Robust soft sensing with causal and injectivity-preserving Graph Neural Network

Graph Neural Networks (GNNs) excel in soft sensing by effectively modeling complex interdependencies among process variables. This study presents a graph-based framework for improved process quality prediction in nonlinear, dynamic industrial systems. We address two key challenges in chemical process soft sensing: (i) unknown graphs where the structure is not available a priori , and (ii) injectivity issues from scalar features. To resolve non-injective aggregation, where distinct neighborhoods become indistinguishable, we expand the input domain to preserve structural uniqueness in both undirected and directed graphs. We also propose a method for learning directed graphs using Sparse Debiased Dynamic Mode Decomposition, which captures temporal dynamics and produces sparse, interpretable, and noise-resilient representations. An end-to-end framework jointly learns the graph structure and GNN parameters, allowing the graph to adapt during training based on the prediction task. The proposed methods are validated through simulations under varying noise levels and a benchmark case study involving a Sulfur Recovery Unit, demonstrating strong robustness and predictive performance. • Introduces an injectivity-preserving GNN framework for industrial soft sensing. • Proposes sparse, debiased DMD to infer spatiotemporal causal graph structures. • Jointly learns graph topology and GNN weights via an end-to-end loss formulation. • Demonstrates robustness under varying noise through extensive numerical simulations. • Validates superior predictive performance on benchmark Sulfur Recovery Unit process.