Learning Stable Deep Dynamics Models (2001.06116v1)

Abstract: Deep networks are commonly used to model dynamical systems, predicting how the state of a system will evolve over time (either autonomously or in response to control inputs). Despite the predictive power of these systems, it has been difficult to make formal claims about the basic properties of the learned systems. In this paper, we propose an approach for learning dynamical systems that are guaranteed to be stable over the entire state space. The approach works by jointly learning a dynamics model and Lyapunov function that guarantees non-expansiveness of the dynamics under the learned Lyapunov function. We show that such learning systems are able to model simple dynamical systems and can be combined with additional deep generative models to learn complex dynamics, such as video textures, in a fully end-to-end fashion.

• The paper introduces a novel framework that integrates a Lyapunov function directly into the architecture to certify global stability.
• It employs input convex neural networks to design the Lyapunov function, ensuring global exponential stability without falling into local minima.
• Empirical validation on systems ranging from pendulums to video texture generation shows improved performance and robustness over standard models.

Overview of Learning Stable Deep Dynamics Models

The paper "Learning Stable Deep Dynamics Models" by Gaurav Manek and J. Zico Kolter addresses the challenge of modeling dynamical systems using deep neural networks with formal guarantees of stability over the entire state space. While deep networks hold substantial promise for predicting system evolution, traditional approaches face difficulties when proving the stability of these systems. The authors introduce an innovative method for ensuring global stability by jointly learning the dynamics and a Lyapunov function, integrating non-expansiveness as a hard constraint directly into the model architecture.

Key Contributions

1. Stability Framework: Unlike soft stability constraints commonly employed in loss functions, the proposed model incorporates stability through the Lyapunov function directly in its architecture. This ensures stability across the entire state space, making this approach fundamentally distinct from existing methods that focus on local regions or specific datasets.
2. Lyapunov Function Design: The authors employ input convex neural networks (ICNN) to construct the Lyapunov function, ensuring global exponential stability. These networks, defined through positive definite functions, avoid local minima and affirm the descent properties necessary for stability certification. This Lyapunov-based approach is novel in the context of deep learning models for dynamical systems.
3. Empirical Validation: The paper demonstrates the model's efficacy for simple dynamical systems such as pendulums and complex outputs like video texture generation. In both cases, the stability constraints lead to improved performance compared to generic neural network models.

Theoretical Implications

This work outlines a new pathway for neural network-based modeling of dynamical systems by embedding stability into the system's core design. The theoretical guarantee of stability ensures the trained models' robustness and reliability, addressing concerns about neural network model safety previously highlighted in dynamic system contexts.

Practical Applications

• Autonomous Control Systems: Stable modeling techniques can be leveraged for developing control systems in reinforcement learning settings where stability is paramount.
• Complex Video Textures: Achieving stable dynamics in latent space, as demonstrated for video texture generation, is critical for realistic simulation and predictive modeling in multimedia applications.
• Robotic Systems: This approach offers potential extensions to control strategies in robotic systems, ensuring stable operations across diverse tasks and environments.

Future Developments

Further exploration could expand stable model integration with control policies or reinforcement learning frameworks. Additionally, enhancing the computational efficiency of this approach and exploring its applicability in high-dimensional systems could yield broader practical benefits.

Overall, this paper significantly contributes to the ongoing discourse on stable model architectures in dynamic systems, providing both theoretical advancements and empirical illustrations of effectiveness. As deep learning continues to intersect with traditional fields like control theory, the proposed method offers a robust blueprint for future research endeavors in AI-driven dynamical modeling.