Interpretable prediction of sample size–dependent fatigue crack formation lifetime using deep symbolic regression and polycrystalline plasticity models

Fatigue Indicator Parameters (FIPs), derived from cyclic intragranular and intergranular mechanical variables using the Crystal Plasticity Finite Element Method (CPFEM), can serve as surrogate measures of the driving force for fatigue crack formation within the first grain or nucleant phase. Simulating larger sample (i.e., increasing the number of grains) using CPFEM generally result in higher maximum FIP values, indicating a greater likelihood of fatigue crack initiation. However, the substantial computational demands of CPFEM limit its practical application in investigating the sample size effect on maximum FIPs. This study employs the recently developed Deep Symbolic Regression (DSR) algorithm to generate interpretable expressions linking sample size with the statistical characteristics of maximum FIPs in duplex Ti–6Al–4 V with random texture. These data–driven expressions obtained through DSR are systematically compared with predictions derived from the statistically grounded Extreme Value Theory (EVT), which suggests that the entire FIP dataset exceeding a threshold x₀​ converges to Gumbel distribution. The strong agreements found between DSR and EVT expressions not only validates the mathematical underpinnings of EVT but also demonstrates how EVT can elucidate the physical insights revealed by DSR. Building on this, we introduce a novel method, i.e., Regrouping of Maximum FIPs (RMF), to improve prediction reliability by mitigating the influence of the threshold x₀ in EVT. Finally, by leveraging the statistical distribution of maximum FIPs derived from DSR, we forecast the sample size–dependent Fatigue Crack Formation Lifetime (FCFL), providing a robust tool for engineering applications.