Causality-driven multi-stage physics-informed neural networks for high-fidelity simulation of key variable fields in industrial reactors

Industrial reactors, such as smelting kettles and polymerization reactors, form the critical cyber-physical systems (CPS) at the forefront of modern process manufacturing. The accurate simulation of their internal multi-physics fields is fundamental to achieving state observability, intelligent control, and ultimately, smart manufacturing. Constrained by scarce measurable data, a complex reaction mechanism, and strong multi-physics coupling, existing models struggle to accurately simulate the key variable fields. To address this, we propose, for the first time, a causality-driven multi-stage physics-informed neural network (CD-MSPINN). First, we derive theoretical constraints from the partial differential equations and boundary conditions within the reactor, ensuring that the simulation rigorously obeys the physics of industrial reactors and substantially mitigates the data-scarcity problem. Second, we design a causality-driven multi-stage training strategy to decouple the multi-physics fields and propose a causality-guided neural network weight-initialization method that balances industrial first-principles with training efficiency. Validation based on on-site polyester fiber polymerization data and two public turbulent-flow industrial system datasets demonstrates that CD-MSPINN significantly outperforms existing state-of-the-art algorithms in accuracy, computational time, and data dependency, confirming its effectiveness and superiority.