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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), xR(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 {Lm(x(k)) - f(x(k))}2 and is minimized with respect to the Neural Network parameters p. The resulting modelm(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.