Robustly Invertible Nonlinear Dynamics and the BiLipREN: Contracting Neural Models with Contracting Inverses

TL;DR

This paper introduces the BiLipREN, a neural model with guaranteed invertibility and robustness, based on contraction and Lipschitz properties, enabling reliable inverse reconstruction in nonlinear dynamical systems.

Contribution

The paper proposes bi-Lipschitz recurrent equilibrium networks (biLipREN), a novel neural model that is inherently invertible and robust, with applications in nonlinear dynamics and system identification.

Findings

biLipREN is robustly invertible by design

The model maintains stability and invertibility through contraction and Lipschitz constraints

Numerical examples demonstrate the effectiveness of the approach

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.