3977331148909333) - cos( e^( 0.5839073910575703)))))) / cos( cos( (x - cos( e^( 0.5839073910575703))))))))) + 3.951688988841374))) + (sin( e^( 0.5839073910575703)) / (x - cos( e^( 0.5839073910575703))))))))) / -3.9294994016950726)) Notsimplified (MaximanorMatlabcouldsimplify) Fitnesspenaltyforlength

  17. Sources Adapted Tiny_GP Java code RiccardoPoli. Found at http://www.gp-field-guide.org.uk/ Burgess, Glenn. “Finding Approximate Analytic Solutions to Differential Equations” commissioned by Department of Defense (Australia) in 1999 Koza, John. Genetic Programming RiccardoPoli

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