# Physics-Informed Graph Neural Networks for Attack Path Prediction
## 5 Conclusions  
In this paper, we investigated potential applications of Physics-Informed Graph Neural Networks (PIGNNs) for automated attack path prediction. In Section 1, we presented the motivations for this research. We developed three main research objectives: **(1)** building and releasing a dataset for experiments with Deep Neural Network (DNN)-based Attack Path Analysis (APA), **(2)** experimenting with the PIGNN for three risk assessment learning problems (inductive, deductive, and hybrid), and **(3)** reaching SoTA performance, which is evaluated against EPSS, the current SoTA for learning-based attack prediction.  
Section 2 presented our methods for crafting our dataset along with an exploratory data analysis of potential biases and limitations. Then, we introduced our methods for building physics-informed neural networks. By leveraging graph-theory-based properties, we defined a physics-informed loss function, denoted by \(\mathcal{L}_{\text{pinn}}\), which penalizes the prediction model when the predicted path contradicts domain knowledge and expected path properties. We used this loss function to train two distinct architectures. The first is based on a Graph Neural Network (GNN) and multilayer perceptron (MLP) and aims at predicting full attack paths for a given environment, start node and end node. The second architecture we proposed also makes use of a GNN, along with an autoencoder (AE) designed to identify the most likely initial access nodes in the environment given an impact node or an impact node given the initial access node. While the full-path prediction architecture is useful for exploring potential risk scenarios with preset boundaries (e.g., as suggested in many risk management frameworks [4]), the start/end node classification architecture is also useful for incident response and threat analysis (e.g., an adversary has been spotted on a system, and the model can help to determine the initial access of the infection chain).  
Section 3 presented our results after training the models using K-fold cross validation. We obtained a final evaluation F1 score of 0.9308 and \(\text{ROC-AUC}=0.9533\), which is superior to the EPSS performance. For start/end node classification, we obtained an F1 score of 0.9780 and ROC-AUC of 0.9037 for start node prediction and F1 score of 0.8214 and \(\text{ROC-AUC}=0.8081\) for end node prediction. We also included an ablation study, showing that the physics-informed loss function correlates to a model performance increase, and conducted a sensitivity analysis using SHAP. We observed that the full-path model relies more on the topological properties of the graph than the node properties, which is promising for CSKG-based data augmentation strategies. Finally, Section 4 discussed potential research opportunities emerging from this research, along with the limitations of our procedure.  
Conceptualization, M.F.; methodology, M.F.; software, M.F.; validation, M.F., P.-E.A. and M.M.; formal analysis, M.F.; writing--original draft preparation, M.F.; writing--review and editing, M.F., P.-E.A. and M.M.; visualization, M.F.; supervision, P.-E.A. and M.M.; All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
All data and algorithms presented in this paper can be found in the replication package ([[https://github.com/mbdlrocks/PhD_Replication_Package/tree/master/Physics-Informed-GNN%20](https://github.com/mbdlrocks/PhD_Replication_Package/tree/master/Physics-Informed-GNN%20)]([https://github.com/mbdlrocks/PhD_Replication_Package/tree/master/Physics-Informed-GNN%20](https://github.com/mbdlrocks/PhD_Replication_Package/tree/master/Physics-Informed-GNN%20))(PIGNN), accessed on 7 April 2025).  
The authors declare no conflicts of interest.  
Figure A3: Distribution of edge types across graphs.