Physics-informed neural network–transformer for dual-objective prediction and mix optimization of backfill materials

To address the issues of traditional backfill material mix design relying on experience and low efficiency, this study proposes a physics-informed neural network (PINN)–transformer method that integrates physical constraints. A dual-task prediction framework is constructed considering material strength and slump, embedding strength development monotonicity, convexity constraints, and slump rheological principles into model training to improve the accuracy and physical reasonableness of the prediction results. Experimental results show that this method improves the mean absolute error (MAE) metric by 6.0% compared to the transformer in strength prediction and improves the slump prediction MAE metric by 6.5%. A multi-objective mix optimization system is established based on prediction results and economic analysis, proposing three optimization strategies adapted to different engineering requirements. This method breaks through the limitations of traditional empirical design and provides efficient and reliable technical support for scientific mix design and engineering decision-making regarding mine backfill materials.