Artificial neural network methods for boundary integral equations

In this work, we present a deep neural network method for solving two-dimensional boundary value problems (BVPs) formulated in terms of boundary integral equations (BIEs). It is assumed that the boundary is parameterized by Non-Uniform Rational B-Splines (NURBS), commonly used in CAD, and the solution is 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.

The method inherits main advantages of boundary-type methods over the domain type methods: (a) the problem is solved on the boundary only, hence it requires much smaller number of collocation points, leading to significant savings in the computational cost; (b) for unbounded domains, asymptotic behavior of the solution at infinity is taken into account automatically and BIE is formulated on the inner boundary only; (c) high precision due to the exact NURBS-parameterization of the boundary, tight link to CAD and the ability to treat irregular boundaries (cracks, sharp corners, etc). Standard approaches to integration and removal of kernel singularity, used in Boundary Element Methods (BEM), are adopted.

Application of the method to three benchmark problems for the Laplace equation is demonstrated. In all cases, a good agreement with the analytical solutions is achieved.