Inductive Differential Constraint Method: Symbolic Regression‐Based Partial Differential Equation Regularization for Fatigue Crack Growth Prediction

This study introduces the inductive differential constraint method (IDCM), a data‐informed structural regularization framework that enhances neural network predictions under data‐scarce regimes by enforcing invariant differential structures extracted from simulation data. Unlike physics‐informed neural networks, which rely on known governing equations and dense collocation sampling, IDCM constructs data‐informed structural regularization terms from discovered symbolic relations without assuming them to constitute universal physical laws. The key novelty lies in transforming case‐dependent symbolic expressions into parameter‐free differential constraints. The proposed approach employs symbolic regression to identify explicit functional relationships from limited data. Systematic differentiation then eliminates case‐specific parameters, yielding a parameter‐free ordinary differential equation, which is subsequently extended to a partial differential equation. The resulting invariant constraint is incorporated into the training loss and evaluated at independently generated collocation points over the continuous input‐variable space, promoting consistency with the discovered response structure without geometric discretization. Under fixed training settings, application to a symmetric two‐crack configuration under fixed‐amplitude loading demonstrates that IDCM improves interpolation robustness within the considered geometric family, reducing validation loss from to compared to conventional data‐driven neural networks. Validation in this study focuses on unseen intermediate initial vertical distances within the simulated range of ; further evaluation under substantially different crack‐configuration families remains future work. Beyond fracture mechanics, the framework may offer a useful strategy for extracting and enforcing differential structures in physical prediction problems where governing equations are unknown or only partially available, provided that suitable symbolic relations can be identified.