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

Boundary integrated neural networks (BINNs) for 2D elastostatic and piezoelectric problems: Theory and MATLAB code