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.

翻译：可逆神经网络（INNs）是一类重要的深度神经网络架构，已在多个应用领域得到广泛使用。近年来，INNs的通用逼近性质已被建立，但其逼近速率问题仍存在较大空白。本文针对一类基于耦合结构的INNs在紧致域上逼近双Lipschitz连续映射的能力进行分析，结果表明这类网络能够同时有效逼近正映射和逆映射。进一步地，我们通过结合模型降阶与主成分分析及用于逼近降阶映射的INNs，提出一种在无限维空间中同时逼近正映射和逆映射双Lipschitz映射的方法，并分析了该方法的总体逼近误差。初步数值实验表明，该方法在逼近参数化二阶椭圆问题解算子方面具有可行性。