Certified Approximate Reachability (CARe): Formal Error Bounds on Deep Learning of Reachable Sets

2025-03-31

AI Summary

Deep learning approaches for computing reachable sets of continuous-time dynamical systems lack formal error guarantees, undermining their reliability in safety-critical applications. Method: We propose a theoretically rigorous and scalable method centered on constructing an ε-approximate Hamilton–Jacobi partial differential equation (HJ-PDE), which explicitly links neural network training loss to reachable set accuracy. Our framework integrates SMT-based residual verification with a Counterexample-Guided Inductive Synthesis (CEGIS) loop to jointly optimize learning and verification in an error-driven, iterative refinement process. Contribution/Results: This is the first method to provide provably sound reachable sets—i.e., guaranteed over-approximations—learned by neural networks. We demonstrate scalability to high-dimensional systems and derive a global, tight, formal upper bound on the approximation error. The approach significantly enhances the trustworthiness and safety assurance of data-driven reachability analysis.

Technology Category

• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

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.

Problem

• Certify accuracy of learned reachable sets in deep learning
• Bound approximation errors in dynamical system reachability
• Integrate learning and verification for soundness guarantees

Innovation

• Introduces epsilon-approximate HJ-PDE for accuracy
• Uses SMT solvers to certify error bounds
• Employs CEGIS for learning and verification loop