  In this paper, we make the first attempt to apply the boundary integrated
neural networks (BINNs) for the numerical solution of two-dimensional (2D)
elastostatic and piezoelectric problems. BINNs combine artificial neural
networks with the well-established boundary integral equations (BIEs) to
effectively solve partial differential equations (PDEs). The BIEs are utilized
to map all the unknowns onto the boundary, after which these unknowns are
approximated using artificial neural networks and resolved via a training
process. In contrast to traditional neural network-based methods, the current
BINNs offer several distinct advantages. First, by embedding BIEs into the
learning procedure, BINNs only need to discretize the boundary of the solution
domain, which can lead to a faster and more stable learning process (only the
boundary conditions need to be fitted during the training). Second, the
differential operator with respect to the PDEs is substituted by an integral
operator, which effectively eliminates the need for additional differentiation
of the neural networks (high-order derivatives of neural networks may lead to
instability in learning). Third, the loss function of the BINNs only contains
the residuals of the BIEs, as all the boundary conditions have been inherently
incorporated within the formulation. Therefore, there is no necessity for
employing any weighing functions, which are commonly used in traditional
methods to balance the gradients among different objective functions. Moreover,
BINNs possess the ability to tackle PDEs in unbounded domains since the
integral representation remains valid for both bounded and unbounded domains.
Extensive numerical experiments show that BINNs are much easier to train and
usually give more accurate learning solutions as compared to traditional neural
network-based methods.
