We present a data-driven approach to the quantitative verification of probabilistic programs and stochastic dynamical models. Our approach leverages neural networks to compute tight and sound bounds for the probability that a stochastic process hits a target condition within finite time. This problem subsumes a variety of quantitative verification questions, from the reachability and safety analysis of discrete-time stochastic dynamical models, to the study of assertion-violation and termination analysis of probabilistic programs. We rely on neural networks to represent supermartingale certificates that yield such probability bounds, which we compute using a counterexample-guided inductive synthesis loop: we train the neural certificate while tightening the probability bound over samples of the state space using stochastic optimisation, and then we formally check the certificate's validity over every possible state using satisfiability modulo theories; if we receive a counterexample, we add it to our set of samples and repeat the loop until validity is confirmed. We demonstrate on a diverse set of benchmarks that, thanks to the expressive power of neural networks, our method yields smaller or comparable probability bounds than existing symbolic methods in all cases, and that our approach succeeds on models that are entirely beyond the reach of such alternative techniques.

This paper introduces Fossil 2.0, an advanced software tool designed for synthesizing certificates such as Lyapunov and barrier functions for dynamical systems represented by ordinary differential equations and difference equations. Fossil 2.0 features a range of significant enhancements, including improved user interfaces, an expanded library of certificates, controller synthesis capabilities, and an extensible architecture. These advancements are detailed as part of this paper. The core of Fossil is a counterexample-guided inductive synthesis (CEGIS) framework that ensures soundness. The tool employs neural networks as templates to generate candidate functions, which are rigorously validated using a satisfiability modulo theories (SMT) solver. Key improvements over the previous release include support for a broader class of certificates, integration of control law synthesis, and compatibility with discrete-time models.

Neural-based, data-driven analysis and control of dynamical systems have been recently investigated and have shown great promise, e.g. for safety verification or stability analysis. Indeed, not only do neural networks allow for an entirely model-free, data-driven approach, but also for handling arbitrary complex functions via their power of representation (as opposed to, e.g. algebraic optimization techniques that are restricted to polynomial functions). Whilst classical Lyapunov techniques allow to provide a formal and robust guarantee of stability of a switched dynamical system, very little is yet known about correctness guarantees for Neural Lyapunov functions, nor about their performance (amount of data needed for a certain accuracy). We formally introduce Neural Lyapunov functions for the stability analysis of switched linear systems: we benchmark them on this paradigmatic problem, which is notoriously difficult (and in general Turing-undecidable), but which admits existing recently-developed technologies and theoretical results. Inspired by switched systems theory, we provide theoretical guarantees on the representative power of neural networks, leveraging recent results from the ML community. We additionally experimentally display how Neural Lyapunov functions compete with state-of-the-art results and techniques, while admitting a wide range of improvement, both in theory and in practice. This study intends to improve our understanding of the opportunities and current limitations of neural-based data-driven analysis and control of complex dynamical systems.

This paper presents Fossil 2.0, a new major release of a software tool for the synthesis of certificates (e.g., Lyapunov and barrier functions) for dynamical systems modelled as ordinary differential and difference equations. Fossil 2.0 is much improved from its original release, including new interfaces, a significantly expanded certificate portfolio, controller synthesis and enhanced extensibility. We present these new features as part of this tool paper. Fossil implements a counterexample-guided inductive synthesis (CEGIS) loop ensuring the soundness of the method. Our tool uses neural networks as templates to generate candidate functions, which are then formally proven by an satisfiability modulo theories solver acting as an assertion verifier. Improvements with respect to the first release include a wider range of certificates, synthesis of control laws, and support for discrete-time models.

We present a novel method for the safety verification of nonlinear dynamical models that uses neural networks to represent abstractions of their dynamics. Neural networks have extensively been used before as approximators; in this work, we make a step further and use them for the first time as abstractions. For a given dynamical model, our method synthesises a neural network that overapproximates its dynamics by ensuring an arbitrarily tight, formally certified bound on the approximation error. For this purpose, we employ a counterexample-guided inductive synthesis procedure. We show that this produces a neural ODE with non-deterministic disturbances that constitutes a formal abstraction of the concrete model under analysis. This guarantees a fundamental property: if the abstract model is safe, i.e., free from any initialised trajectory that reaches an undesirable state, then the concrete model is also safe. By using neural ODEs with ReLU activation functions as abstractions, we cast the safety verification problem for nonlinear dynamical models into that of hybrid automata with affine dynamics, which we verify using SpaceEx. We demonstrate that our approach performs comparably to the mature tool Flow* on existing benchmark nonlinear models. We additionally demonstrate and that it is effective on models that do not exhibit local Lipschitz continuity, which are out of reach to the existing technologies.

This paper accompanies FOSSIL: a software tool for the synthesis of Lyapunov functions and of barrier certificates (or functions) for dynamical systems modelled as differential equations. Lyapunov functions are formal certificates for stability analysis, whereas barrier functions are formal certificates for the safety of dynamical models. FOSSIL is sound and automatic thanks to a counterexample-guided inductive synthesis loop. This method exploits the flexibility of candidate functions generated by training neural network templates, the formal assertions provided by a verifier (namely, an SMT solver), and finally new procedures to ease the exchange of information between the two mentioned components. We endow the tool with features of usability, scalability, and robustness---all of which are showcased on benchmarks.