1. 1 Constraint Reasoning for Differential Models Jorge Cruz CENTRIA - Centre for Artificial Intelligence DI/FCT/UNL June 2009

2. 2 PRESENTATION OUTLINE Constraint Reasoning for Differential Models Conclusions and Future Work Constraint Reasoning Examples: Drug Design / Epidemic Study

3. 3 Constraint Satisfaction Problem (CSP): set of variables set of domains set of constraints Solution: assignment of values which satisfies all the constraints Continuous CSP (CCSP): Intervals of reals [a,b] Numeric (=, ,  ) Many Constraint Reasoning GOAL Find Solutions; Find an enclosure of the solution space

4. 4 Continuous Constraint Satisfaction Problem (CCSP): Solution: assignment of values which satisfies all the constraints Constraint Reasoning GOAL Find solutions; Find an enclosure of the solution space z [1,5] y [ ,  ] [0,2] x y = x 2 z  x x+y+z  5.25 Interval Domains Numerical Constraints Many Solutions x=1, y=1, z=1... x=1, y=1, z=3.25...

5. 5 Representation of Continuous Domains F-intervalR F [r 1..r 2 ] [f 1.. f 2 ] r [  r ..  r  ] F-box

6. 6 box split Solving CCSPs: Branch and Prune algorithms constraint propagation Safe Narrowing Functions Strategy for isolate canonical solutions provide an enclosure of the solution space

7. 7 Constraint Reasoning (vs Simulation) Uses safe methods for narrowing the intervals accordingly to the constraints of the model Represents uncertainty as intervals of possible values xy [1,5] [0,2] y = x 2 00 Simulation: 11 24 no y  4? x  1? Constraint Reasoning: [1,2] [1,4]

8. 8 How to narrow the domains? Safe methods are based on Interval Analysis techniques xy [1,5] [0,2] y = x 2 x[a,b] x2[a,b]2=x[a,b] x2[a,b]2= [0,  max(a 2,b 2 )  ] [  min(a 2,b 2 ) ,  max(a 2,b 2 )  ] if a  0  b otherwise If x  [0,2] Then y  [0,2] 2 =[0,  max(0 2,2 2 )  ]=[0,4]  y  [1,5]  y  [0,4]  y  [1,5]  [0,4]  y  [1,4]

9. 9 How to narrow the domains? Safe methods are based on Interval Analysis techniques xy [1,5] [0,2] y = x 2 x[a,b] x2[a,b]2=x[a,b] x2[a,b]2= [0,  max(a 2,b 2 )  ] [  min(a 2,b 2 ) ,  max(a 2,b 2 )  ] if a  0  b otherwise NF y=x² : Y’  Y  X 2

10. 10 Newton Method for Finding Roots of Univariate Functions Let f be a real function, continuous in [a,b] and differentiable in (a..b) Accordingly to the mean value theorem:  r 1,r 2  [a,b]   [min(r 1,r 2 ),max(r 1,r 2 )] f(r 1 )=f(r 2 )+(r 1  r 2 )  f’(  ) If r 2 is a root of f then f(r 2 )=0 and so:  r 1,r 2  [a,b]   [min(r 1,r 2 ),max(r 1,r 2 )] f(r 1 )=(r 1  r 2 )  f’(  )  r 1,r 2  [a,b]   [min(r 1,r 2 ),max(r 1,r 2 )] r 2 = r 1  f(r 1 )/f’(  ) And solving it in order to r 2 : Therefore, if there is a root of f in [a,b] then, from any point r 1 in [a,b] the root could be computed if we knew the value of 

11. 11 Newton Method for Finding Roots of Univariate Functions The idea of the classical Newton method is to start with an initial value r 0 and compute a sequence of points r i that converge to a root To obtain r i+1 from r i the value of  is approximated by r i : r i+1 = r i  f(r i )/f’(  )  r i  f(r i )/f’(r i ) r0r0 r1r1 r2r2 r0r0 r0r0 r1r1 r2r2 r1r1 r2r2

12. 12 Newton Method for Finding Roots of Univariate Functions Near roots the classical Newton method has quadratic convergence r0r0 r 1 =+  However, the classical Newton method may not converge to a root!

13. 13 Interval Extension of the Newton Method The idea of the Interval Newton method is to start with an initial interval I 0 and compute an enclosure of all the r that may be roots  r 1,r  [a,b]   [min(r 1,r),max(r 1,r)] r= r 1  f(r 1 )/f’(  ) If r is a root within I 0 then:  r 1  I 0 r  r 1  f(r 1 )/f’(I 0 ) (all the possible values of  are considered )  I0 I0 In particular, with r 1 =c=center(I 0 ) we get the Newton interval function: r  c  f(c)/f’(I 0 ) = N(I 0 ) Since root r must be within the original interval I 0, a smaller safe enclosure I 1 may be computed by: I 1 = I 0  N(I 0 )

14. 14 Interval Extension of the Newton Method The idea of the Interval Newton method is to start with an initial interval I 0 and compute an enclosure of all the r that may be roots r1r1 I0I0 c N(I0)N(I0) I1I1

15. 15 How to narrow the domains? Safe methods are based on Interval Analysis techniques xy [1,5] [0,2] y = x 2 If x  [0,2] and y  [1,5] Then yy  y  [1,5]  [0,4]  y  [1,4] y  x 2 = 0F(Y) = Y  [0,2] 2 F’(Y) = 1  y  Y  x  [0,2] y  x 2 =0  y  Interval Newton method

16. 16 How to narrow the domains? Safe methods are based on Interval Analysis techniques xy [1,5] [0,2] y = x 2 NF y=x² : Y’  Y  y  x 2 = 0F(Y) = Y  [0,2] 2 F’(Y) = 1  y  Y  x  [0,2] y  x 2 =0  y  Interval Newton method

17. 17 How to narrow the domains? Safe methods are based on Interval Analysis techniques xy [1,5] [0,2] y = x 2 NF y=x² : Y’  Y  X 2 NF y=x² : X’  (X  Y ½ )  (X  Y ½ ) + NF y=x² : Y’  Y  NF y=x² : X’  X  contractility Y’  Y X’  X correctness  y  Y y  Y’  ¬  x  X y=x 2  x  X x  X’  ¬  y  Y y=x 2

18. 18 [1,5] [ ,  ] [0,2] z y x y = x 2 z  x x+y+z  5.25 NF y=x² : Y’  Y  X 2 NF x+y+z  5.25 : X’  X  ([ , 5.25 ]  Y  Z) NF x+y+z  5.25 : Y’  Y  ([ , 5.25 ]  X  Z) NF x+y+z  5.25 : Z’  Z  ([ , 5.25 ]  X  Y) NF y=x² : X’  (X  Y ½ )  (X  Y ½ ) + NF z  x : X’  X  (Z  [ 0,  ]) NF z  x : Z’  Z  (X  [ 0,  ])  [1,4]  [1,2]     [ , 3.25 ]  [ 1, 3.25 ] Solving a Continuous Constraint Satisfaction Problem Constraint Propagation

19. 19 [1,5] [ ,  ] [0,2] z y x y = x 2 z  x x+y+z  5.25  [1,4]  [1,2]    [ , 3.25 ]  [ 1, 3.25 ] Solving a Continuous Constraint Satisfaction Problem Constraint Propagation  [ 1, 3.25 ]  [ 1,  3.25 ]       NF y=x² : Y’  Y  X 2 NF x+y+z  5.25 : X’  X  ([ , 5.25 ]  Y  Z) NF x+y+z  5.25 : Y’  Y  ([ , 5.25 ]  X  Z) NF x+y+z  5.25 : Z’  Z  ([ , 5.25 ]  X  Y) NF y=x² : X’  (X  Y ½ )  (X  Y ½ ) + NF z  x : X’  X  (Z  [ 0,  ]) NF z  x : Z’  Z  (X  [ 0,  ])

20. 20 z y x y = x 2 z  x x+y+z  5.25 [ 1, 3.25 ] Solving a Continuous Constraint Satisfaction Problem Constraint Propagation [ 1, 3.25 ] [ 1,  3.25 ] x y z  3.25 y = 3.25 y = x 2  x+y+z  5.25  z  2-  3.25 <  3.25 z  x 111  113.25  1.52.25 1.5  x + Branching Stopping Criterion

21. 21 How to deal with change in dynamic models? Classical constraint methods do not address differential models directly Typically through differential equations Differential model: v(t)=v(0)e t solution x 1 = x 0 e Constraints: Variables: x 0, x 1 Domains: [0.5,1]  [-1,2] Constraint model: x 0  [0.5,2/e] x 1  [0.5e,2] And without using the solution form? (non linear models)

22. 22 Constraint Reasoning for Differential Models All functions from [0,1] to R

23. 23 Functions s from [0,1] to R such that: Constraint Reasoning for Differential Models

24. 24 Functions s from [0,1] to R such that: Constraint Reasoning for Differential Models

25. 25 Functions s from [0,1] to R such that: I0I0 I1I1 Constraint Reasoning for Differential Models

26. 26 Extended Continuous Constraint Satisfaction Problem z [1,5] y [ ,  ] [0,2] x y = x 2 z  x x+y+z  5.25 z y CSDP ODE system Implicit representation of the trajectory Trajectory properties Explicit representation of its properties which can be integrated with the other constraints Developed safe methods for narrowing the intervals representing the possible property values NF CSDP :Y’ ... Z’ ...

27. 27 Trajectory Properties maximum minimum time  k k area  k k first  k k timeMaximum continuous function t value

28. 28 Solving a CSDP Use Narrowing functions for pruning the domains through propagation 0 0 6 1.5 s 1 ( t)  TR 1 t 0 0 6 1.5 t s 2 ( t)  TR 2 Maintain a safe trajectory enclosure x 1  I 1 x 2  I 2 x 3  I 3 x 4  I 4 x 5  I 5... and safe enclosures for each trajectory property:

29. 29 Solving a CSDP x  Ix  I s  TR 6 0 0 1.5 t TR Maximum Narrowing Functions I a b I  I  [a,b]I  I  [a,b] where a is the maximum lower bound of the point enclosures within [1,3] b is the maximum upper bound of the gap enclosures within [1,3]  t p  [1,3] TR (t p )  TR (t p )  [ ,c]  [t p1,t p2 ]  [1,3] TR ([t p1,t p2 ])  TR ([t p1,t p2 ])  [ ,c] where c is the upper bound of I continuous function

30. 30 0 0 1 1.5 TR t   then: 0.408 s(t)  [1.25]  [0,0.3]  [  1.225,  0.525]=[0.8825,1.25] Assume: s(t)  s(0)  [  0.5,0.5]=[0.75,1.75]  t  [0,1] For: 0.75 1.75 0.3  t  [0,0.3] Interval Picard Operator s(0)  [1.25] s(1)  [ ,  ]  t  [0,1] s(t)  [ ,  ] Solving a CSDP

31. 31 0 0 1 1.5 TR t   Trajectory Narrowing Function 0.3 Interval Picard Operator (gap enclosure): Interval Taylor Series (point enclosure): t i =0t i+1 =0.3 h=0.3  =[0,0.3] s(0)=[1.25] s(  )=[0.8825,1.25]  t  [0,0.3] s(t)  [0.8825,1.25] p=0  s(0.3)  [0.9875,1.0647] p=1  s(0.3)  [1.0069,1.0151] p=2  s(0.3)  [1.0131,1.0138] s(0)  [1.25] s(1)  [ ,  ]  t  [0,1] s(t)  [ ,  ] Solving a CSDP

32. 32 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[-1.0,2.0] TR =[0,1]  [- ,  ] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF

33. 33 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[-1.0,2.0] TR =[0]  [0.5,1.0]:(0,1]  [- ,  ] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF

34. 34 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[-1.0,2.0] TR =[0]  [0.5,1.0]:(0,1)  [- ,  ]:[1]  [-1.0,2.0] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF Based on an Interval Taylor Series method

35. 35 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[-1.0,2.0] TR =[0]  [0.5,1.0 ]:(0,0.5)  [0.45,1.8]:[0.5]  [0.82,1.65]:(0.5,1)  [- ,  ]:[1]  [-1.0,2.0] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF

36. 36 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[-1.0,2.0] TR =[0]  [0.5,1.0 ]:(0,0.5)  [0.45,1.8]:[0.5]  [0.82,1.65]:(0.5,1)  [0.8,2.9]:[1]  [1.35,2.0] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF

37. 37 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[1.35,2.0] TR =[0]  [0.5,1.0 ]:(0,0.5)  [0.45,1.8]:[0.5]  [0.82,1.65]:(0.5,1)  [0.8,2.9]:[1]  [1.35,2.0] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF

38. 38 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[1.35,2.0] TR =[0]  [0.5,1.0 ]:(0,0.5)  [0.45,1.8]:[0.5]  [0.82,1.22]:(0.5,1)  [0.8,2.1]:[1]  [1.35,2.0] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF

39. 39 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,1.0] I 1 =[1.35,2.0] TR =[0]  [0.5,0.74]:(0,0.5)  [0.45,1.3]:[0.5]  [0.82,1.22]:(0.5,1)  [0.8,2.1]:[1]  [1.35,2.0] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) NF

40. 40 Example: Constraint Satisfaction Differential Problem I 0 =[0.5,0.74] I 1 =[1.35,2.0] TR =[0]  [0.5,0.74]:(0,0.5)  [0.45,1.3]:[0.5]  [0.82,1.22]:(0.5,1)  [0.8,2.1]:[1]  [1.35,2.0] ODE S,[0,1] ( x ODE )Value 0 ( x 0 )Value 1 ( x 1 ) (fixed point)

41. 41 Application to Drug Design Differential model of the drug absorption process: concentration of the drug in the gastro-intestinal tract concentration of the drug in the blood stream drug intake regimen: Periodic limit cycle (p 1 =1.2, p 2 =ln(2)/5): x(t)x(t) y(t)y(t) tt

42. 42 Application to Drug Design Important properties of drug concentration are: maximum minimum time  1.1 area  1.0 y(t) CSDP framework can be used for: Bound the parameters (e.g p1) by imposing bounds on these properties Should be kept between 0.8 and 1.5 Area under curve above 1.0 between 1.2 and 1.3 Cannot exceed 1.1 for more than 4 hours p1  [0.0, 4.0] p1  [1.3, 1.4]

43. 43 Application to Drug Design Important properties of drug concentration are: maximum minimum time  1.1 area  1.0 y(t) CSDP framework can be used for: Compute safe bounds for these properties for chosen parameters Is guaranteedly kept between 0.881 and 1.462 ([0.8,1.5]) Area under curve above 1.0 between 1.282 and 1.3 ([1.2,1.3]) Exceeds 1.1 for 3.908 to 3.967 hours (<4.0) p1  [1.3, 1.4]

44. 44 Application to Epidemic Studies Susceptibles: can catch the disease Infectives: have the disease and can transmit it The SIR model of epidemics: Removed: had the disease and are immune or dead Parameters r a efficiency of the disease transmission recovery rate from the infection

45. 45 Application to Epidemic Studies The SIR model of epidemics: t S(t)S(t) I(t)I(t) R(t)R(t) Population Important questions about an infectious disease are: i max the maximum number of infectives: i max t max the time that it starts to decline: t max t end when will it ends: t end r end how many people will catch the disease: r end

46. 46 Application to Epidemic Studies The SIR model of epidemics: t S(t)S(t) I(t)I(t) R(t)R(t) Population i max t max t end r end CSDP framework can be used for: Bound the parameters according to the information available about the spread of a disease on a particular population (ex: boarding school) Predict the behaviour of an infectious disease from its parameter ranges

47. 47 Conclusions and Future Work the work extends Constraint Reasoning with ODEs it relies on safe methods that do not eliminate solutions it may support decision in applications where one is interested in finding the range of parameters for which some constraints on the ODE solutions are met it is an expressive and declarative constraint approach directions for further research: Explore alternative safe methods Apply to different models Extend to PDEs

48. 48 Bibliography Jorge Cruz. Constraint Reasoning for Differential ModelsConstraint Reasoning for Differential Models Vol: 126 Frontiers in Artificial Intelligence and Applications, IOS Press 2005 Ramon E. Moore. Interval Analysis Prentice-Hall 1966 Eldon Hansen, G. William Walster. Global Optimization Using Interval AnalysisGlobal Optimization Using Interval Analysis Marcel Dekker 2003 Jaulin, L., Kieffer, M., Didrit, O., Walter, E. Applied Interval AnalysisApplied Interval Analysis Springer 2001 Links Interval Computations (http://www.cs.utep.edu/interval-comp/) A primary entry point to items concerning interval computations. COCONUT (http://www.mat.univie.ac.at/~neum/glopt/coconut/) Project to integrate techniques from mathematical programming, constraint programming, and interval analysis.