On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks
This work addresses a gap in approximation rates for INNs, which is incremental but important for applications in domains like computational physics.
The paper tackles the problem of approximating bi-Lipschitz maps using invertible neural networks (INNs), providing an analysis of their capacity and developing an approach for infinite-dimensional spaces, with preliminary numerical results showing feasibility for parameterized elliptic problems.
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.