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

arXiv: https://arxiv.org/abs/2505.03069 · HTML: https://mlsanity.com/paper/2505.03069/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 48 references that are in the map.
- First posted on arXiv 5 May 2025.

## Builds on (most cited first, 13 of 13)

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Lipschitz Bounded Equilibrium Networks](https://mlsanity.com/paper/2010.01732/) (2020)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Potential Based Diffusion Motion Planning](https://mlsanity.com/paper/2407.06169/) (2024)
- [ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows](https://mlsanity.com/paper/2010.13129/) (2020)
- [Generative Modeling of Turbulence](https://mlsanity.com/paper/2112.02548/) (2021)
- [Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networks](https://mlsanity.com/paper/2402.01344/) (2024)
- [Physics-guided neural networks for feedforward control with input-to-state stability guarantees](https://mlsanity.com/paper/2301.08568/) (2023)
- [Graphical Nonlinear System Analysis](https://mlsanity.com/paper/2107.11272/) (2021)

## Built on by (most cited first, 1 of 1)

- [Robustly Invertible Nonlinear Dynamics and the BiLipREN: From Inversion-Based Control to Generative Trajectory Modelling](https://mlsanity.com/paper/2607.10026/) (2026)

## Abstract (arXiv)

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.