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We have developed a framework for constructing neural network dynamic models with provable global stability guarantees

Almost Surely Stable Deep Dynamics

NIPS 2020, (2021)

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Abstract

We introduce a method for learning provably stable deep neural network based dynamic models from observed data. Specifically, we consider discrete-time stochastic dynamic models, as they are of particular interest in practical applications such as estimation and control. However, these aspects exacerbate the challenge of guaranteeing st...More

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Introduction
  • Stability is a critical requirement in the design of physical systems. White-box models based on first principles can explicitly account for stability in their design.
  • Deep neural networks (DNNs) are flexible function approximators, well suited for modeling complicated dynamics.
  • Their black-box design makes both physical interpretation and stability analysis challenging.
  • This paper focuses on the construction of provably stable DNN-based dynamic models.
  • These models are amenable to standard deep learning architectures and training practices, while retaining the asymptotic behavior of the underlying dynamics.
Highlights
  • Stability is a critical requirement in the design of physical systems
  • This paper focuses on the construction of provably stable deep neural networks (DNNs)-based dynamic models
  • The examples here deal with low-dimensional state spaces for convenient visualizations, we note that our method is not restricted to this setting, as the dynamic model is based on DNNs and can take any state dimension
  • We have developed a framework for constructing neural network dynamic models with provable global stability guarantees
  • We showed how convexity can be exploited to give a closed-form stable dynamic model, extended this approach to implicitly-defined stable models
Methods
  • The code for the methods is available here: https://github.com/NPLawrence/stochastic_ dynamics.
  • Further details about the experiments and models can be found in Appendix D.
  • The authors give an example dealing with a chaotic system in Appendix C.
  • The examples here deal with low-dimensional state spaces for convenient visualizations, the authors note that the method is not restricted to this setting, as the dynamic model is based on DNNs and can take any state dimension
Results
  • Results are shown in Figure

    3 and correspond to the matrix A=

    1 0.90 in Eq (16). In the first experiment, the authors use training data from the system (16) in which there is no noise.
  • Results are shown in Figure.
  • 3 and correspond to the matrix A=.
  • The authors use training data from the system (16) in which there is no noise.
  • The MDN gives very small variance in its predictions.
  • The predicted mean refers to the dynamics defined by feeding the means through the MDN as ‘states’.
  • The two plots show predictions corresponding to the system (16) with B = 0.1, where the last plot uses implicit dynamics.
Conclusion
  • The authors have developed a framework for constructing neural network dynamic models with provable global stability guarantees.
  • The authors showed how convexity can be exploited to give a closed-form stable dynamic model, extended this approach to implicitly-defined stable models.
  • The latter case can be reduced to a one-dimensional root-finding problem, making a robust and cheap implementation straightforward.
  • Interesting avenues for future work include applications to control and reinforcement learning
Summary
  • Introduction:

    Stability is a critical requirement in the design of physical systems. White-box models based on first principles can explicitly account for stability in their design.
  • Deep neural networks (DNNs) are flexible function approximators, well suited for modeling complicated dynamics.
  • Their black-box design makes both physical interpretation and stability analysis challenging.
  • This paper focuses on the construction of provably stable DNN-based dynamic models.
  • These models are amenable to standard deep learning architectures and training practices, while retaining the asymptotic behavior of the underlying dynamics.
  • Objectives:

    The authors' goal is to construct a DNN representation of f with global stability guarantees about the origin.
  • Methods:

    The code for the methods is available here: https://github.com/NPLawrence/stochastic_ dynamics.
  • Further details about the experiments and models can be found in Appendix D.
  • The authors give an example dealing with a chaotic system in Appendix C.
  • The examples here deal with low-dimensional state spaces for convenient visualizations, the authors note that the method is not restricted to this setting, as the dynamic model is based on DNNs and can take any state dimension
  • Results:

    Results are shown in Figure

    3 and correspond to the matrix A=

    1 0.90 in Eq (16). In the first experiment, the authors use training data from the system (16) in which there is no noise.
  • Results are shown in Figure.
  • 3 and correspond to the matrix A=.
  • The authors use training data from the system (16) in which there is no noise.
  • The MDN gives very small variance in its predictions.
  • The predicted mean refers to the dynamics defined by feeding the means through the MDN as ‘states’.
  • The two plots show predictions corresponding to the system (16) with B = 0.1, where the last plot uses implicit dynamics.
  • Conclusion:

    The authors have developed a framework for constructing neural network dynamic models with provable global stability guarantees.
  • The authors showed how convexity can be exploited to give a closed-form stable dynamic model, extended this approach to implicitly-defined stable models.
  • The latter case can be reduced to a one-dimensional root-finding problem, making a robust and cheap implementation straightforward.
  • Interesting avenues for future work include applications to control and reinforcement learning
Related work
  • Our work is most similar in spirit to that of Manek and Kolter [30]. However, their proposed approach is for deterministic, continuous-time systems, whereas this paper is concerned with learning from noisy discrete measurements xt, xt+1, . . . (rather than observations of the functions x(·) and x (·)). Discrete-time systems with stochastic elements require completely different analysis. Lyapunov stability theory has been deployed in several other recent machine learning and reinforcement learning works. Richards et al [35] introduce a general neural network structure for representing Lyapunov functions. The approach is used to estimate the largest region of attraction for a fixed deterministic, discrete-time system. Umlauft and Hirche [39] consider the stability of nonlinear stochastic models under certain state transition distributions. However, their approach is constrained to provably stable stochastic dynamics under a quadratic Lyapunov function. Khansari-Zadeh and Billard [25] consider Gaussian mixture models for learning continuous-time dynamical systems but only enforce stability of the means. Wang et al [40] develop dynamical models in which the latent dynamics and observations follow Gaussian Processes; stability analysis is later given by Beckers and Hirche [5, 6]. In reinforcement learning, [7, 17, 13] utilize Lyapunov stability to perform safe policy updates within an estimated region of attraction.
Funding
  • Acknowledgments and Disclosure of Funding We gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Honeywell Connected Plant
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