HiDeNN-TD: Reduced-order hierarchical deep learning neural networks

This paper presents a tensor decomposition (TD) based reduced-order model of the hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-TD method keeps advantages of both HiDeNN and TD methods. The automatic mesh adaptivity makes the HiDeNN-TD more accurate than the finite element method (FEM) and conventional proper generalized decomposition (PGD) and TD, using a fraction of the FEM degrees of freedom. This work focuses on the theoretical foundation of the method. Hence, the accuracy and convergence of the method have been studied theoretically and numerically, with a comparison to different methods, including FEM, PGD, TD, HiDeNN and Deep Neural Networks. In addition, we have theoretically shown that the PGD/TD converges to FEM at increasing modes, and the PGD/TD solution error is a summation of the mesh discretization error and the mode reduction error. The proposed HiDeNN-TD shows a high accuracy with orders of magnitude fewer degrees of freedom than FEM, and hence a high potential to achieve fast computations with a high level of accuracy for large-size engineering and scientific problems. As a trade-off between accuracy and efficiency, we propose a highly efficient solution strategy called HiDeNN-PGD. Although the solution is less accurate than HiDeNN-TD, HiDeNN-PGD still provides a higher accuracy than PGD/TD and FEM with only a small amount of additional cost to PGD.