416 Figure 1 – Convolutional VAE representation model: learning a latent process state from 15-min multivariate observation windows. 5.2. MDN-RNN ARCHITECTURE The dynamics model is a Mixture-Density Recurrent Neural Network (MDN-RNN) that predicts the future latent process state conditioned on recent latent history, operator actions, and exogenous disturbances (Figure 2). At each time step, the input is the concatenation of: (i) the latent vector <>produced by the CVAE encoder from the observation window, (ii) the action variables k>(setpoints/manipulated variables), and (iii) the exogenous variables [>(non-manipulated but influential variables). These inputs are arranged as sequences of length :(15 steps in this work), so the RNN learns the short-term temporal dependencies and multivariable couplings that drive the process evolution. The recurrent backbone is a GRU network with 256 hidden units. Instead of outputting a single point prediction, the network feeds an MDN head that parameterizes a mixture of ;=8diagonal-covariance Gaussian components for the target latent state at horizon ℎ, i.e., <>?F. Concretely, the MDN outputs mixture weights í+, means 7+, and standard deviations 8+defining the conditional density (eq 2) 1(<>?F ∣ history)=Cí+ . +/! °Ö<>?F; 7+,diag(8+#)Ü…([ü 2) Training minimizes the negative log-likelihood (NLL) (eq 3) of the realized target latent state under this mixture (eq. 2), which enables heteroscedastic and potentially multimodal uncertainty. :MO4 = −ñlogÖ1(<>?F ∣ history)Üô…([ü 3) For inference (Figure 2), we use the analytical mixture expectation as a point forecast and estimate P10–P90 bands by Monte Carlo sampling from the predicted mixture (N = 200 samples). Predicted latent states are then decoded through the CVAE decoder to obtain probabilistic forecasts in the original variable space.
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