Fluid antenna systems (FAS) leverage continuous reconfigurable apertures to transcend the spatial limits of fixed-position arrays, yet acquiring full-aperture channel state information (CSI) from sparse observations remains a formidable bottleneck. In this paper, we establish a continuous-field formulation for FAS by recasting channel estimation as a stochastic inverse field-reconstruction problem constrained by the projected Helmholtz model. Through electromagnetic information theory (EIT)-based spectral leakage analysis, we demonstrate that finite-aperture reconstruction induces an intrinsic convolutional relationship in the spectral domain, rendering the estimation task a fundamental frequency-domain inverse deconvolution problem. To solve this, we propose a U-shaped Fourier neural operator (UFNO) specifically tailored for noisy, underdetermined wireless environments. Furthermore, we leverage the Landau--Pollak theorem to derive an aperture-dependent EDoF bound that guides the spatial mode budget, and employ a physics-aware Sobolev loss to promote consistency in both channel values and spatial variations. Unlike conventional learning-based approaches, a single frozen UFNO can be directly evaluated at unseen spatial--frequency resolutions without resolution-specific retraining or explicit output interpolation. Extensive simulations show that our framework significantly outperforms state-of-the-art benchmarks in accuracy while achieving orders-of-magnitude lower inference latency than iterative solvers at high spatial--frequency resolutions.

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