Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models

Abstract

We introduce an automated, formal, counterexample-based approach to synthesise Barrier Certificates (BC) for the safety verification of continuous and hybrid dynamical models. The approach is underpinned by an inductive framework: this is structured as a sequential loop between a learner, which manipulates a candidate BC structured as a neural network, and a sound verifier, which either certifies the candidate’s validity or generates counter-examples to further guide the learner. We compare the approach against state-of-the-art techniques, over polynomial and non-polynomial dynamical models: the outcomes show that we can synthesise sound BCs up to two orders of magnitude faster, with in particular a stark speedup on the verification engine (up to three orders less), whilst needing a far smaller data set (up to three orders less) for the learning part. Beyond improvements over the state of the art, we further challenge the new approach on a hybrid dynamical model and on larger-dimensional models, and showcase the numerical robustness of our algorithms and codebase.

Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods

“…This empirical loss can be minimized using stochastic gradient descent with respect to the weights and biases of the neural network used to represent V . This approach provides the foundation for many certificate-based learning works; early examples include [28], and later works include [3], [29], [30], [31], [32], [33]. Examples that learn CLF or CBF certificates that imply controllers include [34], [35], [36], [37], [9], [17], [38].…”

“…Since this initial work, several other authors have also ad-dressed this issue of finding a certificate for a fixed, pre-defined controller. [50], [29] incorporate an SMT solver to verify the learned Lyapunov certificates and provide counterexamples for training, and [3] takes a similar approach to learning barrier functions (the authors of [50], [29], [3] later unified these methods in a single software framework in [30]). [4] learns a contraction metric in the case when the control policy is known but applies this technique for system identification rather than controller verification (the contraction metric constrains the learned dynamics model to be stabilizable).…”

“…The third common category of verification methods, based on optimization, fills this gap by running periodically throughout the training process to verify the certificate as it is learned (and provide feedback in the form of counterexamples where the certificate is not yet valid). These approaches typically take the form of a learner-verifier architecture, sometimes also referred to as counterexample-guided inductive synthesis (CEGIS [7], [29], [3], [57], [30]). In these architectures (illustrated in Fig.…”

“…SMT problems can be solved using dedicated solvers such as dReal [58]. For examples of this approach, see [7], [29], [3], [30], [50]. Both MILP-and SMT-based methods help address the certificate verification problem, with the added benefit of speeding the training process by providing counter-examples.…”

“…Perhaps the most well known of these certificate functions are Lyapunov functions: pseudo-energy functions that, if shown to be strictly decreasing along trajectories of the system, prove the stability of the system about a fixed point [1]. Other well-known certificates include barrier functions (which prove forward invariance of a set, and thus safety [2], [3]) and contraction metrics (which prove differential stability in trajectory tracking [4], [5]). A summary of these different certificates is shown in Fig.…”