# Physics-Informed Graph Neural Networks for Attack Path Prediction
## 4 Discussion  
The results shown in Section 3 provide valuable insights into the research question outlined in Section 1. Our results indicate that generalizing over environments and adversary characteristics across multiple environments is feasible--even with a limited dataset--using physics-informed learning. Our results show that our approach achieves strong performance across all three risk assessment tasks discussed in Section 2.3: **(1)** deductive reasoning, e.g., identifying the ending of an attack path (consequence) given its starting point (cause), **(2)** inductive reasoning, e.g., determining the starting point of the attack path given its ending, or **(3)** a combination of both, e.g., identifying the full path when both the cause and consequence are fixed.  
Based on our ablation study (see Section 3.2), it appears that incorporating physics-informed loss components enhances training efficiency with limited samples, aligning with existing research on PINNs, as discussed in Section 2.2. In our case, implementing the physics-informed loss increases the F1 score by more than 10%. When studying which features are impactful for the prediction models (see Section 3.3), we observed that edges were more important than most node attributes. This observation is promising for new research opportunities. As discussed in Section 1 and [54], we are currently studying data augmentation strategies based on CSKG. Such data augmentation techniques rely on topological representations of knowledge to represent information, which our network seems to be adapted to. These preliminary results are promising and pave the way for further research. In [54], we discussed ongoing experiments exploring the use of the Common Attack Pattern and Enumeration Catalogue (CAPEC) [106] as a source for data augmentation in a real environment. A key advantage of leveraging graph structures for augmentation is that the graph convolution block presented in Algorithm 1 can process the enriched data directly, without requiring modifications to the existing convolution block.  
Certain design choices and reliance on automated tools currently impose limitations on the full potential of the proposed methods and dataset. Namely, constraints applied during data generation--such as restricting to identity snowball attacks, using fixed-size graphs, and maintaining a limited level of detail--were intentionally set to keep the problem tractable in this first experiment. Having established this proof of concept, we propose expanding the dataset in future research through data augmentation by collecting real-world data and introducing additional nodes and edges to the graph. From a broader perspective, we believe this research contributes insights that extend beyond security risk assessment. Our findings make a compelling case for incorporating more structural  
Figure 9: Sensitivity analysis of node features (_left_) and edge types (_right_) using SHAP for \(\mathcal{M}_{3}\)  
reasoning in the design of deep learning models. While the \"model-agnostic\" nature of neural networks remains a defining characteristic [107], our results suggest that crafting loss functions constrained by domain knowledge can enhance generalization.