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### Constraint Reasoning for Differential Models

Jorge Cruz

CENTRIA-Centre for Artificial Intelligence

DI/FCT/UNL

June 2009

- Constraint Reasoning

- Constraint Reasoning for Differential Models

- Examples: Drug Design / Epidemic Study

- Conclusions and Future Work

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

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

box split

Safe Narrowing Functions

Solving CCSPs:

isolate canonical solutions

Branch and Prune algorithms

Strategy for

provide an enclosure of the solution space

[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

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

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

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)

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’()

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

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

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

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

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]

[,]

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]

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]

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)

[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

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

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

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+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

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][-,]

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][-,]

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]

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]

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]

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]

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]

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]

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]

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

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:

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

- 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.

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