On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks
Bangti Jin Zehui Zhou Jun Zou
Invertible neural networks (INNs) represent an important class of deep neural
network architectures that have been widely used in several applications. The
universal approximation properties of INNs have also been established recently.
However, the approximation rate of INNs is largely missing. In this work, we
provide an analysis of the capacity of a class of coupling-based INNs to
approximate bi-Lipschitz continuous mappings on a compact domain, and the
result shows that it can well approximate both forward and inverse maps
simultaneously. Furthermore, we develop an approach for approximating
bi-Lipschitz maps on infinite-dimensional spaces that simultaneously
approximate the forward and inverse maps, by combining model reduction with
principal component analysis and INNs for approximating the reduced map, and we
analyze the overall approximation error of the approach. Preliminary numerical
results show the feasibility of the approach for approximating the solution
operator for parameterized second-order elliptic problems.