TLDR: Ex-HiDeNN is a novel AI framework that combines deep learning (C-HiDeNN-TD) with symbolic regression (PySR) to automatically discover accurate and human-understandable mathematical equations from complex data. It addresses the ‘black-box’ problem of AI by providing clear, closed-form expressions, guided by a unique ‘separability score’. The framework has demonstrated superior performance and interpretability in diverse engineering applications, including predicting material fatigue life, identifying hardness from microindentation data, and discovering classical constitutive laws for materials, marking a significant step towards more trustworthy AI in science and engineering.
In the rapidly evolving world of data-driven science and computation, artificial intelligence (AI) models have become incredibly powerful at identifying complex relationships within data. However, many of these advanced models, often referred to as ‘black-box’ models, operate without providing clear insights into their decision-making processes. This lack of transparency is a significant hurdle, especially in critical fields like aerospace, healthcare, and engineering, where understanding the ‘why’ behind a prediction is as important as the prediction itself.
Traditional methods like ordinary least-squares regression often fall short in capturing the intricate complexities of real-world systems. While modern neural networks—such as MLPs, CNNs, and GNNs—excel at prediction, their internal workings remain largely opaque. This has led to a growing demand for interpretable AI models, which allow humans to understand how a model arrives at its conclusions.
One promising avenue for interpretability is symbolic regression, a technique that aims to discover closed-form mathematical expressions directly from data. These explicit formulas are completely transparent, allowing for direct comparison with known physical mechanisms and easy identification of dominant effects. However, symbolic regression faces its own challenges, including the exponential growth of the search space for optimal expressions, inconsistency in results, and susceptibility to overfitting to noise, especially in high-dimensional problems.
Introducing Ex-HiDeNN: A Hybrid Approach to Interpretable AI
A new framework, called Explainable Hierarchical Deep Learning Neural Networks (Ex-HiDeNN), has been developed to bridge this gap between powerful predictive models and human interpretability. Ex-HiDeNN is a novel, hybrid architecture that combines the expressive power and separability of a specialized neural network called C-HiDeNN-TD with the inherent interpretability of symbolic regression. This approach aims to automatically and efficiently extract clear, closed-form mathematical expressions from data, mitigating the combinatorial complexity and noise sensitivity often seen in traditional symbolic regression.
At its core, Ex-HiDeNN operates through a two-stage pipeline. The first stage involves the C-HiDeNN-TD, a hierarchical deep learning neural network designed to learn a continuously differentiable representation of the target function. This network is particularly adept at handling multi-dimensional data and can identify underlying separable structures. A key innovation is the calculation of a ‘separability score’ (S⊗) based on the C-HiDeNN-TD’s learned representation. This score indicates how ‘separable’ or ‘multiplicative’ the underlying function is.
In the second stage, Ex-HiDeNN uses a symbolic regression engine, specifically PySR, to discover the closed-form expressions. Crucially, the separability score guides this process. For highly separable data, Ex-HiDeNN seeks a simple product of one-dimensional functions. For moderately separable data, it looks for a sum of multiplicative terms. Even for strongly coupled data, the C-HiDeNN-TD surrogate provides an intelligent sampling strategy that improves upon symbolic regression alone. This adaptive approach allows Ex-HiDeNN to propose highly parsimonious (simple) models for well-structured data and more expressive models for complex systems.
Also Read:
- AI Framework Uncovers Hidden Material Properties and Accelerates Discovery
- MP-ALOE: A Comprehensive Dataset for Advancing Machine Learning Interatomic Potentials
Real-World Engineering Applications
The effectiveness of Ex-HiDeNN has been demonstrated across various benchmark problems and real-world engineering applications:
- Fatigue Life Prediction: Ex-HiDeNN successfully discovered a closed-form equation for fatigue life in additively manufactured steel, even from a highly sparse dataset with 25 input parameters. The resulting equation not only accurately predicted fatigue life but also provided intuitive metallurgical insights, showing how factors like carbon content, tempering temperature, and copper content influence material strength.
- Material Hardness Identification: The framework was applied to predict Vickers Hardness from microindentation data. Ex-HiDeNN derived an equation that achieved exceptional accuracy, outperforming existing methods by a significant margin and explaining nearly all the variance in the hardness data.
- Yield Surface Expression: Ex-HiDeNN was used to learn a classical constitutive law for a Matsuoka-Nakai yield surface, which models pressure-sensitive granular materials. The discovered expression accurately captured the material’s behavior and demonstrated features consistent with established mechanics principles.
- Dynamical Systems Discovery: The framework also showed its capability in discovering governing dynamics from time-series data, such as the well-known chaotic Lorenz system, by fitting derivative data to snapshot data.
While Ex-HiDeNN represents a significant step forward, the researchers acknowledge its current limitations, including its dependence on the quality of the initial C-HiDeNN-TD surrogate and the inherent challenges of symbolic regression. Future work aims to integrate alternative, more robust surrogate models like KHRONOS and explore gradient-based expression discovery methods that can directly leverage the differentiable nature of the surrogates.
By providing human-understandable, accurate, and actionable mathematical forms, Ex-HiDeNN paves the way for more trustworthy AI in scientific and engineering applications. This ability to derive explicit expressions facilitates direct physical interpretation and seamless integration with existing theoretical frameworks and simulation pipelines. For more details, you can refer to the full research paper here.
