# HiDeNN-PGD: reduced-order hierarchical deep learning neural networks

arXiv: https://arxiv.org/abs/2105.06363 · HTML: https://mlsanity.com/paper/2105.06363/

## Key facts

- Cited by 3 papers in the map (data through 7 Oct 2026).
- Builds on 5 of its 45 references that are in the map.
- First posted on arXiv 13 May 2021.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)

## Built on by (most cited first, 3 of 3)

- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2401.03492/) (2024)
- [Explainable Hierarchical Deep Learning Neural Networks (Ex-HiDeNN)](https://mlsanity.com/paper/2507.05498/) (2025)

## Abstract (arXiv)

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.