Download Neural Network Methods for boundary value problems with irregular

Neural Network Methods for
Boundary Value Problems with
Irregular Boundaries
I. E. Lagaris, A. Likas, D. G. Papageorgiou
University of Ioannina
Ioannina - GREECE
Why Neural Networks ?
• Have already been successfully
used on problems with regular
boundaries†.
• Analytic, closed form solution.
• Highly efficient on parallel
hardware.
†I. E. Lagaris, A. Likas and D. I. Fotiadis, IEEE TNN 9 (1998) pp 987-1000
What is an Irregular boundary ?
• A boundary that has not a simple
geometrical shape.
• A boundary that is described as a set of
distinct points that belong to it.
Difficulties
• Complex shapes pose severe problems to the
existing solution techniques.
• Extensions of methods that would apply to
problems with simple geometry are not trivial.
• We here present such an extension, to a
method based on Neural Networks.
Statement of the problem
• Solve the equation: L(x) = f(x), xR(N)
subject to Dirichlet or Neumann BCs.
• L is a differential non-linear operator
• The bounding hypersurface may be
either simple or complex.
The Case of Simple Boundaries
• If the boundary is a hypercube then for the
case of Dirichlet BCs we have developed
the following solution model.
m(x) = B(x) + Z(x)N(x,p)
• B(x) satisfies the boundary conditions.
• Z(x) is zero only on the boundary.
• N(x,p) is a Neural Network.
The Z-function
In the case of an orthogonal hyperbox
the Z-function is readily constructed as:
Z(x) = i (xi -ai )(xi -bi )
where xi is the ith component of x that lies in
the interval [ai , bi ].
In the case of irregular boundaries
the Z-function is not easily constructed.
The Procedure
• Let x(k) be points in the bounded domain.
• The “Error” is defined as:
E(p) = k {Lm(x(k)) - f(x(k))}2
and is minimized with respect to the
Neural Network parameters p.
The resulting modelm(x) = B(x) + Z(x)N(x,p)
is an approximate solution.
Modifications for Irregular
Boundaries.
• The Z-function is not easy to construct.
• The Dirichlet BCs are cast as:
m(X(i)) = bi
where X(i) are points on the boundary.
There are two options that we examined
for constructing the solution model.
Constrained Optimization
• The model is written as: m(x) = N(x,p)
• The Error to be optimized is taken as:
(k )
(k ) 2
( j)
2
{
L

(
x
)

f
(
x
)}


{

(
X
)

b
}
 m
 m
j
k
j
Where μ > 0 is a penalty parameter.
X(j) are boundary points.
x(k) are points in the solution domain.
RBF-Correction
• The model is made up, as a sum of two
Networks, a perceptron N(x,p) and a
Radial Basis Functions (RBF) Network.
m ( x )  N ( x , p)   a j e
  ( x  X ( j ) )2
j
• αj are chosen so as to satisfy the BCs exactly.
• λ is chosen so as to ease the numerics.
Pros and Cons
• The constrained optimization approach is
very efficient compared to the RBF synergy
approach, since to determine the RBF
coefficients, a linear system must be solved
every time.
• The RBF correction guarantees exact
satisfaction of the BCs, which is not the case
in the constrained optimization approach.
Procedure
• Use the constrained optimization to obtain a
solution that satisfies the BCs approximately.
• The obtained solution m(x) = N(x,p) may be
corrected via the RBF approach to exactly
satisfy the BCs.
m ( x )  N ( x , p)   a j e
  ( x  X ( j ) )2
j
• The correction will be small and local,
centered around the boundary points.
Experimental Results
We experimented with several domains.
• A star with six corners.
• A cardioid
• A part of a hollow sphere.
•Boundary points : 109
•Domain points :
391
•Boundary points: 100
•Domain points: 500
Example I
 ( x, y )  e
2
( x, y)
4
 1 x  y 
(1  x 2  y 2 )2
2
2
• The analytic solution is: log(1  x 2  y 2 )
• We solved this problem with Dirichlet BCs
in the star shaped domain.
• We plot the difference between the model
and the analytic solution.
Accuracy | m ( x, y )  exact ( x, y ) |
Example II
• The same (highly non-linear) example
inside the cardioid domain.
 ( x, y )  e
2
( x, y)
4
 1 x  y 
(1  x 2  y 2 )2
2
2
• We solved it for both Dirichlet and
Neumann BCs by extending the method
appropriately.
Accuracy | m ( x, y )  exact ( x, y ) |
Discussion
• Similar results hold for the 3-D problem.
• We tested the generalization of the model
by comparing it to the analytic solution in
points other than the training points.
• The conclusion is that the deviation is in
the same range as for the training points.
Tools
• For the optimization procedure we used the
Merlin 3.0 Optimization Package.
• Special linear solvers may be employed for
the calculation of the “error” gradient in the
case of the RBF synergy approach.
• Implementation in parallel machines or on
the so called “neuroprocessors” will greatly
contribute to the acceleration of the method.
Conclusions
• The method we presented is suitable for
handling complex boundaries with little effort.
• We demonstrated its applicability by solving
highly non-linear PDEs.
• Currently we are working on a method to
construct a suitable Z-function for irregular
boundaries.