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

arXiv 2607.10026 ↗First posted 10 Jul 2026eess.SY cs.LG cs.SYcited by 0

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Abstract (arXiv)

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.

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References · 26

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14 more
  1. 2021Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustnesscited by 45
  2. 2024Potential Based Diffusion Motion Planningcited by 41
  3. 2020Modeling Continuous Stochastic Processes with Dynamic Normalizing Flowscited by 37
  4. 2020ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flowscited by 33
  5. 2019Learning Surrogate Lossescited by 21
  6. 2020Learning Surrogates via Deep Embeddingcited by 11
  7. 2024Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networkscited by 9
  8. 2019Scaled Relative Graph: Nonexpansive operators via 2D Euclidean Geometrycited by 5
  9. 2021Recurrent Neural Network-based Internal Model Control design for stable nonlinear systemscited by 5
  10. 2025EB-MBD: Emerging-Barrier Model-Based Diffusion for Safe Trajectory Optimization in Highly Constrained Environmentscited by 4
  11. 2023Physics-guided neural networks for feedforward control with input-to-state stability guaranteescited by 3
  12. 2021Graphical Nonlinear System Analysiscited by 2
  13. 2025Robustly Invertible Nonlinear Dynamics and the BiLipREN: Contracting Neural Models with Contracting Inversescited by 1
  14. 2025React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-RENcited by 1

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