Interpretable, dimensionally consistent surrogate scaling laws for underwater explosion bubble dynamics via symbolic regression

Abstract The widely used Cole empirical scaling law for underwater explosion (UNDEX) bubble dynamics suffers from limited accuracy, caused by fixed power-law exponents from narrow experimental ranges, fully decoupled charge-weight and depth dependence, and the absence of a self-diagnostic mechanism. This study proposes a three-stage data-driven paradigm combining machine-learning (ML) surrogate modeling, SHapley Additive exPlanations (SHAP) analysis, and symbolic regression (SR) to derive improved scaling laws for key UNDEX quantities: maximum bubble radius ( $$R_{\max }$$ ), first oscillation period ( $$T_1$$ ), and peak shock pressure ( $$P_{\text {peak}}$$ ). 858 high-fidelity simulations are generated with a validated unified bubble theory solver across a broad parameter range. Three ensemble ML surrogates are trained via five-fold cross-validation, with XGBoost achieving the best performance, substantially outperforming the Cole formula for $$T_1$$ and $$P_{\text {peak}}$$ . SHAP analysis identifies $$\pi _W = W^{1/3}/D$$ as the dominant controlling parameter for $$R_{\max }$$ and $$T_1$$ , revealing nonlinear depth dependence missed by Cole’s law to guide SR searches. PySR discovers three concise analytical formulas: the $$T_1$$ law corrects Cole’s depth exponent and introduces a charge–depth coupling term, reducing test-set MAPE from 10.86% to 0.88%; the $$P_{\text {peak}}$$ formula adjusts the decay exponent, cutting MAPE from 23.18% to 6.47%. Residual analysis confirms these physically interpretable corrections eliminate Cole’s inherent systematic depth-dependent bias, providing experimentally testable hypotheses for future validation. Furthermore, a preliminary cross-model validation against an independently formulated Keller–Miksis/JWL model—whose constants are not calibrated to the training data—confirms that the discovered corrections are not specific to the training theory ( $$T_1$$ agreement within 3%), and a comparison with the classical experiment-calibrated similitude relations of Swisdak and Zamyshlyaev further shows that the discovered laws improve upon all of these alternatives.