Constructing artificial boundary condition of dispersive wave systems by deep learning neural network

To solve one dimensional dispersive wave systems in an unbounded domain, a uniform way to establish localized artificial boundary conditions is proposed. The idea is replacing the half-infinite interval outside the region of interest with a super element which exhibits the same dynamics response. Instead of designing the detailed mechanical structures of the super element, we directly reconstruct its stiffness, mass, and damping matrices by matching its frequency-domain reaction force with the expected one. An artificial neural network architecture is thus specifically tailored for this purpose. It comprises a deep learning part to predict the response of generalized degrees of freedom under different excitation frequencies, along with a simple linear part for computing the external force vectors. The trainable weight matrices of the linear layers correspond to the stiffness, mass, and damping matrices we need for the artificial boundary condition. The training data consists of input frequencies and the corresponding expected frequency domain external force vectors, which can be readily obtained through theoretical means. In order to achieve a good result, the neural network is initialized based on an optimized spring-damper-mass system. The adaptive moment estimation algorithm is then employed to train the parameters of the network. Different kinds of equations are solved as numerical examples. The results show that deep learning neural networks can find some unexpected optimal stiffness/damper/mass matrices of the super element. By just introducing a few additional degrees of freedom to the original truncated system, the localized artificial boundary condition works surprisingly well.