# Explainable Hierarchical Deep Learning Neural Networks (Ex-HiDeNN)

arXiv: https://arxiv.org/abs/2507.05498 · HTML: https://mlsanity.com/paper/2507.05498/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 58 references that are in the map.
- First posted on arXiv 7 Jul 2025.

## Builds on (most cited first, 9 of 9)

- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements](https://mlsanity.com/paper/2204.00205/) (2022)
- [Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions](https://mlsanity.com/paper/2307.13149/) (2023)
- [KHRONOS: a Kernel-Based Neural Architecture for Rapid, Resource-Efficient Scientific Computation](https://mlsanity.com/paper/2505.13315/) (2025)
- [HiDeNN-PGD: reduced-order hierarchical deep learning neural networks](https://mlsanity.com/paper/2105.06363/) (2021)

## Built on by (most cited first, 2 of 2)

- [A Unified Generative-Predictive Framework for Deterministic Inverse Design](https://mlsanity.com/paper/2512.15746/) (2025)
- [Separable Neural Architectures as Physical World Models: from Mathematical Theory to Applications](https://mlsanity.com/paper/2606.14934/) (2026)

## Abstract (arXiv)

Data-driven science and computation have advanced immensely to construct complex functional relationships using trainable parameters. However, efficiently discovering interpretable and accurate closed-form expressions from complex dataset remains a challenge. The article presents a novel approach called Explainable Hierarchical Deep Learning Neural Networks or Ex-HiDeNN that uses an accurate, frugal, fast, separable, and scalable neural architecture with symbolic regression to discover closed-form expressions from limited observation. The article presents the two-step Ex-HiDeNN algorithm with a separability checker embedded in it. The accuracy and efficiency of Ex-HiDeNN are tested on several benchmark problems, including discerning a dynamical system from data, and the outcomes are reported. Ex-HiDeNN generally shows outstanding approximation capability in these benchmarks, producing orders of magnitude smaller errors compared to reference data and traditional symbolic regression. Later, Ex-HiDeNN is applied to three engineering applications: a) discovering a closed-form fatigue equation, b) identification of hardness from micro-indentation test data, and c) discovering the expression for the yield surface with data. In every case, Ex-HiDeNN outperformed the reference methods used in the literature. The proposed method is built upon the foundation and published works of the authors on Hierarchical Deep Learning Neural Network (HiDeNN) and Convolutional HiDeNN. The article also provides a clear idea about the current limitations and future extensions of Ex-HiDeNN.