Explainable Hierarchical Deep Learning Neural Networks (Ex-HiDeNN)

2025-07-07

AI Summary

Addressing the challenge of discovering interpretable closed-form expressions from complex data—where accuracy and interpretability are often mutually exclusive—this paper proposes Ex-HiDeNN. The method first decouples variables via separability detection, then employs a two-stage modeling pipeline integrating hierarchical deep neural networks (HiDeNN/Conv-HiDeNN) with symbolic regression. This yields a lightweight, scalable, and inherently interpretable hybrid framework that explicitly outputs analytical expressions while maintaining high predictive accuracy. On benchmark tasks such as dynamical system identification, Ex-HiDeNN achieves significantly lower prediction errors than state-of-the-art symbolic regression and black-box deep learning methods. Furthermore, it demonstrates superior performance—achieving new state-of-the-art results—in three real-world engineering physics applications: fatigue equation discovery, material hardness inversion, and yield surface modeling. These results validate Ex-HiDeNN’s effectiveness, robustness, and generalizability for scientific machine learning and physics-informed modeling.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining

Abstract

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.

Problem

• Discover interpretable closed-form expressions from complex datasets
• Develop efficient neural architecture for symbolic regression tasks
• Apply method to engineering problems for accurate predictions

Innovation

• Hierarchical deep learning with symbolic regression
• Embedded separability checker for efficiency
• Accurate closed-form expression discovery