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