  We propose an automatic and formally sound method for synthesising Lyapunov
functions for the asymptotic stability of autonomous non-linear systems.
Traditional methods are either analytical and require manual effort or are
numerical but lack of formal soundness. Symbolic computational methods for
Lyapunov functions, which are in between, give formal guarantees but are
typically semi-automatic because they rely on the user to provide appropriate
function templates. We propose a method that finds Lyapunov functions fully
automatically$-$using machine learning$-$while also providing formal
guarantees$-$using satisfiability modulo theories (SMT). We employ a
counterexample-guided approach where a numerical learner and a symbolic
verifier interact to construct provably correct Lyapunov neural networks
(LNNs). The learner trains a neural network that satisfies the Lyapunov
criteria for asymptotic stability over a samples set; the verifier proves via
SMT solving that the criteria are satisfied over the whole domain or augments
the samples set with counterexamples. Our method supports neural networks with
polynomial activation functions and multiple depth and width, which display
wide learning capabilities. We demonstrate our method over several non-trivial
benchmarks and compare it favourably against a numerical optimisation-based
approach, a symbolic template-based approach, and a cognate LNN-based approach.
Our method synthesises Lyapunov functions faster and over wider spatial domains
than the alternatives, yet providing stronger or equal guarantees.

Systems and Control;Machine Learning;Logic in Computer Science;Systems and Control