415 percent solids (PV/SP), inlet water flow, mill speed, power, impact intensity, load width and toe position, filling level, feed/discharge pressures, and pebble tonnage. Signals were grouped into observations (process states/outputs), actions (operator/setpoint variables), and exogenous inputs (short-term non-manipulable conditions). Data were time-ordered and preprocessed with (i) optional operating-regime filtering, (ii) outlier clipping, (iii) removal of missing values in critical variables with limited within-segment interpolation (≤1-minute gap), and (iv) segmentation into continuous blocks with at least 1 hour of valid data. To avoid temporal leakage, segments were split chronologically into train/validation/test (with a window gap when required). The CVAE was trained only on observation windows to learn a latent process state, and the MDN-RNN was trained to forecast 5/15/30-minute horizons using latent history plus action and exogenous variables. 5. METHOD This section describes the proposed probabilistic digital twin as a representation–dynamics pipeline. First, a convolutional variational autoencoder (CVAE) learns a compact latent state of the SAG milling process from multivariate observation windows. Second, a mixture-density recurrent neural network (MDNRNN) models the stochastic evolution of this latent state conditioned on recent history, operator actions (setpoints), and exogenous drivers. This separation enables plant-grade multihorizon forecasting and whatif scenario simulation while providing uncertainty estimates through predictive distributions. 5.1. CVAE ARCHITECTURE The representation model is a convolutional variational autoencoder (CVAE) (figure 1) trained to encode the short-term process state into a low-dimensional latent space. Inputs are 15-minute windows sampled at 1-minute resolution, arranged as a tensor of shape 15-time steps × 13 observation variables. Observation variables correspond to process states/outcomes (e.g., throughput, pressures, power, fill level, and size-fraction measurements) and are used exclusively in this stage so that the learned representation captures the nonlinear process state without directly incorporating operator setpoints. The encoder consists of residual convolutional blocks that capture local temporal dependencies and multivariable coupling. It parameterizes a diagonal Gaussian posterior üR(< ∣ () = °(7, 8)with latent dimensionality 4S =8 (i.e., 7 ∈ℝTand log8∈ℝT). The decoder mirrors the encoder using transposed convolutions to reconstruct the observed window, producing (¶with the same dimensionality as the input. Training optimizes a β-ELBO (β-VAE objective), where the Kullback–Leibler (KL) divergence term is weighted to control the trade-off between reconstruction fidelity and latent regularization: ℒ(®,©)=™U+(<∣()[−log1A(( ∣ <)] + G O.N çüR(< ∣ () ∥ 1(<)é…([ü 1) In practice, using G≠1(with warm-up when applicable) improves training stability and mitigates posterior collapse, yielding a latent space that remains informative for the downstream dynamics model. The CVAE is assessed using reconstruction error and the magnitude of the KL term on train/validation and heldout test data to verify consistency and generalization of the learned state representation.
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