🤖 AI Summary
This work proposes a Riemannian geometry–based semantic modeling approach to structurally represent scientific literature and uncover its knowledge associations and evolutionary trajectories. Documents are embedded into a two-dimensional knowledge manifold using character-level n-gram TF-IDF vectors, and knowledge positioning, directional gradient analysis, and synthetic research directions are achieved through constrained stress minimization, smoothed particle hydrodynamics (SPH) interpolation, Gaussian process regression, and geodesic optimization. The method innovatively integrates Riemannian geometry with SPH interpolation for semantic modeling of scholarly texts, enabling cross-domain conceptual bridging and the generation of semantically coherent synthetic papers. Evaluated on datasets from composite materials and aerospace engineering, the approach successfully reproduces established research clusters and yields geodesic paths that exhibit plausible conceptual transitions, thereby demonstrating its effectiveness and novelty.
📝 Abstract
We present the knowledge manifold: a Riemannian geometric space in which a corpus of documents is arranged according to semantic positional relationships derived from character n-gram TF-IDF representations. The framework proceeds in five tightly coupled stages. First, each document is converted to a character-level n-gram TF-IDF vector (4-7 grams, up to 250,000 features, L2-normalized) and embedded in a two-dimensional knowledge map via constrained stress minimization with repulsion, variance, and centering regularizers. Second, knowledge at an arbitrary query point is estimated through Smoothed Particle Hydrodynamics (SPH) interpolation using a cubic-spline kernel, yielding an interpolated TF-IDF feature vector that can be linguistically characterized. Third, directional knowledge gradients at 0, 45, and 90 degrees are computed from the SPH interpolation map, and pairwise directional similarity is quantified via inner product and cosine similarity. Fourth, a Gaussian Process Regression (GPR) model, with a Constant x RBF + White kernel fitted on a 10-dimensional SVD projection, provides a Bayesian posterior mean, uncertainty estimate, and per-document contribution rate at the query point. Fifth, geodesics in the knowledge space are obtained by minimizing a discrete Riemannian path energy derived from the SPH-induced metric tensor, using L-BFGS-B with seven deterministic initial-path candidates. We apply the formulation to a corpus of 20 papers in fiber-reinforced composite materials and aerospace structural mechanics, showing that the semantic map recovers meaningful research clusters, geodesic paths reveal natural conceptual bridges between distant topics, and SPH/GPR interpolation enables the generation of virtual knowledge: hypothetical paper abstracts describing unstudied but geometrically predicted research directions.