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

1

Constraint Reasoning for Differential Models

Jorge Cruz

CENTRIA - Centre for Artificial Intelligence

DI/FCT/UNL

June 2009

http://ssdi.di.fct.unl.pt/~jc/

http://centria.di.fct.unl.pt/





http://di.fct.unl.pt/

2

PRESENTATION OUTLINE

• Constraint Reasoning for Differential Models

• Conclusions and Future Work

• Constraint Reasoning

• Examples: Drug Design / Epidemic Study

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

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 = x2

z x

x+y+z 5.25

Interval Domains

Numerical Constraints

Many Solutionsx=1, y=1, z=1 ...

x=1, y=1, z=3.25 ...

5

Representation of Continuous Domains

F-interval R

F

[r1..r2]

[f1 .. f2]

r

[r..r]

F-box

6

box split

Solving CCSPs:

Branch and Prune algorithms

constraint propagation

Safe Narrowing Functions

Strategy forisolate canonical solutions

provide an enclosure of the solution space

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

x y

[1,5][0,2]y = x2

0 0Simulation:1 12 4

no

y4?x1?

Constraint Reasoning: [1,2] [1,4]

8

How to narrow the domains?

Safe methods are based on Interval Analysis techniques

x y

[1,5][0,2]y = x2

x[a,b] x2[a,b]2=[0,max(a2,b2)]

[min(a2,b2),max(a2,b2)]

if a0b

otherwise

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]

9



x y

[1,5][0,2]y = x2

x[a,b] x2[a,b]2=[0,max(a2,b2)]

[min(a2,b2),max(a2,b2)]

if a0b

otherwise

NFy=x²: Y’ YX2

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:

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’()

r1,r2[a,b] [min(r1,r2),max(r1,r2)] r2= r1f(r1)/f’()

And solving it in order to r2:

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

11


-20

-10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10

x

f(x)

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)

r0r1r2

r0

r0

r1r2

r1

r2

12


-20

-10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10

x

f(x)

Near roots the classical Newton method has quadratic convergence

r0

r1=+

However, the classical Newton method may not converge to a root!

13

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’()

If r is a root within I0 then: r1I0

r r1f(r1)/f’(I0) (all the possible values of are considered)

I0

In particular, with r1=c=center(I0) we get the Newton interval function: r cf(c)/f’(I0) = N(I0)

Since root r must be within the original interval I0, a smaller safe enclosure I1 may be computed by:

I1 = I0 N(I0)

14

Interval Extension of the Newton Method

-20

-10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10

x

f(x)

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

I0c

N(I0)

I1

15



x y

[1,5][0,2]y = x2

If x[0,2] and y[1,5]

Then y

y[1,5] [0,4] y[1,4]

],[],[

]),([N 401

203351

2

y x2 = 0 F(Y) = Y [0,2]2 F’(Y) = 1

)Y('F

))Y(c(F)Y(c)Y(N yY x[0,2] yx2=0 y

Interval Newton method

16



x y

[1,5][0,2]y = x2

NFy=x²: Y’ Y

1

2X)Y(c)Y(c

y x2 = 0 F(Y) = Y [0,2]2 F’(Y) = 1

)Y('F

))Y(c(F)Y(c)Y(N yY x[0,2] yx2=0 y

Interval Newton method

17



x y

[1,5][0,2]y = x2

NFy=x²: Y’ YX2

NFy=x²: X’ (XY½)(XY½)+

NFy=x²: Y’ Y

1

2X)Y(c)Y(c

NFy=x²: X’ X

X

)X(cY)X(c

2

2

contractility

Y’ Y

X’ X

correctness

yY yY’ ¬xX y=x2

xX xX’ ¬yY y=x2

18

[1,5]

[,]

[0,2] z

y

x

y = x2

z x

x+y+z 5.25

NFy=x²: Y’ YX2

NFx+y+z5.25: X’ X([,5.25]YZ)

NFx+y+z5.25: Y’ Y([,5.25]XZ)

NFx+y+z5.25: Z’ Z([,5.25]XY)


NFzx: X’ X(Z[0,])

NFzx: Z’ Z(X[0,])

[1,4]

[1,2]

[,3.25]

[1,3.25]

Solving a Continuous Constraint Satisfaction Problem

Constraint Propagation

19

[1,5]

[,]

[0,2] z

y

x

y = x2

z x

x+y+z 5.25

[1,4]

[1,2]

[,3.25]

[1,3.25]



[1,3.25]

[1,3.25]

NFy=x²: Y’ YX2

NFx+y+z5.25: X’ X([,5.25]YZ)

NFx+y+z5.25: Y’ Y([,5.25]XZ)

NFx+y+z5.25: Z’ Z([,5.25]XY)


NFzx: X’ X(Z[0,])

NFzx: Z’ Z(X[0,])

20

z

y

x

y = x2

z x

x+y+z 5.25

[1,3.25]



[1,3.25]

[1,3.25]

x y z

3.25 y = 3.25y = x2 x+y+z 5.25 z 2- 3.25

<3.25z x

1 1 1 1 1 3.25

1.5 2.25 1.5 x

+ Branching

Stopping Criterion

21

How to deal with change in dynamic models?

Classical constraint methods do not address differential models directly

Typically through differential equations

vdt

dv

]0.1,5.0[)0( v

]0.2,0.1[)1( v

Differential model:

v(t)=v(0)etsolution

x1 = x0eConstraints:

Variables: x0, x1

Domains: [0.5,1][-1,2]

Constraint model:

x0[0.5,2/e] x1[0.5e,2]And without using the solution form?

(non linear models)

22

-2

-1

0

1

2

3

0 0,5 1


vdt

dv

]0.1,5.0[)0( v

]0.2,0.1[)1( v

All functions from [0,1] to R

23

vdt

dv

]0.1,5.0[)0( v

]0.2,0.1[)1( v

Functions s from [0,1] to R such that:

-2

-1

0

1

2

3

0 0,5 1

)()(

]1,0[ tsdt

tdst


24

vdt

dv

]0.1,5.0[)0( v

]0.2,0.1[)1( v


-2

-1

0

1

2

3

0 0,5 1

]0.1,5.0[)0( s

)()(

]1,0[ tsdt

tdst


25

vdt

dv

]0.1,5.0[)0( v

]0.2,0.1[)1( v


]0.1,5.0[)0( s

)()(

]1,0[ tsdt

tdst

]0.2,0.1[)1( s

-2

-1

0

1

2

3

0 0,5 1

I0

I1


26

Extended Continuous Constraint Satisfaction Problem

z

[1,5]

y

[,]

[0,2]x

y = x2

z x

x+y+z 5.25

vdt

dv

)0(v)1(v 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

NFCSDP: Y’ ...Z’ ...

27

Trajectory Properties

maximum

minimum

timek

k

areakk

firstk

k

timeMaximum

continuous function

t

value

28

Solving a CSDP

Use Narrowing functions for pruning the domains through propagation

00

6

1.5

s1( t)TR1

t

0

0

6

1.5

t

s2( t)TR2

Maintain a safe trajectory enclosure

x1 I1

x2 I2

x3 I3

x4 I4

x5 I5

...

and safe enclosures for each trajectory property:

29

Solving a CSDP

x I

s TR

60

0

1.5

t

TR

Maximum Narrowing Functions

Ia

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]

tp[1,3] TR (tp) TR (tp) [,c]

[tp1,tp2][1,3] TR ([tp1,tp2]) TR ([tp1,tp2]) [,c] where c is the upper bound of I

)(max31

tst

continuous function

30

00

1

1.5

TR

t

then: ]525.0,225.1[]75.1,75.0[7.0)(

dt

tds

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

)(7.0)(

tsdt

tds

s(0) [1.25]

s(1) [, ]

t[0,1] s(t) [, ]

Solving a CSDP

31

00

1

1.5

TR

t

Trajectory Narrowing Function

0.3

Interval Picard Operator (gap enclosure):

Interval Taylor Series (point enclosure):

)()!1(

)(!

)()( )1(

1

1)(

1

p

p

k

p

ik

k

ii sp

hts

k

htsts

ti=0 ti+1=0.3 h=0.3 =[0,0.3]

s(0)=[1.25]

)()7.0()()( tsts kk

s()=[0.8825,1.25]

t[0,0.3] s(t) [0.8825,1.25]

]25.1,8825.0[)7.0()!1(

3.0]25.1[)7.0(

!

3.0]25.1[)3.0( 1

1

1

p

p

k

pk

k

pks

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]

)(7.0)(

tsdt

tds

s(0) [1.25]

s(1) [, ]

t[0,1] s(t) [, ]

Solving a CSDP

32

Example: Constraint Satisfaction Differential Problem

I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

I1=[-1.0,2.0]

TR =[0,1][-,]

ODES,[0,1](xODE) Value0(x0) Value1(x1)

NF

33


I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

I1=[-1.0,2.0]

TR =[0][0.5,1.0]:(0,1][-,]


NF

34


I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

I1=[-1.0,2.0]

TR =[0][0.5,1.0]:(0,1)[-,]:[1][-1.0,2.0]


NF

Based on an Interval Taylor Series method

35


I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

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

36


I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

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

37


I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

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

38


I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

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

39


I0=[0.5,1.0]

-2

-1

0

1

2

3

0 0,5 1

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]


NF

40


I0=[0.5,0.74]

-2

-1

0

1

2

3

0 0,5 1

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]


(fixed point)

41

Application to Drug Design

)()()(

1 tDtxpdt

tdx )()(

)(21 typtxp

dt

tdy

Differential model of the drug absorption process:

concentration of the drug in the gastro-intestinal tractconcentration of the drug in the blood stream

drug intake regimen:0.65.0

5.00.0

0

2)(

t

t

if

iftD

Periodic limit cycle (p1=1.2, p2=ln(2)/5):

0

0,5

1

0 1 2 3 4 5 60,5

1

1,5

0 1 2 3 4 5 6

x(t)y(t)

t t

42


Important properties of drug concentration are:

maximum

minimum

time1.1

area1.0

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 1.5Area 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


Important properties of drug concentration are:

maximum

minimum

time1.1

area1.0

y(t)


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

Application to Epidemic Studies

Susceptibles: can catch the diseaseInfectives: have the disease and can transmit it

The SIR model of epidemics:

)()()(

tItrSdt

tdS )()()(

)(taItItrS

dt

tdI )(

)(taI

dt

tdR

Removed: had the disease and are immune or dead

Parametersr

a

efficiency of the disease transmission

recovery rate from the infection

45

0

200

400

600

800

0 2 4 6 8 10 12 14 16 18 20 22 24



t

S(t)

I(t)

R(t)

Population

Important questions about an infectious disease are:

imax

the maximum number of infectives: imax

tmax

the time that it starts to decline: tmax

tend

when will it ends: tend

rend

how many people will catch the disease: rend

46

0

200

400

600

800

0 2 4 6 8 10 12 14 16 18 20 22 24



t

S(t)

I(t)

R(t)

Population

imax

tmax tend

rend


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

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

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.

http://www.iospress.nl/loadtop/load.php?isbn=1586035320

http://www.ebooks.dekker.com/eBookCover.asp?eBookID=0824758706&type=ISBN

http://www.springer.com/east/home/generic/search/results?SGWID=5-40109-22-2093571-0

http://www.cs.utep.edu/interval-comp/

http://www.mat.univie.ac.at/~neum/glopt/coconut/

Documents

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