High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations

Understand

In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality.

• Specifically, we introduce a concept which we refer to as approximation spaces of artificial neural networks and we present several tools to handle those spaces.
• Roughly speaking, approximation spaces consist of sequences of functions which can, in a suitable way, be approximated by ANNs without curse of dimensionality in the sense that the number of required ANN parameters to approximate a function of the sequence with an accuracy $\varepsilon > 0$ grows at most polynomially both in the reciprocal $1/\varepsilon$ of the required accuracy and in the dimension $d \in \mathbb{N} = \{1, 2, 3, \ldots \}$ of the function.
• We show that these approximation spaces are closed under various operations including linear combinations, formations of limits, and infinite compositions.