Neural Network Methods for Boundary Value Problems with Irregular Boundaries

1. Neural Network Methods for Boundary Value Problems with Irregular Boundaries I. E. Lagaris, A. Likas, D. G. Papageorgiou University of Ioannina Ioannina - GREECE

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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 theith component of x that lies in the interval [ai , bi ]. In the case of irregular boundaries theZ-functionis not easily constructed.

8. 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.

9. Modifications for Irregular Boundaries. • The Z-function is not easy to construct. • The Dirichlet BCs are cast as: m(X(i)) = bi whereX(i)are points on the boundary. There are two options that we examined for constructing the solution model.

10. Constrained Optimization • The model is written as: m(x) = N(x,p) • The Error to be optimized is taken as: Where μ > 0 is a penalty parameter. X(j) are boundary points. x(k) are points in the solution domain.

11. 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. αj are chosen so as to satisfy the BCs exactly. λ is chosen so as to ease the numerics.

12. 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.

13. 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. • The correction will be small and local, centered around the boundary points.

14. Experimental Results We experimented with several domains. • A star with six corners. • A cardioid • A part of a hollow sphere.

15. Boundary points : 109 Domain points : 391

16. Boundary points: 100 Domain points: 500

17. Example I The analytic solution is: We solved this problem with Dirichlet BCs in the star shaped domain. We plot the difference between the model and the analytic solution.

18. Analytic Solution

20. RBF Correction

21. Example II • The same (highly non-linear) example inside the cardioid domain. • We solved it for both Dirichlet and Neumann BCs by extending the method appropriately.

23. 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.

24. 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.

25. 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.