$\mathrm{E}^{2}\text{AG}$: Entropy-Regularized Ensemble Adaptive Graph for Industrial Soft Sensor Modeling

Adaptive graph neural networks (AGNNs) have achieved remarkable success in industrial process soft sensing by incorporating explicit features that delineate the relationships between process variables. This article introduces a novel GNN framework, termed entropy-regularized ensemble adaptive graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\mathbf{E}^{\mathbf{2}}\mathbf{AG})$</tex>, aimed at enhancing the predictive accuracy of AGNNs. Specifically, this work pioneers a novel AGNN learning approach based on mirror descent, which is central to ensuring the efficiency of the training procedure and consequently guarantees that the learned graph naturally adheres to the row-normalization requirement intrinsic to the message-passing of GNNs. Subsequently, motivated by multi-head self-attention mechanism, the training of ensembled AGNNs is rigorously examined within this framework, incorporating an entropy regularization term in the learning objective to ensure the diversity of the learned graph. After that, the architecture and training algorithm of the model are then concisely summarized. Finally, to ascertain the efficacy of the proposed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{E}^{\mathbf{2}}\mathbf{AG}$</tex> model, extensive experiments are conducted on real-world industrial datasets. The evaluation focuses on prediction accuracy, model efficacy, and sensitivity analysis, demonstrating the superiority of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{E}^{\mathbf{2}}\mathbf{AG}$</tex> in industrial soft sensing applications.