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

TL;DR

The paper defines robust invertibility for nonlinear dynamics through contraction and bi-Lipschitz properties, and introduces strong input-output monotonicity as a practical building block. It then presents BiLipREN, a bi-Lipschitz recurrent equilibrium network whose forward dynamics and contracting inverse are guaranteed, and shows how static and dynamic orthogonal layers enable expressive, invertible compositions including nonlinear inner–outer factorizations. Theoretical results establish conditions under which BiLipREN is contracting and bi-Lipschitz, with an explicitly invertible partner model, and experiments illustrate nonlinear inner–outer factorization and robust inversion under noise. This framework offers a principled path to robust, invertible learning-based dynami cal models with controllable inversion properties for applications in control, prediction, and generative modeling.

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.

Figures (7)

• Figure 1: The Scaled Relative Graph chaffey2023graphical is defined as a set of complex numbers with $z(u, v) := \left\{ \frac{\| y - z \|}{\| u - v \|} e^{\pm j \angle (u - v, y - z)} \;\middle|\; y = \bm{G}(u), z = \bm{G}(v) \right\}$, where $\angle (u, y) := \text{acos} {(\operatorname{Re} \langle u , y \rangle}{\|u\|^{-1} \|y\|^{-1})} \in [0, \pi]$, illustrated with green dots. This SRG depicts the incremental input-output properties of a $(\mu,\nu)$-biLipschitz operator $\bm{G}$. The ring (blue area) is for all $(\mu,\nu)$-biLipschitz operator $\bm{G}$ while the small circle (red area) is for the operator $\bm{G}$ satisfying the strong input-output monotonicity condition \ref{['eq:in_mono']}.
• Figure 2: Open loop simulation of the outer system, the original system and the composed system under the Gaussian noise (Top), the impulse response of the inner system (Bottom).
• Figure 3: Four coupled nonlinear mass spring dampers.
• Figure 4: One of output sequences using a biLipREN and a contracting REN to fit a 4 coupled mass spring damper.
• Figure 5: Normalized training error (left) and test error (right) of biLipREN and contracting REN.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5
• Theorem 1
• proof
• Definition 6
• Lemma 1