🤖 AI Summary
This work addresses the challenge of robustly modeling invertibility for nonlinear dynamical systems. We introduce “bi-Lipschitz invertibility”—a novel definition requiring both forward and inverse mappings to be contractive (i.e., incrementally exponentially stable) and Lipschitz continuous. To realize this property, we propose BiLipREN, a neural architecture built upon recurrent equilibrium networks (RENs), incorporating orthogonal linear transformations and implicit layers, with explicit constraints enforcing bidirectional contraction. Its parameterization inherently supports minimum-phase/all-pass decomposition. We provide rigorous theoretical proof that BiLipREN strictly satisfies bi-Lipschitz invertibility. Numerical experiments demonstrate its strong robustness in reconstructing initial states and outputs under perturbations, significantly improving invertible stability and output distinguishability for nonlinear dynamical systems.
📝 Abstract
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.