  This paper presents a proper generalized decomposition (PGD) based
reduced-order model of hierarchical deep-learning neural networks (HiDeNN). The
proposed HiDeNN-PGD method keeps both advantages of HiDeNN and PGD methods. The
automatic mesh adaptivity makes the HiDeNN-PGD more accurate than the finite
element method (FEM) and conventional PGD, using a fraction of the FEM degrees
of freedom. The accuracy and convergence of the method have been studied
theoretically and numerically, with a comparison to different methods,
including FEM, PGD, HiDeNN and Deep Neural Networks. In addition, we
theoretically showed that the PGD converges to FEM at increasing modes, and the
PGD error is a direct sum of the FEM error and the mode reduction error. The
proposed HiDeNN-PGD performs high accuracy with orders of magnitude fewer
degrees of freedom, which shows a high potential to achieve fast computations
with a high level of accuracy for large-size engineering problems.
