DP PINN a dual path physics embedded neural network for cr 2027 Minerals En

The crusher product particle size distribution is a critical technical indicator in mineral processing, governing downstream grinding efficiency, separation recovery, and circuit economics. Existing breakage matrix models struggle to adapt to variations in ore properties and operating conditions, whereas pure data-driven methods lack physical consistency and depend on large datasets. Physics-informed neural networks (PINNs) embed mechanisms through differential operators and are effective for continuous differential equations; for discrete breakage matrix models, however, they can only impose soft algebraic constraints, which regularize fitting but may not ensure robust extrapolation. To address this limitation, we propose a dual-path physics-embedded neural network (DP-PINN) that hard-embeds the classification function C, breakage function B, and breakage matrix model as differentiable computational layers in an independent physics reference path. This path operates in parallel with a data-driven main path, jointly optimizing mechanism constraints and data fitting, while a size-class-adaptive fusion gate balances the two predictions by operating condition. Validation on industrial data from a tungsten ore processing plant shows that DP-PINN achieves a coefficient of determination (R2) of 0.978 ± 0.015 and a mean absolute error (MAE) of 0.83% ± 0.17% on the test set (mean ± standard deviation over ten repeated random splits), outperforming data-driven, physics-based, and standard-PINN baselines on average across three evaluation metrics under the repeated random-split protocol. The embedded physical parameters remain trainable, enabling adaptation to varying ore properties and equipment conditions without offline recalibration and providing a practical route to high-precision particle-size prediction under limited data in comminution circuits.