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

תקציר מקורי באנגליתarXiv:2609.15355v1 Announce Type: cross Abstract: 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 $X(t)=\sum_{d\geq1}\xi_d\nu_d(t)$, 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 Fr\'echet sensitivity of the target functional. Our constructive analysis combines coordinate truncation, anisotropic partitioning, local Taylor approximation, and ReLU network realization, while allowing unrestricted interactions among the retained coordinates. We establish a general nonasymptotic upper bound for the uniform approximation error and a complementary