RELU NEURAL NETWORK APPROXIMATION TO SMOOTH FUNCTIONAL OPERATOR: DIMENSIONAL DECAY AND ERROR ANALYSIS

We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by ReLU neural networks. A key feature in deep learning for functional data is the varying importance of different coordinates/dimensions. Representing the functional input in a basis expansion, we quantify the importance of each coordinate through both the magnitude of its corresponding basis scores and the directional sensitivity of the target functional. Our 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 pseudo-dimension-based lower bound for the worst-case approximation error. Under generalized exponential coordinate decay $w_d{s}_d\asymp \exp (-c{d}^{\rho })$, with $\rho >0$, the upper and lower bounds match at the leading order, which is stretched-exponential in the logarithm of the network size budget, and thus yield the nearly optimal approximation rate. This work characterizes neural network approximation error for general smooth infinite-dimensional functional operator explicitly through the joint dimensional decay of coordinate magnitudes and directional sensitivities.