Approximation of High-Dimensional Functions by ReLU Networks: Curse or No Curse

We present a unified framework linking Barron and Sobolev spaces to analyze the approximation properties of neural networks with ReLU-type activation functions. Within this framework, we establish both classical and new sharp approximation results. We show that for functions in an appropriate Barron space, ReLU networks can achieve high approximation accuracy without suffering from the curse of dimensionality. The same approximation rate is obtained in a corresponding Sobolev space when using linearized ReLU networks, and these results are compared with those for classical global polynomial approximations. A key observation is that the relevant Barron and Sobolev spaces have comparable complexity, as measured by metric entropy. As a consequence, a piecewise linear finite element space expressed in terms of ReLU networks can avoid the curse of dimensionality for sufficiently smooth functions, whereas the classical linear finite element space cannot. Finally, we introduce a bit-centric perspective showing that parameter count alone is not a reliable measure of approximation complexity. Together, these results clarify the mathematical connections between finite element analysis and deep learning, and provide insights into scientific machine learning.