Evaluation of PINL's performance. (a)–(c) Effects of the integration of the structural knowledge on the performance of PINL. When no prior knowledge is available, i.e., model-free digital twin (MFDT), the structural code A is randomly generated to construct the graph skeleton. (d) Effects of network size on training time obtained from 20 trails across various ICSs. (e)–(h) Comparisons with representative machine learning-based DTs. (e)–(g) Accuracy comparison in prediction, decision-making, and anomaly detection. The proposed PINL scheme achieves comparable accuracy in capturing system dynamics and control policies in contrast to FNN-DT and STGNN-DT. (h) Model training time. A longer training time indicates greater computational demands and a lengthy testing process for parameter adjustments. Notably, the PINL has the least training time, underscoring its potential for practical implementation in ICSs with minimal effort.