Finding a control Lyapunov function (CLF) in a dynamical system with a controller is an effective way to guarantee stability, which is a crucial issue in safety-concerned applications. Recently, deep learning models representing CLFs have been applied into a learner-verifier framework to identify satisfiable candidates. However, the learner treats Lyapunov conditions as complex constraints for optimisation, which is hard to achieve global convergence. It is also too complicated to implement these Lyapunov conditions for verification. To improve this framework, we treat Lyapunov conditions as inductive biases and design a neural CLF and a CLF-based controller guided by this knowledge. This design enables a stable optimisation process with limited constraints, and allows end-to-end learning of both the CLF and the controller. Our approach achieves a higher convergence rate and larger region of attraction (ROA) in learning the CLF compared to existing methods among abundant experiment cases. We also thoroughly reveal why the success rate decreases with previous methods during learning.

Learning-based neural network (NN) control policies have shown impressive empirical performance. However, obtaining stability guarantees and estimates of the region of attraction of these learned neural controllers is challenging due to the lack of stable and scalable training and verification algorithms. Although previous works in this area have achieved great success, much conservatism remains in their frameworks. In this work, we propose a novel two-stage training framework to jointly synthesize a controller and a Lyapunov function for continuous-time systems. By leveraging a Zubov-inspired region of attraction characterization to directly estimate stability boundaries, we propose a novel training-data sampling strategy and a domain-updating mechanism that significantly reduces the conservatism in training. Moreover, unlike existing works on continuous-time systems that rely on an SMT solver to formally verify the Lyapunov condition, we extend state-of-the-art neural network verifier $α,\!β$-CROWN with the capability of performing automatic bound propagation through the Jacobian of dynamical systems and a novel verification scheme that avoids expensive bisection. To demonstrate the effectiveness of our approach, we conduct numerical experiments by synthesizing and verifying controllers on several challenging nonlinear systems across multiple dimensions. We show that our training can yield region of attractions with volume $5 - 1.5\cdot 10^{5}$ times larger compared to the baselines, and our verification on continuous systems can be up to $40-10{,}000$ times faster compared to the traditional SMT solver dReal. Our code is available at https://github.com/Verified-Intelligence/Two-Stage_Neural_Controller_Training.

Learning-based methods hold the promise of solving hard nonlinear control problems in robotics.

Langevin algorithms are popular Markov chain Monte Carlo methods that are often used to solve high-dimensional large-scale sampling problems in machine learning. The most classical Langevin Monte Carlo algorithm is based on the overdamped Langevin dynamics. There are many variants of Langevin dynamics that often show superior performance in practice. In this paper, we provide a unified approach to study the acceleration of the variants of the overdamped Langevin dynamics through the lens of large deviations theory. Numerical experiments using both synthetic and real data are provided to illustrate the efficiency of these variants.

Despite advances in learning-based methods, finding valid Lyapunov functions for nonlinear dynamical systems remains challenging. Current neural network approaches face two main issues: challenges in scalable verification and limited interpretability. To address these, we propose an end-to-end framework using transformers to construct analytical Lyapunov functions (local), which simplifies formal verification, enhances interpretability, and provides valuable insights for control engineers. Our framework consists of a transformer-based trainer that generates candidate Lyapunov functions and a falsifier that verifies candidate expressions and refines the model via risk-seeking policy gradient. Unlike Alfarano et al. (2024), which utilizes pre-training and seeks global Lyapunov functions for low-dimensional systems, our model is trained from scratch via reinforcement learning (RL) and succeeds in finding local Lyapunov functions for high-dimensional and non-polynomial systems. Given the analytical nature of the candidates, we employ efficient optimization methods for falsification during training and formal verification tools for the final verification. We demonstrate the efficiency of our approach on a range of nonlinear dynamical systems with up to ten dimensions and show that it can discover Lyapunov functions not previously identified in the control literature.