The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems (1808.00924v2)

Abstract: Learning algorithms have shown considerable prowess in simulation by allowing robots to adapt to uncertain environments and improve their performance. However, such algorithms are rarely used in practice on safety-critical systems, since the learned policy typically does not yield any safety guarantees. That is, the required exploration may cause physical harm to the robot or its environment. In this paper, we present a method to learn accurate safety certificates for nonlinear, closed-loop dynamical systems. Specifically, we construct a neural network Lyapunov function and a training algorithm that adapts it to the shape of the largest safe region in the state space. The algorithm relies only on knowledge of inputs and outputs of the dynamics, rather than on any specific model structure. We demonstrate our method by learning the safe region of attraction for a simulated inverted pendulum. Furthermore, we discuss how our method can be used in safe learning algorithms together with statistical models of dynamical systems.

• The paper introduces a neural network-based Lyapunov function that adaptively certifies stability for safe learning in nonlinear dynamical systems.
• It leverages neural networks to capture larger regions of attraction compared to traditional SOS or LQR-based approaches, as demonstrated on an inverted pendulum.
• The adaptive certification framework enhances safe exploration in reinforcement learning and robust autonomous control in uncertain environments.

Overview of the Lyapunov Neural Network for Safe Learning

In the paper "The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems" by Richards et al., the authors address a pressing challenge in robotics and artificial intelligence: enabling safe learning in dynamic and uncertain environments. The paper introduces a method to construct neural network-based Lyapunov functions as adaptive safety certificates for nonlinear dynamical systems. This approach is of particular relevance for safety-critical applications, such as autonomous vehicles, where the exploration typically required by learning algorithms must be bounded by safety constraints.

Methodology

The core contribution lies in the synthesis of Lyapunov functions using neural networks, offering both adaptability and rigorous stability guarantees. Traditional methods for identifying Lyapunov functions often restrict them to particular function classes, such as sum-of-squares (SOS) polynomials, which may not suitably approximate the region of attraction (ROA) for complex, nonlinear dynamics. The neural network Lyapunov candidate introduced here overcomes these limitations by leveraging the universal approximation capabilities of neural networks, trained through a novel algorithm to capture the largest certifiable safe region.

The proposed method constructs a Lyapunov function parameterized by a neural network, ensuring the function is positive-definite and Lipschitz continuous throughout the state space. This structurally guarantees that the candidate satisfies the necessary conditions for stability certification. The algorithm iteratively trains the neural network to expand and re-shape the level sets of the Lyapunov function, closely matching the true ROA and thereby maximizing the certifiable safe set.

Key Results

The paper demonstrates its methodology using a simulated inverted pendulum, a benchmark problem for dynamical system control. Compared to traditional approaches—such as Lyapunov functions derived from linear-quadratic regulators (LQR) and SOS polynomial functions—the neural network Lyapunov function more effectively captures the true ROA. The quantitative results indicate that the new approach covers a significantly larger proportion of the safe state space compared to the other methods.

Implications

Practically, this work opens paths for more efficient and safer exploration strategies in reinforcement learning (RL) by providing more precise safety boundaries during the learning process. The adaptive potential of the neural network Lyapunov functions could be integrated with statistical learning models, such as Gaussian processes, to provide both deterministic and probabilistic safety guarantees. This enables resilient and adaptable control solutions in uncertain, real-world environments, striking a balance between exploration and exploitation in RL contexts.

Future Directions

There are several avenues for future research based on this work. First, the integration of these neural network Lyapunov functions with real-time learning systems, particularly in high-dimensional robotic applications, would benefit from further exploration. Additionally, investigating the robustness of these networks against model uncertainties and environmental perturbations could enhance their practical utility. There is also potential to refine the proposed algorithm to ensure monotonic convergence towards the maximum certifiable safe region, potentially through advanced optimization and sampling techniques.

In summary, the work provides a robust and flexible framework for safe learning in dynamical systems using neural network-generated Lyapunov functions. By securely expanding the regions where learning and exploration are permitted, this method holds significant promise for advancing the field of safe artificial intelligence.