Interpretable machine learning with prediction uncertainty quantification for d₃₃ in (K_0.5Na_0.5) NbO₃-based lead-Free piezoelectric ceramics

The accelerated discovery of high-performance lead-free piezoelectric ceramics is hindered by the vast compositional space and the limited interpretability of conventional machine learning (ML) models. Here, we propose a physics-informed and interpretable ML framework with integrated uncertainty quantification to predict and understand the piezoelectric coefficient d₃₃ of (K_0.5Na_0.5) NbO₃ (KNN)-based ceramics. A curated dataset of 1113 experimental samples is used to construct 65 descriptors by decoupling A-site and B-site ionic contributions. Pearson correlation analysis reduces these to an optimized 11-dimensional feature set for training deep neural networks, Wide & Deep networks, and residual networks. A Bayesian neural network further provides predictive uncertainty, which quantitatively reflects the confidence of machine-learning-based d₃₃ predictions rather than experimental measurement uncertainty. To achieve physical interpretability, SHapley Additive exPlanations (SHAP) are combined with the Sure Independence Screening and Sparsifying Operator (SISSO) to derive a compact analytical descriptor revealing that sintering temperature, B-site electronic anisotropy, and A-site ionic displacement jointly govern d₃₃. The proposed framework achieves high accuracy (R² ≈ 0.81) while offering transparent design rules for next-generation lead-free piezoelectrics.