关于可逆神经网络对双 Lipschitz 映射的逼近
On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks
AI总结：
本文分析基于耦合的 INN 对紧域上双 Lipschitz 映射的正、逆映射同时逼近能力，并结合模型约简、PCA 与 INN 提出无限维情形的逼近方法及误差分析，数值验证其可用于参数化二阶椭圆问题解算子。
摘要：
可逆神经网络（INN）代表了一类重要的深度神经网络架构，已在多种应用中得到广泛使用。INN 的通用逼近性质近来也已得到确立。然而，INN 的逼近速率在很大程度上仍然缺失。在本工作中，我们分析了一类基于耦合的 INN 在紧域上逼近双 Lipschitz 连续映射的能力，结果表明它能够同时很好地逼近正向映射和逆向映射。此外，我们开发了一种在无限维空间上逼近双 Lipschitz 映射的方法，该方法通过将模型约简、主成分分析以及用于逼近约简映射的 INN 相结合，来同时逼近正向映射和逆向映射；同时，我们分析了该方法的总体逼近误差。初步数值结果表明，该方法用于逼近参数化二阶椭圆问题的解算子具有可行性。
英文摘要：
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.
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