Artificial Neural Network Method Based on Boundary Integral Equations

Video recording:

Speaker: Han Zhang (University of New South Wales (UNSW Sydney), Australia)
Title: Artificial Neural Network Method Based on Boundary Integral Equations
Time: Wednesday, 2022.06.29, 10:00 a.m. (CET)
Place: fully virtual (contact Dr. Jakub Lengiewicz to register)
Format: 30 min. presentation + 30 min. discussion

Abstract: Artificial Neural Networks (ANNs) have aroused great interest of researchers as an option to solve or approximate complex engineering problems. In Anitescu et al. [1], it has been used to solve boundary value problem (BVP) for second order partial differential equations (PDEs) based on minimizing the combined error in PDE at multiple collocation points inside the domain and at the boundary conditions. In such an approach, ANNs have shown the potential of being an alternative of finite element method (FEM) which is commonly used for solving BVPs.

 The boundary element method (BEM) has been a common alternative of FEM in solving BVPs. In many practical applications, BVPs could be transformed to a boundary integral equation (BIE) and then solved by BEM. BEM has several commendable advantages over domain-type methods, such as lowering the dimension of the problem to reduce the computational cost and avoiding domain truncation error in exterior domains. In Simpson et al. [2], an isogeometric boundary element method (IGABEM) was proposed, which combines BEM with Non-Uniform Rational B-Splines (NURBS) and further improves the accuracy of BEM.

 In this work, we proposed a new approach for solving BVPs formulated as BIEs. The approach consists in using deep neural network to approximate the solution of BIEs. Loss function is set to estimate the error at the collocation points and is subsequently minimized by the appropriate weights and biases. The approach preserves the main advantage of IGABEM, i.e. using NURBS to describe the boundary, hence keeping the exact geometry and providing the tight link with CAD. The approach inherits the main advantages of the boundary-type methods and allows to reduce the computational cost by using collocation points on the boundary only, which is very beneficial in practical applications where only the boundary data measurements are available.

 Application of the method to some benchmark problems for the Laplace equation is demonstrated. The results are presented in terms of the loss function convergence plots and error norms. A detailed parametric study is presented to evaluate the performance of the method. It is shown that the method is highly accurate for Dirichlet, Neumann and mixed problems in continuous domain, however the accuracy deteriorates in presence of corners.

 In future studies, accuracy of the method can be improved by varying the choice of the activation function and the network architecture. The method can be extended to other applications by changing the BIE. In the current study, the method was implemented for 2D applications, but extension to 3D is also possible and it is an objective of future works.

 [1] Anitescu et al. Artificial neural network methods for the solution of second order boundary value problems. 2019
 [2] Simpson et al. A two-dimensional isogeometric boundary element method for elastostatic analysis. 2012