Robustly Invertible Nonlinear Dynamics and the BiLipREN: From Inversion-Based Control to Generative Trajectory Modelling

2026-07-10

AI Summary

Nonlinear dynamical systems often lack robust invertibility with respect to disturbances and initial state mismatches, limiting their reliability in control and generative modeling. This work proposes a robustly invertible nonlinear dynamical system framework that, for the first time, constructs a causal inverse system by composing strongly input–output monotone dynamic layers with static orthogonal layers. The resulting recurrent neural network ensures both forward and inverse dynamics are contracting and bi-Lipschitz, and yields a nonlinear minimum-phase/all-pass decomposition. A differentiable parameterization is achieved via bi-Lipschitz recurrent equilibrium networks (BiLipRENs), which significantly enhance performance in trajectory optimization, robust control, and complex distribution generation across tasks such as data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows.

Abstract

This paper proposes a new notion of robust invertibility for nonlinear dynamical systems, and introduces constructive parameterizations of recurrent neural network which are robustly invertible by design. We define robust invertibility as the existence of a causal inverse system such that both the forward and inverse systems are contracting and have bounded incremental input-output gains (the system is bi-Lipschitz), implying that both forward prediction and input reconstruction are robust to signal perturbations and initial-state mismatch. We construct robustly invertible recurrent models via series composition of static orthogonal layers and dynamic layers satisfying a strong input-output monotonicity property, and provide a differentiable neural network parameterizations in the form of the bi-Lipschitz recurrent equilibrium network (BiLipREN). Additionally, composition with dynamic orthogonal layers yields a nonlinear minimum-phase/all-pass (a.k.a. inner--outer) factorization. We illustrate the utility of the framework through a series of application examples in data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows, illustrating its utility for robust control, trajectory optimization, and generative modeling of complex trajectory distributions.

Problem

• robust invertibility
• nonlinear dynamical systems
• bi-Lipschitz
• recurrent neural networks
• input reconstruction

Innovation

• robust invertibility
• bi-Lipschitz
• recurrent neural networks
• contraction theory
• normalizing flows