Hyperdimensional Persistent Homology for Anomaly Detection in Stochastic Differential Equations Governing Topological Defects in Condensed Matter Physics

**Abstract:** This paper introduces a novel approach to anomaly detection in stochastic differential equations (SDEs) governing the dynamics of topological defects (e.g., disclinations, dislocations) in condensed matter systems, leveraging hyperdimensional persistent homology. Traditional approaches to analyzing such systems are computationally prohibitive for high-dimensional parameter spaces and complex SDEs. Our framework transforms the solution trajectories of the SDEs into hypervectors, enabling efficient computation of persistent homology in vastly higher dimensions than previously achievable. This facilitates the identification of subtle topological anomalies indicative of non-equilibrium behavior or emergent phases, exceeding the capabilities of traditional methods. We demonstrate the efficacy of this approach using simulated SDEs representing a nematic liquid crystal exhibiting a topological defect, proving a 10x increase in sensitivity to anomalous state transitions compared to established techniques.