Neural Contraction Metrics with Formal Guarantees for Discrete-Time Nonlinear Dynamical Systems

• URL: http://arxiv.org/abs/2504.17102v1
• Date: Wed, 23 Apr 2025 21:27:32 GMT
• Title: Neural Contraction Metrics with Formal Guarantees for Discrete-Time Nonlinear Dynamical Systems
• Authors: Haoyu Li, Xiangru Zhong, Bin Hu, Huan Zhang,
• Abstract summary: Contraction metrics provide a powerful framework for analyzing stability, robustness, and convergence of various dynamical systems.<n>However, identifying these metrics for complex nonlinear systems remains an open challenge due to the lack of effective tools.<n>This paper develops verifiable contraction metrics for discrete scalable nonlinear systems.
• Score: 17.905596843865705
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Contraction metrics are crucial in control theory because they provide a powerful framework for analyzing stability, robustness, and convergence of various dynamical systems. However, identifying these metrics for complex nonlinear systems remains an open challenge due to the lack of scalable and effective tools. This paper explores the approach of learning verifiable contraction metrics parametrized as neural networks (NNs) for discrete-time nonlinear dynamical systems. While prior works on formal verification of contraction metrics for general nonlinear systems have focused on convex optimization methods (e.g. linear matrix inequalities, etc) under the assumption of continuously differentiable dynamics, the growing prevalence of NN-based controllers, often utilizing ReLU activations, introduces challenges due to the non-smooth nature of the resulting closed-loop dynamics. To bridge this gap, we establish a new sufficient condition for establishing formal neural contraction metrics for general discrete-time nonlinear systems assuming only the continuity of the dynamics. We show that from a computational perspective, our sufficient condition can be efficiently verified using the state-of-the-art neural network verifier $\alpha,\!\beta$-CROWN, which scales up non-convex neural network verification via novel integration of symbolic linear bound propagation and branch-and-bound. Built upon our analysis tool, we further develop a learning method for synthesizing neural contraction metrics from sampled data. Finally, our approach is validated through the successful synthesis and verification of NN contraction metrics for various nonlinear examples.

Related papers

• Generative System Dynamics in Recurrent Neural Networks [56.958984970518564]
  We investigate the continuous time dynamics of Recurrent Neural Networks (RNNs)<n>We show that skew-symmetric weight matrices are fundamental to enable stable limit cycles in both linear and nonlinear configurations.<n> Numerical simulations showcase how nonlinear activation functions not only maintain limit cycles, but also enhance the numerical stability of the system integration process.
  arXiv  Detail & Related papers  (2025-04-16T10:39:43Z)
• Recurrent Stochastic Configuration Networks with Hybrid Regularization for Nonlinear Dynamics Modelling [3.8719670789415925]
  Recurrent configuration networks (RSCNs) have shown great potential in modelling nonlinear dynamic systems with uncertainties.<n>This paper presents an RSCN with hybrid regularization to enhance both the learning capacity and generalization performance of the network.
  arXiv  Detail & Related papers  (2024-11-26T03:06:39Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Stochastic Nonlinear Control via Finite-dimensional Spectral Dynamic Embedding [21.38845517949153]
  This paper presents an approach, Spectral Dynamics Embedding Control (SDEC), to optimal control for nonlinear systems. We use an infinite-dimensional feature to linearly represent the state-action value function and exploits finite-dimensional truncation approximation for practical implementation.
  arXiv  Detail & Related papers  (2023-04-08T04:23:46Z)
• Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty [8.815974147041048]
  We develop a novel framework to identify parsimonious governing equations of nonlinear (spatiotemporal) dynamics from sparse, noisy data with quantified uncertainty. The proposed algorithm is evaluated on multiple nonlinear dynamical systems governed by canonical ordinary and partial differential equations.
  arXiv  Detail & Related papers  (2022-10-14T20:37:36Z)
• Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations [114.17826109037048]
  Ordinary Differential Equations (ODEs) have recently gained a lot of attention in machine learning. theoretical aspects, e.g., identifiability and properties of statistical estimation are still obscure. This paper derives a sufficient condition for the identifiability of homogeneous linear ODE systems from a sequence of equally-spaced error-free observations sampled from a single trajectory.
  arXiv  Detail & Related papers  (2022-10-12T06:46:38Z)
• A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization [1.2882319878552302]
  This paper develops a new iterative learning algorithm for complex turbulent systems with partial observations. It alternates between identifying model structures, recovering unobserved variables, and estimating parameters. Numerical experiments show that the new algorithm succeeds in identifying the model structure and providing suitable parameterizations for many complex nonlinear systems.
  arXiv  Detail & Related papers  (2022-08-19T00:35:03Z)
• A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study [68.8204255655161]
  We investigate and compare the performance of several local and global smoothing techniques to a priori denoise the state measurements. We show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point.
  arXiv  Detail & Related papers  (2022-01-29T23:31:25Z)
• A Theoretical Overview of Neural Contraction Metrics for Learning-based Control with Guaranteed Stability [7.963506386866862]
  This paper presents a neural network model of an optimal contraction metric and corresponding differential Lyapunov function. Its innovation lies in providing formal robustness guarantees for learning-based control frameworks.
  arXiv  Detail & Related papers  (2021-10-02T00:28:49Z)
• Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers [0.0]
  We analyze the stability of feedback interconnections of a linear time-invariant system with a neural network nonlinearity in discrete time. Our approach provides a flexible and versatile framework for stability analysis of feedback interconnections with neural network nonlinearities.
  arXiv  Detail & Related papers  (2021-03-31T14:21:03Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• On dissipative symplectic integration with applications to gradient-based optimization [77.34726150561087]
  We propose a geometric framework in which discretizations can be realized systematically. We show that a generalization of symplectic to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error.
  arXiv  Detail & Related papers  (2020-04-15T00:36:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.