提出可逆神经动力模型BiLipREN，保证正向与逆向都稳定且对扰动不敏感。
Robustly Invertible Nonlinear Dynamics and the BiLipREN: Contracting Neural Models with Contracting Inverses
- 基于收缩性与增量稳定性设计可逆神经网络，确保正反向动态均稳定。
- 模型对输入扰动和初始条件变化具有鲁棒性，输出可准确还原输入序列。
- 适合需要高可靠性逆过程的系统建模，如状态估计与控制设计。
本文从收缩性与增量稳定性角度研究非线性动力系统的可逆性，提出一种新型可逆递归神经模型——BiLipREN。若一个非线性状态空间模型存在可实现的逆模型，且正向模型及其逆模型均为收缩型（即增量指数稳定）并满足Lipschitz连续性（增量增益有界），则称其为鲁棒可逆。该双Lipschitz性质确保了对输入扰动的鲁棒性，以及在初始条件和测量输出受微小扰动时仍能可靠重建输入序列。在此基础上，我们提出参数化神经动态模型：双Lipschitz递归平衡网络（biLipREN），其可逆性由构造保证。此外，biLipREN可与正交线性系统组合，构建更通用的双Lipschitz动态模型，例如非线性最小相位/全通（内/外）分解的类比。通过数值示例展示了该方法的有效性。
原文摘要 · Abstract (English)
We study the invertibility of nonlinear dynamical systems from the perspective of contraction and incremental stability analysis and propose a new invertible recurrent neural model: the BiLipREN. In particular, we consider a nonlinear state space model to be robustly invertible if an inverse exists with a state space realisation, and both the forward model and its inverse are contracting, i.e. incrementally exponentially stable, and Lipschitz, i.e. have bounded incremental gain. This property of bi-Lipschitzness implies both robustness in the sense of sensitivity to input perturbations, as well as robust distinguishability of different inputs from their corresponding outputs, i.e. the inverse model robustly reconstructs the input sequence despite small perturbations to the initial conditions and measured output. Building on this foundation, we propose a parameterization of neural dynamic models: bi-Lipschitz recurrent equilibrium networks (biLipREN), which are robustly invertible by construction. Moreover, biLipRENs can be composed with orthogonal linear systems to construct more general bi-Lipschitz dynamic models, e.g., a nonlinear analogue of minimum-phase/all-pass (inner/outer) factorization. We illustrate the utility of our proposed approach with numerical examples.
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