Kernel-induced statistical representations for continuous stiffness field inversion via Karhunen-Loève expansion

Learning representations of continuous physical fields from sparse observations remains challenging due to the mismatch between high-dimensional spatial variations and limited measurement information. In this work, continuous stiffness inversion is formulated as a structured latent representation learning problem rather than a direct field regression. A Karhunen-Loève (KL) expansion derived from a spatial covariance kernel is introduced as a mathematical representation of continuously varying stiffness fields, projecting the spatial field into a compact coefficient space. The resulting latent variables provide a constrained parameterization of admissible field variations and reduce the ambiguity of the inverse mapping from dynamic responses to spatial properties. By learning a direct mapping from sparse frequency response functions (FRFs) to KL coefficients, the proposed framework reconstructs continuous stiffness fields without intermediate modal feature extraction. Experiments on a simply supported Euler-Bernoulli beam benchmark demonstrate accurate field reconstruction (Field Error=4.06%, $R^2=0.9779$) under 5% measurement noise. The results illustrate how statistical field representations can facilitate continuous inference of physical parameters from sparse dynamic observations.