A Hierarchical Deep Symbolic AI Method for Closed‐Form, Interpretable Constitutive Modeling

ABSTRACT Recent advances in advanced manufacturing have enabled the development of multifunctional materials such as soft composites, whose complex nonlinear behaviors demand accurate and interpretable constitutive modeling. Traditional phenomenological formulations rely heavily on assumed functional forms and expert intuition, requiring extensive trial and error and offering limited generality. In contrast, data‐driven approaches based on neural networks provide strong predictive capability but often require large datasets and lack interpretability, stability, and physical consistency. To address these challenges, we propose a hierarchical deep symbolic artificial intelligence (HDSAI) method for discovering constitutive laws directly from data while providing closed‐form analytical expressions. The method constructs constitutive relations hierarchically in a layer‐by‐layer manner using symbolic mathematical operators as building blocks. Each layer combines and transforms the outputs from the previous one, progressively enriching functional complexity while maintaining analytical transparency. A layer‐wise evolutionary and selection algorithm is introduced to optimize the symbolic operators within and across layers, efficiently balancing model accuracy and parsimony. The HDSAI method offers three key advantages: (1) deeper symbolic composition while avoiding uncontrolled symbolic‐tree explosion; (2) flexible control of model complexity via progressive symbolic evolution; and (3) enhanced stability and generalization compared with existing symbolic regression and neural‐network‐based models. Benchmark studies against state‐of‐the‐art symbolic regression including GPTIPS, PySR, and LoHM methods demonstrate that the proposed HDSAI method not only achieves higher numerical accuracy and stability but also successfully rediscovers the exact ground‐truth constitutive expressions that existing methods fail to capture. This capability underscores the effectiveness of the hierarchical architecture in navigating complex search spaces and preserving physically interpretable functional forms. Systematic investigations of key parameters, including population size, layer depth, data availability, and noisy level of the data, further reveal that the hierarchical design enhances convergence robustness and expressiveness, enabling HDSAI to efficiently identify compact, physically meaningful constitutive laws with superior interpretability and generalization.