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ReLU Neural Network Approximation to Smooth Functional Operator: Dimensional Decay and Error Analysis

arXiv · AI, language, vision and robotics · article · Sep 14, 2026 · UTC

We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by deep ReLU neural networks. Writing the functional input as $X(t)=\sum_{d\geq1}ξ_dν_d(t)$, we quantify the importance of coordinate $d$ through $w_ds_d$, where $s_d$ bounds the magnitude of the corresponding basis score and $w_d$ controls the directional Fréchet sensitivity of the target functional. Our constructive analysis combines coordinate truncation, anisotropic partitioning, local Taylor approximation, and ReLU network realization, while allowing unrestricted inter

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