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 (setpoints/manipulated variables), and (iii) the exogenous variables (nonmanipulated 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., +ℎ. Concretely, the MDN outputs mixture weights , means , and standard deviations defining the conditional density (eq 2) ( +ℎ ∣ history)=∑ =1 ( +ℎ; ,diag( 2))…( 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. = −[log( ( +ℎ ∣ history))]…( 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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