276 Views

Download Presentation
## Constraint Reasoning for Differential Models

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Constraint Reasoning for Differential Models**Jorge Cruz CENTRIA-Centre for Artificial Intelligence DI/FCT/UNL June 2009**PRESENTATION OUTLINE**• Constraint Reasoning • Constraint Reasoning for Differential Models • Examples: Drug Design / Epidemic Study • Conclusions and Future Work**Continuous CSP (CCSP):**Intervals of reals [a,b] Numeric (=,,) Solution: Many assignment of values which satisfies all the constraints Find Solutions; Find an enclosure of the solution space GOAL Constraint Reasoning Constraint Satisfaction Problem (CSP): set of variables set of domains set of constraints**[1,5]**y y = x2 x+y+z 5.25 x Many Solutions [0,2] x=1, y=1, z=1 ... x=1, y=1, z=3.25 ... z z x [,] Solution: assignment of values which satisfies all the constraints Find solutions; Find an enclosure of the solution space GOAL Constraint Reasoning Continuous Constraint Satisfaction Problem (CCSP): Interval Domains Numerical Constraints**[r1..r2]**r [r..r] [f1 .. f2] F-box Representation of Continuous Domains F-interval R F**constraint propagation**box split Safe Narrowing Functions Solving CCSPs: isolate canonical solutions Branch and Prune algorithms Strategy for provide an enclosure of the solution space**[0,2]**[1,5] y = x2 x y no Simulation: 0 0 1 1 x1? y4? 2 4 Constraint Reasoning: [1,2] [1,4] Constraint Reasoning (vs Simulation) Represents uncertainty as intervals of possible values Uses safe methods for narrowing the intervals accordingly to the constraints of the model**[0,max(a2,b2)]**ifa0b [0,2] [1,5] x[a,b]x2[a,b]2= y = x2 [min(a2,b2),max(a2,b2)] otherwise x y If x[0,2] Then y[0,2]2 =[0,max(02,22)]=[0,4] y[1,5] y[0,4] y[1,5] [0,4] y[1,4] How to narrow the domains? Safe methods are based on Interval Analysis techniques**[0,max(a2,b2)]**ifa0b [0,2] [1,5] x[a,b]x2[a,b]2= y = x2 [min(a2,b2),max(a2,b2)] otherwise x y NFy=x²: Y’ YX2 How to narrow the domains? Safe methods are based on Interval Analysis techniques**Accordingly to the mean value theorem:**r1,r2[a,b] [min(r1,r2),max(r1,r2)] f(r1)=f(r2)+(r1 r2)f’() If r2 is a root of f then f(r2)=0 and so: r1,r2[a,b] [min(r1,r2),max(r1,r2)] f(r1)=(r1 r2)f’() And solving it in order to r2: r1,r2[a,b] [min(r1,r2),max(r1,r2)] r2= r1f(r1)/f’() Newton Method for Finding Roots of Univariate Functions Let f be a real function, continuous in [a,b] and differentiable in (a..b) Therefore, if there is a root of f in [a,b] then, from any point r1 in [a,b] the root could be computed if we knew the value of **r0**r1 r2 r0 r1 r2 r0 r1 r2 Newton Method for Finding Roots of Univariate Functions The idea of the classical Newton method is to start with an initial value r0 and compute a sequence of points ri that converge to a root To obtain ri+1 from ri the value of is approximated by ri: ri+1= rif(ri)/f’() rif(ri)/f’(ri)**r1=+**r0 Newton Method for Finding Roots of Univariate Functions Near roots the classical Newton method has quadratic convergence However, the classical Newton method may not converge to a root!**If r is a root within I0 then:**In particular, with r1=c=center(I0) we get the Newton interval function: I0 r1I0rr1f(r1)/f’(I0) r cf(c)/f’(I0) = N(I0) (all the possible values of are considered) Since root r must be within the original interval I0, a smaller safe enclosure I1 may be computed by: I1= I0 N(I0) Interval Extension of the Newton Method The idea of the Interval Newton method is to start with an initial interval I0 and compute an enclosure of all the r that may be roots r1,r[a,b] [min(r1,r),max(r1,r)] r= r1f(r1)/f’()**c**I0 N(I0) I1 Interval Extension of the Newton Method The idea of the Interval Newton method is to start with an initial interval I0 and compute an enclosure of all the r that may be roots r1**y x2 = 0**F(Y) = Y [0,2]2 F’(Y) = 1 [0,2] [1,5] y = x2 x y yY x[0,2] yx2=0 y Interval Newton method If x[0,2] and y[1,5] Then y y[1,5] [0,4] y[1,4] How to narrow the domains? Safe methods are based on Interval Analysis techniques**y x2 = 0**F(Y) = Y [0,2]2 F’(Y) = 1 [0,2] [1,5] y = x2 x y yY x[0,2] yx2=0 y Interval Newton method NFy=x²: Y’ Y How to narrow the domains? Safe methods are based on Interval Analysis techniques**contractility**correctness [0,2] [1,5] y = x2 NFy=x²: Y’ YX2 x y Y’ Y yY yY’ ¬xX y=x2 + NFy=x²: X’ (XY½)(XY½) X’ X xX xX’ ¬yY y=x2 NFy=x²: X’ X NFy=x²: Y’ Y How to narrow the domains? Safe methods are based on Interval Analysis techniques**[1,4]**y NFy=x²: Y’ YX2 + NFy=x²: X’ (XY½)(XY½) y = x2 x NFx+y+z5.25: X’ X([,5.25]YZ) x+y+z 5.25 NFx+y+z5.25: Y’ Y([,5.25]XZ) z z x [1,2] NFx+y+z5.25: Z’ Z([,5.25]XY) [,3.25] NFzx: X’ X(Z[0,]) [1,3.25] NFzx: Z’ Z(X[0,]) Solving a Continuous Constraint Satisfaction Problem Constraint Propagation [1,5] [0,2] [,] **[1,3.25]**y NFy=x²: Y’ YX2 + NFy=x²: X’ (XY½)(XY½) y = x2 x NFx+y+z5.25: X’ X([,5.25]YZ) x+y+z 5.25 NFx+y+z5.25: Y’ Y([,5.25]XZ) z z x NFx+y+z5.25: Z’ Z([,5.25]XY) [1,3.25] NFzx: X’ X(Z[0,]) NFzx: Z’ Z(X[0,]) Solving a Continuous Constraint Satisfaction Problem Constraint Propagation [1,4] [1,5] [0,2] [1,2] [,] [,3.25] [1,3.25] **Stopping Criterion**y 1.5 1 1 2.25 1.5 1 1 1 3.25 y = x2 y = x2 y = 3.25 3.25 3.25 <3.25 x x+y+z 5.25 x+y+z 5.25 z 2- z x z z x [1,3.25] Solving a Continuous Constraint Satisfaction Problem Constraint Propagation + Branching [1,3.25] x y z x [1,3.25]**Differential model:**Constraint model: Variables: x0, x1 solution v(t)=v(0)et Domains: [0.5,1][-1,2] Constraints: x1 = x0e x1[0.5e,2] x0[0.5,2/e] How to deal with change in dynamic models? Typically through differential equations Classical constraint methods do not address differential models directly And without using the solution form? (non linear models)**Constraint Reasoning for**Differential Models All functions from [0,1] to R**Constraint Reasoning for**Differential Models Functions s from [0,1] to R such that:**Constraint Reasoning for**Differential Models Functions s from [0,1] to R such that:**I1**I0 Constraint Reasoning for Differential Models Functions s from [0,1] to R such that:**CSDP**[1,5] ODE system y y y = x2 Trajectory properties z x+y+z 5.25 NFCSDP: Y’ ... Z’ ... x [0,2] z z x Implicit representation of the trajectory [,] 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 Extended Continuous Constraint Satisfaction Problem**maximum**k areak value k k minimum t firstk timeMaximum timek Trajectory Properties continuous function**Solving a CSDP**Maintain a safe trajectory enclosure 1.5 and safe enclosures for each trajectory property: s1(t)TR1 0 t 0 6 1.5 x1 I1 x2 I2 s2(t)TR2 x3 I3 x4 I4 x5 I5 0 0 6 t ... Use Narrowing functions for pruning the domains through propagation**b**1.5 a TR 0 0 t 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] tp[1,3] TR(tp)TR(tp) [,c] where c is the upper bound of I [tp1,tp2][1,3] TR([tp1,tp2])TR([tp1,tp2]) [,c] Solving a CSDP I x I s TR continuous function 6 Maximum Narrowing Functions**1.75**0.75 0.3 0.408 s(t) s(0)[0.5,0.5]=[0.75,1.75] t[0,1] For: Assume: t[0,0.3] then: s(t) [1.25] [0,0.3][1.225,0.525]=[0.8825,1.25] Solving a CSDP s(0) [1.25] s(1) [, ] 1.5 t[0,1]s(t) [, ] TR 0 1 t 0 Interval Picard Operator**ti=0**ti+1=0.3 =[0,0.3] h=0.3 s(0)=[1.25] s()=[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] Solving a CSDP s(0) [1.25] s(1) [, ] 1.5 t[0,1]s(t) [, ] TR Interval Picard Operator (gap enclosure): t[0,0.3]s(t) [0.8825,1.25] 0 Interval Taylor Series (point enclosure): 1 t 0.3 0 Trajectory Narrowing Function**NF**Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[-1.0,2.0] TR =[0,1][-,]**NF**Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[-1.0,2.0] TR =[0][0.5,1.0]:(0,1][-,]**NF**Based on an Interval Taylor Series method Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[-1.0,2.0] TR =[0][0.5,1.0]:(0,1)[-,]:[1][-1.0,2.0]**NF**Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[-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]**NF**Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[-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]**NF**Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[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]**NF**Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[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]**NF**Example: Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) I0=[0.5,1.0] I1=[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]**Example:**Constraint Satisfaction Differential Problem ODES,[0,1](xODE) Value0(x0) Value1(x1) (fixed point) I0=[0.5,0.74] I1=[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]**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 (p1=1.2, p2=ln(2)/5): y(t) x(t) t t Application to Drug Design**CSDP framework can be used for:**Bound the parameters (e.g p1) by imposing bounds on these properties p1[0.0 , 4.0] p1[1.3 , 1.4] Application to Drug Design Important properties of drug concentration are: maximum y(t) area1.0 minimum time1.1 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**CSDP framework can be used for:**Compute safe bounds for these properties for chosen parameters Application to Drug Design Important properties of drug concentration are: maximum y(t) area1.0 minimum time1.1 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]**The SIR model of epidemics:**Susceptibles: can catch the disease Infectives: have the disease and can transmit it Removed: had the disease and are immune or dead r efficiency of the disease transmission Parameters recovery rate from the infection a Application to Epidemic Studies**rend**imax the maximum number of infectives: imax tmax tend the time that it starts to decline: tmax when will it ends: tend how many people will catch the disease: rend Application to Epidemic Studies The SIR model of epidemics: Population S(t) R(t) I(t) t Important questions about an infectious disease are:**imax**tmax tend Application to Epidemic Studies The SIR model of epidemics: Population S(t) R(t) rend I(t) t 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**directions for further research:**Explore alternative safe methods Apply to different models Extend to PDEs Conclusions and Future Work • the work extends Constraint Reasoning with ODEs • 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 • it relies on safe methods that do not eliminate solutions**Bibliography**• Jorge Cruz.Constraint 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 Analysis • Marcel Dekker 2003 • Jaulin, L., Kieffer, M., Didrit, O., Walter, E.Applied 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.