The Hierarchical Deep Learning Neural Network (HIDENN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HIDENN for representing Finite Element Method (or HIDENN-FEM in short) is established. In HIDENN-FEM, weights and biases are functions of the nodal positions, hence the training process in HIDENN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear function, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method (RKPM), and among others. In HIDENN-FEM, building enrichment function through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HIDENN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HIDENN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs.

Bio:Professor W.K. Liu is the Walter P. Murphy Professor of Northwestern University, Director of Global Center on Advanced Material Systems and Simulation, President and Past President of the International Association for Computational Mechanics (IACM) (President (2014-2018) Past President (2018-2024)), Past Chair (2017-2018) (Chair 2015-2016) of the US National Committee on TAM and Member of Board of International Scientific Organizations, both within the US National Academies. Liu’s selected honors include the Distinguished Achievement Team Award for industry-university-government partnerships from the DOE Vehicle Technologies Office;Japan Society of Computational Engineering Sciences Grand Prize; IACM Gauss-Newton Medal (highest honor) and Computational Mechanics Award;ASME Dedicated Service Award, ASME Robert Henry Thurston Lecture Award, ASME Gustus L. Larson Memorial Award, ASME Pi Tau Sigma Gold Medal and ASME Melville Medal; John von Neumann Medal (highest honor) and Computational Structural Mechanics Award from USACM. Fellow of ASME, ASCE, USACM, AAM, and IACM.