Certified Approximate Reachability (CARe): Formal Error Bounds on Deep Learning of Reachable Sets
This addresses the lack of formal error bounds in deep learning for reachability analysis, which is critical for safety-critical applications like autonomous systems, though it is incremental in combining existing verification techniques with learning.
The paper tackles the problem of providing accuracy guarantees for deep learning-based reachable set computations in continuous-time dynamical systems, introducing an epsilon-approximate Hamilton-Jacobi PDE and using SMT solvers to certify error bounds, resulting in the first approach to offer soundness guarantees for learned reachable sets.
Recent approaches to leveraging deep learning for computing reachable sets of continuous-time dynamical systems have gained popularity over traditional level-set methods, as they overcome the curse of dimensionality. However, as with level-set methods, considerable care needs to be taken in limiting approximation errors, particularly since no guarantees are provided during training on the accuracy of the learned reachable set. To address this limitation, we introduce an epsilon-approximate Hamilton-Jacobi Partial Differential Equation (HJ-PDE), which establishes a relationship between training loss and accuracy of the true reachable set. To formally certify this approximation, we leverage Satisfiability Modulo Theories (SMT) solvers to bound the residual error of the HJ-based loss function across the domain of interest. Leveraging Counter Example Guided Inductive Synthesis (CEGIS), we close the loop around learning and verification, by fine-tuning the neural network on counterexamples found by the SMT solver, thus improving the accuracy of the learned reachable set. To the best of our knowledge, Certified Approximate Reachability (CARe) is the first approach to provide soundness guarantees on learned reachable sets of continuous dynamical systems.