From Graphs to Gradients: Physics-Inspired Structural Attribution for Cyber-Physical IoT Systems and Beyond
06:00 · July 8, 2026 · arXiv cs.AI RSS
Interpretable explanation methods in Artificial Intelligence aim to uncover the underlying causes and their effects, enabling a deeper understanding of why a system behaves in a certain way under different inputs. Unlike traditional explainability methods, which mainly highlight correlations between input and output variables, causal explanation focuses on interventional questions. By doing so, it provides more robust insights, helping users understand automated decisions, especially in high-risk domains. Recovering an explicit directed causal structure, however, is often impractical in large-scale, hybrid cyber-physical systems with feedback loops and partial observability. This paper introduces a novel framework inspired by statistical mechanics that instead models variable dependencies through an undirected, energy-based representation of cyber-physical IoT systems. Our approach enables rigorous dependency-aware attribution by analysing how variations in the energy landscape reflect the influence of individual components, without recovering a directed causal graph. It also supports reasoning about perturbation effects across hybrid interactions, providing reliable explanations of abnormal behaviours. We empirically examined our framework through simulations on an industrial IoT testbed with hybrid continuous and discrete variables, demonstrating higher attribution accuracy, improved robustness and better scalability than state-of-the-art graph-based approaches. While the attributions are not intended to fully recover the system's generative dynamics, they provide valuable, dependency-aware explanations supporting both human interpretation and downstream predictive and diagnostic tasks. Although demonstrated in industrial IoT security, our framework also applies to other high-dimensional cyber-physical and socio-technical systems requiring principled, structural explanations.
Summary
Interpretable explanation techniques in AI seek to move beyond simple correlations between inputs and outputs by addressing interventional questions about how specific changes affect system behaviour. In large-scale cyber-physical IoT environments, however, the presence of feedback loops, hybrid continuous-discrete variables and partial observability makes recovery of an explicit directed causal graph both unreliable and computationally prohibitive. Traditional graph-based methods therefore encounter scalability limits and sensitivity to model misspecification when applied to industrial settings.
The framework presented here draws on statistical mechanics to represent system dependencies through an undirected energy-based model. Configurations consistent with normal operation receive low energy values, while abnormal or perturbed states map to higher energies. Attribution is performed by examining how local perturbations and global free-energy variations alter the overall landscape, thereby quantifying the influence of individual components without reconstructing a directed causal structure. This formulation also permits reasoning about the propagation of effects across hybrid interactions.
Empirical evaluation on the SWaT industrial testbed demonstrates that the resulting attributions achieve higher accuracy in identifying root causes of abnormal behaviour, maintain robustness under input perturbations and scale more effectively than state-of-the-art graph-based explainers as system dimensionality increases. Although the attributions do not aim to recover the full generative dynamics of the underlying system, they supply dependency-aware explanations that support both human interpretation and downstream diagnostic tasks. The same approach extends in principle to other high-dimensional cyber-physical and socio-technical domains where principled structural explanations are required.
Why it matters
The research directly supports the Dutch AI market's strong emphasis on transparent, ethical AI and its application in high-tech industrial IoT sectors. It offers advanced researchers a novel, scalable methodology for explainability in complex cyber-physical systems, which is highly applicable to Dutch smart industry initiatives.
