Font Size: 

Boundary Value Problems (BVPs), which consist of a set of Partial Differential Equations (PDEs)together with the prescribed boundary conditions, are conventionally used to model a variety of phenomena in engineering and natural sciences. Common to most of the PDEs encountered in practical applications is that they cannot be solved analytically and require various approximation techniques. In this work, we present a deep neural network method for solving two-dimensional boundary value problems formulated in terms of Boundary Integral Equations (BIEs). The method inherits the main advantage of the isogeometric boundary element method (IGABEM), i.e. the exact  boundary parameterisation by Non-Uniform Rational B-Splines (NURBS) which are commonly used in computer-aided design (CAD). Thesolution is then approximated by a deep neural network with unknown weights and biases. The network is trained to minimize the loss function, which is formulated as an error in the BIE at a set of collocation points.In comparison with the domain type physics informed neural networks, it has the following main advantages: (a) it requires much smaller number of collocation points since the problem is solved on the domain boundary only, which leads to savings in the computational cost; (b) higher precision due to the exact parameterisation of the boundary, tight link to CAD and the ability to treat irregular boundaries (cracks, sharp corners, etc). Application of the method to some benchmark problems for the Laplace equation and linear elasticity is demonstrated. A detailed parametric study is presented to evaluate the performance of the method.

neural networks, boundary integral equations