Decision trees have gained popularity as interpretable surrogate models for learning-based control policies.

Neural networks (NNs) playing the role of controllers have demonstrated impressive empirical performances on challenging control problems. However, the potential adoption of NN controllers in real-life applications also gives rise to a growing concern over the safety of these neural-network controlled systems (NNCSs), especially when used in safety-critical applications. In this work, we present POLAR-Express, an efficient and precise formal reachability analysis tool for verifying the safety of NNCSs. POLAR-Express uses Taylor model arithmetic to propagate Taylor models (TMs) across a neural network layer-by-layer to compute an overapproximation of the neural-network function. It can be applied to analyze any feed-forward neural network with continuous activation functions. We also present a novel approach to propagate TMs more efficiently and precisely across ReLU activation functions. In addition, POLAR-Express provides parallel computation support for the layer-by-layer propagation of TMs, thus significantly improving the efficiency and scalability over its earlier prototype POLAR. Across the comparison with six other state-of-the-art tools on a diverse set of benchmarks, POLAR-Express achieves the best verification efficiency and tightness in the reachable set analysis.

Deep Reinforcement Learning (DRL) has achieved impressive performance in robotics and autonomous systems (RASs). A key impediment to its deployment in real-life operations is the spuriously unsafe DRL policies--unexplored states may lead the agent to make wrong decisions that may cause hazards, especially in applications where end-to-end controllers of the RAS were trained by DRL. In this paper, we propose a novel quantitative reliability assessment framework for DRL-controlled RASs, leveraging verification evidence generated from formal reliability analysis of neural networks. A two-level verification framework is introduced to check the safety property with respect to inaccurate observations that are due to, e.g., environmental noises and state changes. Reachability verification tools are leveraged at the local level to generate safety evidence of trajectories, while at the global level, we quantify the overall reliability as an aggregated metric of local safety evidence, according to an operational profile. The effectiveness of the proposed verification framework is demonstrated and validated via experiments on real RASs.

We study the verification problem for closed-loop dynamical systems with neural-network controllers (NNCS). This problem is commonly reduced to computing the set of reachable states. When considering dynamical systems and neural networks in isolation, there exist precise approaches for that task based on set representations respectively called Taylor models and zonotopes. However, the combination of these approaches to NNCS is non-trivial because, when converting between the set representations, dependency information gets lost in each control cycle and the accumulated approximation error quickly renders the result useless. We present an algorithm to chain approaches based on Taylor models and zonotopes, yielding a precise reachability algorithm for NNCS. Because the algorithm only acts at the interface of the isolated approaches, it is applicable to general dynamical systems and neural networks and can benefit from future advances in these areas. Our implementation delivers state-of-the-art performance and is the first to successfully analyze all benchmark problems of an annual reachability competition for NNCS.

Gaussian ODE filtering is a probabilistic numerical method to solve ordinary differential equations (ODEs). It computes a Bayesian posterior over the solution from evaluations of the vector field defining the ODE. Its most popular version, which employs an integrated Brownian motion prior, uses Taylor expansions of the mean to extrapolate forward and has the same convergence rates as classical numerical methods. As the solution of many important ODEs are periodic functions (oscillators), we raise the question whether Fourier expansions can also be brought to bear within the framework of Gaussian ODE filtering. To this end, we construct a Fourier state space model for ODEs and a `hybrid' model that combines a Taylor (Brownian motion) and Fourier state space model. We show by experiments how the hybrid model might become useful in cheaply predicting until the end of the time domain.