# Interpretable Machine Learning with Prediction Uncertainty Quantification for d33 in (K0.5Na0.5) NbO3-Based Lead-Free Piezoelectric Ceramics

## TL;DR

This paper introduces an interpretable machine learning framework with uncertainty quantification to predict and understand the piezoelectric properties of lead-free ceramics.

## Contribution

A physics-informed ML framework with uncertainty quantification and interpretability for predicting d33 in KNN-based ceramics.

## Key findings

- The framework achieves high accuracy (R2 ≈ 0.81) in predicting the piezoelectric coefficient d33.
- Sintering temperature, B-site electronic anisotropy, and A-site ionic displacement are key factors governing d33.
- Uncertainty quantification reflects the confidence of ML predictions, not experimental measurement errors.

## Abstract

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 (K0.5Na0.5) 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.